Numerical modelling of sound propagation to closed ...

THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Numerical modelling of sound propagation to closed urban courtyards

MAARTEN HORNIKX

Department of Civil and Environmental Engineering Division of Applied Acoustics, Vibroacoustic Group CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden, 2009

Numerical modelling of sound propagation to closed urban courtyards MAARTEN HORNIKX ISBN 978-91-7385-298-2

c MAARTEN HORNIKX, 2009

Doktorsavhandlingar vid Chalmers tekniska högskola Ny serie nr 2979 ISSN 0346-718X Department of Civil and Environmental Engineering Division of Applied Acoustics, Vibroacoustic Group Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone + 46 (0) 31-772 2200

Cover: Upper figure: Geometry of an urban street canyon and closed courtyard. Lower figures: Snapshots of the sound field, generated by an impulsive source (indicated by the red dot) in the street canyon at t = 0 s. Results are for t = 0.125 s, t = 0.250 s and t = 0.500 s and have been calculated with the extended Fourier PSTD method. Printed by Chalmers Reproservice Göteborg, Sweden, 2009

Numerical modelling of sound propagation to closed urban courtyards MAARTEN HORNIKX Department of Civil and Environmental Engineering Division of Applied Acoustics, Vibroacoustic Group Chalmers University of Technology

Abstract Because modern urban environments suffer from excessive levels of road traffic noise, access to closed courtyards is essential in order to offer urban sound environments of high quality with regard to health and perceived sound. To reach this quality, which is described by the quiet side definition, the 24-hour equivalent noise level should be below 45 dB(A). Since many courtyards do not necessarily fulfil this requirement, noise abatement schemes are of interest. Current engineering prediction methods, however, fail to accurately predict the noise level from road traffic in closed courtyards. The 2.5-dimensional equivalent source method (ESM), an accurate frequency-domain method, has therefore been developed to predict sound propagation to a closed courtyard and to evaluate noise abatement schemes in the yard. This method, an extension of the two-dimensional (2-D) ESM, allows for a point source or incoherent line source in a 3-D environment where the geometry is invariant in one direction. For two real-life courtyard geometries, the averaged effects of various noise abatement schemes are presented, mainly based on in- and near-courtyard absorption and screen treatments. Since it is also of interest to study both 3-D and time-domain effects, the 3-D extended Fourier pseudospectral time-domain (PSTD) method has been developed. This method accurately predicts sound propagation through an inhomogeneous moving atmosphere to a closed courtyard. It extends the Fourier PSTD method by allowing to model propagation media with discontinuous properties. By only requiring two spatial points per wavelength, the method is also computationally more efficient than other stateof-the-art, similarly-accurate methods in urban acoustics. It has been validated for typical outdoor sound propagation cases as well as for the closed courtyard geometry. A scale model study has been executed to validate the two developed prediction methods, and to explain the characteristics of the acoustic soundscape in courtyards. Measurements and calculations show that higherorder façade reflections greatly influence the equivalent sound level and sound decay properties in closed courtyards. It explains the relatively constant sound levels in time and space in closed urban courtyards, as well as the significance of noise from distant traffic. Keywords: road traffic noise, urban noise abatement schemes, urban canyon, courtyard, scale model study, equivalent sources method, pseudospectral method, 2.5-D geometry.

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Thesis publications The thesis is based on the work contained in the following Papers, which are appended to the thesis: Paper I Hornikx, M. and Forssén, J., "A scale model study of parallel urban canyons," Acust. Acta Acust. 94, 265-281, (2008). Erratum, Acust. Acta Acust. 94, 641, (2008). Paper II Hornikx, M. and Forssén, J., "The 2.5-dimensional equivalent sources method for directly exposed and shielded urban canyons," J. Acoust. Soc. Am. 122, 2532-2541, (2007). Paper III Hornikx, M. and Forssén, J., "Noise abatement schemes for shielded canyons," Appl. Acoust. 70, 267-283, (2009). Paper IV Hornikx, M. and Waxler, R., "An eigenfunction expansion method to efficiently evaluate spatial derivatives for media with discontinuous properties," In Proc. of Acoustics ’08 Conf., 29 June - 4 July, Paris, France, (2008). Paper V Hornikx, M. and Waxler, R., "An extended Fourier pseudospectral timedomain (PSTD) method for fluid media with discontinuous properties," To be submitted to J. Comp. Acoust., (2009). Paper VI Hornikx, M., Waxler, R. and Forssén, J., "The extended Fourier pseudospectral time-domain (PSTD) method for atmospheric sound propagation," To be submitted to J. Acoust. Soc. Am., (2009). The part of the work done by Hornikx in all Papers amounts to 90 %. Errata to Papers I, II, III and IV are included.

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Other publications The following Papers were published in the period toward this thesis. Because of overlapping contents or less relevance, these Papers are not appended to the thesis. Schiff, M., Hornikx, M. and Forssén, J., "Excess attenuation for sound propagation over an urban canyon," Submitted to Appl. Acoust., (2009). Hornikx, M., Waxler, R. and Forssén, J., "Developments in the eigenfunction expansion method for atmospheric sound propagation," In Proc. of 13th Long Range Sound Propagation Symposium, Lyon, France, (2008). Abstract only Schiff, M., Hornikx, M. and Forssén, J., "A numerical study of sound propagation over urban canyons," In Proc. of Acoustics ’08 Conf., 29 June - 4 July, Paris, France, (2008). Forssén, J. and Hornikx, M., "Characteristics of road traffic noise level statistics for shielded areas," In Proc. of Acoustics ’08 Conf., 29 June - 4 July, Paris, France, (2008). Abstract only Hornikx, M. and Forssén, J., "Improving the shielding of road traffic noise in courtyards: treatments with vertical and horizontal screens," In Proc. of ICA, 19th Int. Cong. on Acoust., 2-7 September, Madrid, Spain, (2007). Hornikx, M. and Forssén, J., "Improving the shielding of road traffic noise in courtyards: absorption treatments," In Proc. of InterNoise 2007, 28-31 August, Istanbul, Turkey, (2007). Forssén, J. and Hornikx, M., "Statistics of road traffic noise levels in shielded urban areas," In Proc. of InterNoise 2007, 28-31 August, Istanbul, Turkey, (2007). Hornikx, M. and Forssén, J., "Scale model measurements and numerical modelling for directly exposed and shielded urban street canyons," In Proc. of Baltic-Nordic Acoust. meeting, Gothenburg, Sweden, (2006). Forssén, J. and Hornikx, M., "Statistics of road traffic noise in shielded urban areas: An initial study of A-weighted levels," In Proc. of Baltic-Nordic Acoust. meeting, Gothenburg, Sweden, (2006).

vii Hornikx, M. and Forssén, J., "Sound abatement schemes for real life shielded urban street canyons," In Proc. of 12th Long Range Sound Propagation Symposium, New Orleans, Louisiana, (2006). Forssén, J. and Hornikx, M., "Statistics of A-weighted road traffic noise levels in shielded urban areas," Acust. Acta Acust. 92, 998-1008, (2006). Hornikx, M., Forssén, J. and Kropp, W., "Scale model measurements and numerical modelling for directly exposed and shielded urban street canyons," In Proc. of Euronoise, Tampere, Finland, Paper nr. 236, (2006). Forssén, J. and Hornikx, M., "Statistics of road traffic noise in shielded urban areas: an initial study of A-weighted levels," In Proc. of Euronoise, Tampere, Finland, Paper nr. 237, (2006). Hornikx, M., "Scale model measurements for directly exposed and shielded urban street canyons," ARC symposium, NTNU Trondheim, Norway, (2006). Abstract only Hornikx, M., Forssén, J. and Kropp, W., "A scale model study of parallel urban street canyons," 149th meeting Acoust. Soc. Am. joint with Canad. Acoust. Ass., Vancouver, Canada, J. Acoust. Soc. Am. 117, 2417, (2005). Abstract only Hornikx, M., "Towards a parabolic equation for modeling urban sound propagation," In Proc. of 11th Long Range Sound Propagation Symposium, Fairly, Vermont, (2004). Hornikx, M., "An advanced parabolic equation method to model outdoor sound propagation," German acoustical society, Workshop Physikalishe Akustik, Bad Honnef, Germany, (2004). Abstract only Hornikx, M. and Thorsson, P., "On the statistical approach of sound propagation through urban areas," In Proc. of the 10th Int. Conf. on Sound and Vib., Stockholm, Sweden, 1421-1428, (2003).

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Acknowledgements

I always want to look forward to the good parts of life that are ahead of me. Sometimes, though, it is time to look back and let one’s thoughts float away over the past while enjoying a suitable drink. At this point, it is more than worthwhile to look back. I have the feeling of finalizing a part of my life. I have lived the past five years as I had them in my mind: as my image of being a PhD student, adopting the PhD student life as a lifestyle. I have utilized this time by being immersed in my subject of urban sound propagation, a subject that did not leave me and pushed me to studies with interesting results. It has been a time where I could meet international colleagues and develop myself in performing research, and most of all a time where I learned a lot. All following pages are a result of this period. For having had the possibility to contribute to science, I am grateful to various people. First of all, I would like to thank my supervisors at Chalmers, Jens Forssén and Wolfgang Kropp. From my first day as a PhD student, I felt confidence from you, something that I could not miss. Jens, thanks for introducing me to the international outdoor sound propagation community and for always competently answering my questions and reading my work. Wolfgang, you became a kind of mentor to me during the years, and I appreciate that most. During my studies, I spent three months in Oxford, Mississippi, at the National Center for Physical Acoustics. I did not have any expectations, but I never regret having travelled there. I am very grateful to Roger Waxler for hosting me there and for the cooperation we started. It became an important period with regard to my scientific work, as the thesis testifies. The Division of Applied Acoustics has over the years always been like a warm home, where I felt comfortable in the atmosphere created by my colleagues. For this, I am grateful to you colleagues. I would especially like to thank Börje and Gunilla, for being the core of the Division and for helping me with all kinds of matters. The work behind this thesis was mostly funded by the Swedish Foundation for Strategic Environmental Research (MISTRA). Besides my working environment, I owe some words of thanks. First of all, I would like to direct these to my parents. They have heard most of my everyday PhD student-life complaints, and always supported me in my academic endeavours - dankjewel mama en papa! And I would like to thank all family and

x friends who have visited me here in Sweden. I have highly appreciated your visits, which kept the link with my origins. The five past years also gave me a new home: Sweden, and especially Göteborg, has become a part of me. I want especially to thank Martin, Charlotte, Christoph, Stig and Astrid for making the years in Göteborg extra joyful. During the last months, I regularly answered the ’how are you doing’ question thus: I do not have a real life any longer. I feel guilty for the last months, of not having taken time for people. But now it is time to pay that back and to look forward again. Although my nearest future lies in my parents’ roots, Belgium, I’m pleased to be still involved at Applied Acoustics and not to leave completely. Göteborg, May 2009 Maarten Hornikx

Contents I Background and aim

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1 Road traffic noise health impact and soundscapes 1.1 Road traffic noise and human response . . . . . . . . . . . . 1.2 Descriptors of the soundscape . . . . . . . . . . . . . . . . . . 1.3 Noise policy and guideline levels . . . . . . . . . . . . . . . . 1.4 Impact of road traffic noise: exposed people and social costs

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2 A soundscape concept: the quiet side 2.1 The quiet side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 State of the art of prediction and modifying the acoustic soundscape in closed urban courtyards . . . . . . . . . . . . . . . . . . . 2.2.1 Prediction of sound propagation to closed courtyards . . . 2.2.2 Possible noise abatement schemes . . . . . . . . . . . . . .

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3 Aim and strategy 3.1 Research needs applying the quiet side concept to courtyards 3.2 Thesis aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Overview of appended Papers and additional thesis results . .

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II Numerical modelling of sound propagation to closed courtyards 25 4 Motivation of selected modelling approaches 4.1 Modelling demands . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Selected methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 The 2.5-D Equivalent Sources Method (ESM) 5.1 The 2.5-D ESM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents 5.2

Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.3

Incoherent line sources . . . . . . . . . . . . . . . . . . . . . . . . .

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5.4

Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 The 3-D extended Fourier pseudospectral time-domain (PSTD) method 49 6.1

6.2

PS methods to calculate spatial derivatives . . . . . . . . . . . . .

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Fourier PS method . . . . . . . . . . . . . . . . . . . . . . .

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6.1.2

Chebyshev PS method . . . . . . . . . . . . . . . . . . . . .

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6.1.3

Extended Fourier PS method . . . . . . . . . . . . . . . . .

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The extended Fourier PSTD method . . . . . . . . . . . . . . . . .

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6.2.1

6.3

Computing the spatial derivatives of the linearized Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.2.2

Moving inhomogeneous medium . . . . . . . . . . . . . .

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6.2.3

Time iteration . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.2.4

Perfectly matched layer (PML) . . . . . . . . . . . . . . . .

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6.2.5

Implementation demonstration of a 2-D canyon . . . . . .

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6.2.6

Numerical efficiency . . . . . . . . . . . . . . . . . . . . . .

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Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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III Abatement of road traffic noise in courtyards

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7 The acoustic soundscape in closed courtyards

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7.1

Signatures of the acoustic soundscape . . . . . . . . . . . . . . . .

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7.2

Fundamentals of the acoustic soundscape . . . . . . . . . . . . . .

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8 Noise abatement schemes for closed courtyards

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IV Conclusions and Further work

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9 Conclusions

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9.1

Scale model study . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The 2.5-D ESM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

9.3

Noise abatement schemes . . . . . . . . . . . . . . . . . . . . . . . 101

9.4

The 3-D extended Fourier PSTD method . . . . . . . . . . . . . . . 102

10 Further work

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Contents

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Appendix A State of the art of wave-based solution methods A.1 Frequency-domain methods . . . . . . . . . . . A.1.1 Boundary element methods . . . . . . . A.1.2 Finite element methods . . . . . . . . . A.1.3 Trefftz-based methods . . . . . . . . . . A.2 Time-domain methods . . . . . . . . . . . . . . A.2.1 Spatial derivative operator methods . . A.2.2 Temporal derivative operator methods A.2.3 Direct methods . . . . . . . . . . . . . .

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B Aspects considered for a successful scale model study

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C Aspects of PS methods C.1 Fourier PS methods: finite differences with an infinite accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Boundary conditions in the Chebyshev PS method . . . C.3 The k-space method . . . . . . . . . . . . . . . . . . . . .

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Bibliography

order of . . . . . . 131 . . . . . . 133 . . . . . . 137

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Contents

Part I Background and aim

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Chapter 1

Road traffic noise health impact and soundscapes

1.1 Road traffic noise and human response The urban sound environment is a composition of natural sounds such as whistling wind through leaves, singing birds and running water, human-induced sounds originating from construction sites, factories and road traffic, and other technological sounds such as ringing mobile phones. Some of the sounds created by human activities are, according to the current EU legislation, defined as environmental noise [46]: "Environmental noise is an unwanted or harmful outdoor sound created by human activities, including noise emitted by means of transport, road traffic, rail traffic, air traffic, and from sites of industrial activity, to which humans are exposed in particular in built-up areas, in public parks or other quiet areas in an agglomeration, in quiet areas in open country, near schools, hospitals and other noise sensitive buildings and areas." Road traffic is the major source from the environmental noise sources, and will therefore be the noise source focused upon in this thesis. Since environmental noise is defined as unwanted and harmful, it can cause a number of adverse health effects. When considering the World Health Organization’s (WHO) definition of human health [131]: "a state of complete physical, mental and social well-being, not merely the absence of disease and infirmity", the following adverse health effects caused by environmental noise can be identified: 3

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1. Road traffic noise health impact and soundscapes

◦ Annoyance. This is the most widespread of all the adverse effects of noise [44]. Annoyance can express itself in different ways, such as fear, uncertainty and mild anger [146, 163]. Although people frequently exposed to noise to some extent develop strategies of adapting and coping with the problem, subconscious physical reactions such as raised blood pressure levels will not diminish over time unless the noise itself is abated [108]. ◦ Sleep disturbance. Three types of effects of noise on sleep can be distinguished: effects on sleeping behaviour, effects on performance and mood during the following day, and long-term effects on well-being and health as for example insomnia [170]. ◦ Disturbed cognitive functioning. Exposure to environmental noise can impair an adult’s cognitive functions such as information processing, understanding and learning [146]. The noise levels must be high, or the task complex or cognitively demanding to have this effect [130]. ◦ Stress. Noise-induced stress can trigger the production of certain hormones, which may lead to a variety of effects, including increased blood pressure. Over a prolonged period of exposure, these effects may in their turn increase the risk of cardiovascular diseases (e.g. heart diseases) [44]. ◦ Hearing impairment. Prolonged, cumulative exposure to sound levels above 70 dB(A) may lead to irreversible loss of hearing [134]. These adverse health effects of noise are not distributed uniformly across society, and groups such as children, the elderly, the sick and the poor suffer most [44]. Moreover, the degree to which noise leads to annoyance and stress depends partly on individual characteristics, in particular a person’s attitude and sensitivity to noise. Finally, external factors like physical and social environment and life-style influence the relation between noise and personal health and well-being [44]. Since other sounds than environmental noise are also present in the urban environment, environmental noise does not necessarily dominate the urban sound environment. As a consequence, the urban sound environment needs not to be perceived as annoying or not. It can, for example, be expressed as measured in metric space along dimensions such as pleasant-unpleasant, exciting-boring, eventful-uneventful and chaotic-tranquil [22]. In the Swedish soundscape support to health (SSH) project, the urban sound environment has been described by a soundscape, defined as: "The sound variations in space and time caused by the topography of the builtup city and its different sound sources."

1.2. Descriptors of the soundscape

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Moreover, two type of soundscapes are discerned: acoustic soundscapes, which can be assessed by physical measuring instruments, and perceived soundscapes, which can be assessed by perceptual scaling methods utilizing persons [144]. Within the SSH project, the soundscape concept has been utilized to create positive soundscapes, rather than adopting the more conventional approach of reducing environmental noise only. The terminology of acoustic and perceived soundscapes has been used throughout this thesis.

1.2 Descriptors of the soundscape The urban sound environment, with road traffic noise as the dominant noise contributor, thus induces adverse effects on health and well-being. We will here elaborate on descriptors of the soundscape, affecting these health effects and the perceived soundscape quality. The following descriptors can be distinguished: ◦ Time-averaged sound levels. Often used descriptors are day, night or twentyfour hour equivalent sound levels. ◦ Time variation in the sound levels, with aspects as intermittency and the maximum sound level. ◦ Frequency content of the acoustic soundscape. ◦ Source identification.

◦ Sound field related to the observer’s position. These effects are not unambiguous and could be arranged differently. For health effects such as long-term sleep disturbance and cardiovascular problems, noise exposure-response relations have been established [2]. The equivalent noise level is then an appropriate descriptor. The degree of annoyance triggered by traffic noise is also first of all determined by the equivalent noise level. The higher the noise level, the more people are annoyed [49, 163]. Equivalent sound levels are indeed used as guideline levels for noise, as reported on in Sec. 1.3. The degree of annoyance depends on other noise characteristics as well. Duration and intermittency of noise for example influence the degree of annoyance [108]. Also, the larger the high-frequency component of the noise, the greater the annoyance. Further, it has been indicated that annoyance from road-traffic sounds with a wide variation in the low-frequency content is not optimally captured by the traditional A-weighted equivalent sound level LAeq [116]. Sleep disturbance is also influenced by other noise characteristics than the equivalent sound level. People are far more sensitive to intermittent noise than to continuous noise [130]. For example, an accelerating car will disturb a person’s sleep

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more than a continuous traffic flow. In addition, the alarm function of the sense of hearing may lead to awakening if the noise contains information perceived to be of relevance, even if the noise level is low. This means that unfamiliar noises are far more likely to disturb sleep than familiar, regular patterns of noise. However, there is never complete habituation, particularly with respect to heart-rate acceleration [146]. Equivalent sound levels do not capture all the perceived soundscape properties. Sound source identification, for example, is found to be a stronger predictor of soundscape quality than equivalent sound levels [115]. In situations with equivalent sound levels between 50 and 55 dB(A), a soundscape design that promotes positive sounds from nature may be efficient in improving soundscapes [115], which supports the concept that a pleasing soundscape is not necessarily created by reducing noise, nor created by providing quietness. The pleasing soundscape would indeed consist of a composition of sounds from people and their activities, from animals and other parts of nature itself, dominated by natural sounds [21, 144]. Regarding the location of the observer, it has been found that the perceived loudness1 indoors is considerably larger at the noisy side of an apartment than at the quiet side, despite the small difference in the A-weighted sound level. Also the indoor-outdoor difference in perceived loudness at the road-traffic noise exposed façade of an apartment building is small, although the corresponding difference in A-weighted sound level is large [21]. The latter could be attributed to the fact that the mix of positive and adverse characteristics of soundscapes makes residents accept higher sound levels at outdoor than indoor places [22].

1.3 Noise policy and guideline levels The WHO recognizes community noise, including traffic noise, as a serious public health problem, and has therefore published guideline levels on community noise [20]. These noise levels are shown in Table 1.1 and for reference, the Swedish guideline values are also included. Some of the descriptors of the soundscape as described in Sec. 1.2 are, however, not captured by these guidelines. Also, research indicates that these guidelines are not strict enough for a high soundscape quality; see e.g. [22]. The WHO values are only guidelines and do not have any legal force. For the legal framework, it is of relevance to regard the European noise policy on environmental noise, which is characterized 1 Sound loudness is a term describing the strength of the ear’s perception of a sound. It is related to sound pressure level, yet not identical.

1.3. Noise policy and guideline levels

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Table 1.1: Guideline equivalent and maximum noise levels according to WHO [20] and the Swedish Parliament [86]. Values concern the living environment. LAeq is the equivalent Aweighted noise level, with given time base. LAmax is the maximum A-weighted noise level.

Outdoor living area Dwellings, indoor Outside bedroom Inside bedroom

LAeq LAmax LAeq LAeq LAmax LAeq LAmax

WHO Level Time base (dB(A)) (h) 2 3 50 /55 16 (day) 356 16 (day) 457 8 (night) 607 8 (night) 8 30 8 (night) 458 8 (night)

Sweden Level Time base (dB(A)) (h) 4 55 24 705 24 30 24 45 8 (night)

by two cornerstones: ◦ Source-specific emission-related directives. Successive directives, starting back in the 1970s, have laid down specific noise emission limits for most road vehicles and for many types of outdoor equipment in order to control noise pollution. ◦ The Environmental Noise Directive (END) of 2002. The END started with the European Commission’s published green paper concerning the abatement of noise and future noise policy, and defined the aim that [52]: "no person should be exposed to noise levels which endanger health and quality of life". The END focuses on a common approach to address environmental noise, to be executed at the national, regional and local levels according to the principle of shared responsibility. Member states must report to the Union about the state of the acoustic environment and about their action plans. However, the END does not set any European Union (EU) limits or target values on environmental noise that would bind member states to consider implementing noise abatement measures. Member states are responsible for the setting of such values and follow different approaches. Some have set legally binding limits or targets whereas others publish recommended values. Also, some states set limits for existing sources, whereas others limit levels for new transport infrastructures or new buildings. Moreover, values adopted vary from one state to 2

Criterion: Moderate annoyance. Criterion: Serious annoyance. 4 At the façade. 5 At the patio. 6 Criterion: Speech intelligibility and moderate annoyance. 7 Criterion: Sleep disturbance. 8 Criterion: Sleep disturbance with open window. 3

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1. Road traffic noise health impact and soundscapes another, and they do not necessarily correspond to the WHO recommended values of Table 1.1.

Besides the current legislation, the EU has funded a network called CALM, which publishes on the European noise policy. In the strategy paper of the CALM network, a plan for future research to reduce environmental noise in Europe is contained [132]. The vision derived and proposed by CALM, for the development of noise research that targets up until the year 2020, is to "avoid harmful effects of noise exposure from all sources and preserve quiet areas". The future research should support the END, and the structure of the noise research strategy is split into perception and emission-related research. As regards perception-related research, some of the presented research needs are: ◦ Advanced computation and measurement methods for more accurate assessment of noise exposure. ◦ Definition and identification of urban and rural quiet areas: - Identification of most appropriate indicators and limit values. - Parameters influencing the public’s perception of quiet areas. ◦ Improvements in dose-effect relationships for Lden 9 and Lnight 10 : - Sleep disturbance (awakening) due to road and railway noise. - Effects of a quiet side of a building and of quiet areas in the neighbourhood. ◦ Additional noise indicators considering specific effects: - Effect of the maximum noise level Lmax . - Effect of quiet periods. These research needs indeed establish the notion that equivalent noise levels do not sufficiently represent soundscape perception and agree with the overview in Sec. 1.2 by complementing the existing guideline noise levels with more indicators to characterize the soundscape. In the strategy paper of the CALM network, a goal has been set to reduce noise from road traffic by 10 dB(A) in 2020. In order to reach this goal, the system "road traffic noise" has to be considered in a holistic way. The activities in research and technological development must cover the following three technical fields of acoustics in the respective ranking: 9 Lden is a 24-hour equivalent A-weighted noise level where the index stands for "dayevening-night", including a 5 dB penalty weighting for evening and a 10 dB penalty weighting for night. 10 Lnight is the night-time equivalent A-weighted noise level.

1.4. Impact of road traffic noise: exposed people and social costs

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◦ The noise source: measures at the source such as quiet road surfaces, lownoise tyres, measures at the vehicle, driving behaviour and traffic flow management [132]. ◦ The noise propagation: measures between noise source and receiver, such as noise barriers and soft ground surfaces. ◦ The noise reception: measures close to the observer, such as sound-proof windows. The large importance of measures at the noise source is motivated by costeffective considerations and the ’polluter pays’ principle. On the other hand, a disadvantage of these at-source measures is that penetration in the vehicle fleet takes several years for tyres and almost a decade for motor vehicles [44]. This aspect, together with the principle of responsibility at all levels of authorities as pointed out in the END and the holistic perspective expressed by CALM, stresses the necessity for action in all three fields of acoustics.

1.4 Impact of road traffic noise: exposed people and social costs Since the EU noise policy does not impose a tight regulation on road traffic noise sources, it is no surprise that more than 44% of the EU2511 population (about 210 million people) has been estimated to be regularly exposed to road traffic noise levels over 55 dB(A) [44]. More than 54 million people are exposed to road traffic noise levels over 65 dB(A). As equivalent noise levels may still trigger adverse effects like annoyance and sleep disturbance below 55 dB(A), the actual number of people exposed to road traffic noise levels that are potentially dangerous to their health may thus be higher than the above-mentioned numbers. Millions of people also experience health effects due to road traffic noise. For example, about 57 million people are annoyed by road traffic noise, 42% of them seriously [44]. This means that about 12% of the European population suffers from annoyance due to road traffic noise. A preliminary analysis shows that each year, over 231,000 people in the EU25 are affected by cardiovascular diseases that can be traced to road traffic noise. About 20% of these people, almost 50,000, suffer a lethal heart attack, thereby dying prematurely [44]. It is clear that noise legislation needs to be improved in order to reduce these numbers. 11

EU25 refers to the 27 EU member states except Cyprus and Malta.

10


The social cost of road traffic noise in the EU2212 is estimated to be at least e38 billion per year, which is approximately 0.4% of total gross domestic product in the EU22. As this estimate only takes into account effects related to noise levels above 55 dB(A), it probably underestimates the actual costs.

12 EU22 refers to the 27 EU member states except Cyprus, Estonia, Latvia, Lithuania and Malta.

Chapter 2

A soundscape concept: the quiet side

"...the Romans banned wheeled traffic from the Forum because of the noise and congestion that it caused – an early example of traffic noise regulations. Delivery carts were however allowed at night. This did not worry the lawmakers, many of whom lived in houses with solid outer walls, and whose windows opened on to quiet inner courtyards."[50]

2.1 The quiet side Most of the European population exposed to excessive road traffic noise levels lives in urban areas. Given the current long-term trend of increasing traffic flows and the weak noise reduction for individual sources [44], it is not very likely that, despite all efforts, the equivalent noise level at building façades directly exposed to road traffic noise will be reduced below the WHO guideline equivalent noise levels in the nearest decade. Kihlman among others therefore proposed a solution strategy to offer urban citizens access to relatively quiet courtyards [95]. These courtyards are shielded from direct exposure to road traffic noise, and therefore form an effective way of creating a side with a low maximum as well as equivalent noise level in densely built urban areas. When inspecting maps of urban city centres, closed courtyards are often present, or could be created by filling in openings in building blocks. In the soundscape 11

12

2. A soundscape concept: the quiet side

support to health (SSH) project, the quiet side is used as a soundscape concept and is defined as [144]: "An urban area with LAeq,24h < 45 dB(A) due to traffic and fans (and similar sources)1 . The quiet side should moreover be of high acoustical, functional and visual quality." The quiet side constitutes a positive soundscape to which citizens have access in urban areas. This "access" consists both of staying at the outside space and having the possibility of opening windows toward it. Important in the quiet side concept is that it not only is characterized by a low equivalent sound level, but moreover offers a soundscape with attractive aspects such as grass and trees (which in turn attract natural life such as birds and butterflies). It has been shown that people are less annoyed in a courtyard with such an attractive environment than in an unattractive one [64]. Öhrström et al. give evidence on health benefits of having quiet outdoor sections bordering a dwelling. When the LAeq due to road traffic noise does not exceed 60 dB(A) at the directly exposed side of the dwelling, 80 % of the people are not annoyed and protected from adverse health effects as stress and sleep disturbance if they also have access to a side with LAeq,24h < 45 dB(A) [185]. Quiet sides may further be restorative, which has a positive effect on health and well-being; see e.g. [93]. The quiet areas could solve a large part of the urban road traffic noise problem, and fit with the reported research needs from the CALM network in Sec. 1.3. In the city of Gothenburg, a city located at the west coast of Sweden with currently approximately 500,000 inhabitants, the environmental department has performed 687 indicative measurements of the equivalent sound level in courtyards [7]. Figure 2.1 shows a histogram of the measured LAeq , with 60 % of the cases not meeting the quiet side LAeq,24h < 45 dB(A) requirement. It is obviously not self-evident that courtyards constitute a quiet side. Given the clear health benefits of quiet sides, it is thus necessary to be able to predict the noise level in closed urban courtyards, as well as have tools to change this noise level in order to ensure that closed courtyards can be considered as quiet sides. Such tools can be noise mitigation methods in the courtyards. Current prediction and mitigation methods are therefore reviewed in the following Sections.

1

The LAeq,24h level here concerns the free field value.

2.2. State of the art of prediction and modifying the acoustic soundscape in closed urban courtyards 13 0.4

PDF (−)

0.3 0.2 0.1 0 20

30

40

50 60 LAeq (dB(A))

70

80

Figure 2.1: Results of 687 indicative LAeq measurements in courtyards in the city of Gothenburg [7].

2.2 State of the art of prediction and modifying the acoustic soundscape in closed urban courtyards 2.2.1 Prediction of sound propagation to closed courtyards As the Environmental Noise Directive (END) demands noise maps of larger European agglomerations, prediction software with implemented calculation methods, such as NORD2000 and the Harmonoise engineering method have been used to produce these maps [89, 117]. These methods however typically predict 10 dB(A) too low equivalent sound levels in closed courtyards, mainly since the multiple façade reflections are not accurately taken care of [96]. Therefore, more accurate prediction methods are necessary to calculate the equivalent sound level in closed courtyards. Mainly, two accurate models have been developed for predicting sound levels in shielded urban areas2 . Van Renterghem and Botteldooren presented an application of the finite-difference time-domain method (FDTD) to the problem of two parallel urban canyons3 [165]. The FDTD model successfully solves the linearized Euler equations (LEE), the governing equations for sound propagation in a moving medium where mean and turbulent atmospheric wind components are allowed for. Impedance boundary conditions are included and the boundary is modelled explicitly in the numerical domain discretized by an equidistant grid. Since typically 10 spatial grid points per wavelength are needed for an ac2 Approximate solutions, such as a ray based method, could also be developed for this purpose. The number of reflection and diffraction contributions that need to be included for accurate prediction with such methods will however become prohibitive, see e.g. [181]. 3 An urban canyon is a street continuously lined by (closed) buildings on both street sides.

14


curate prediction, the drawbacks of the method are its computation time and storage requirements. The method was therefore implemented to solve a twodimensional (2-D) problem, and used at short distances. For larger distances, a coupling with the computationally faster Parabolic Equation (PE) method was successfully implemented [166]. The FDTD was then used in the geometrically complex environment close to source and receiver, and the PE was used to propagate the field over large intermediate distances where the geometry has a lower complexity. Heimann applied a FDTD solution of the LEE to a 3-D case of a street shielded from direct road traffic noise in a parallel street. The modelled domain dimensions were limited up to 46 m and the frequency to 250 Hz [73]. The second developed method is the frequency-domain equivalent sources method (ESM), which solves the Helmholtz equation. This method divides the geometrical domain into sub-domains, placing equivalent sources at the subdomain interfaces and is based on Green’s functions between equivalent sources. The ESM has been developed for interior and exterior problems; see e.g. [88]. It has been applied to the geometry of two parallel street canyons by Ögren and Kropp [183]. Diffraction is included implicitly, whereas impedance surfaces and surface irregularities are modelled explicitly by adding equivalent sources and further sub-domains if needed. The advantage of the model compared to the FDTD is the faster computation time. An inhomogeneous and moving atmosphere above the canyons can be modelled by using different Green’s functions in this region, as was done by modelling atmospheric turbulence [182]. With regard to the acoustic soundscape from road traffic noise in closed courtyards, near as well as distant sources should be included for correct prediction [183]. Due to computational limits, accurate prediction methods such as the FDTD and ESM are not suitable for noise mapping of larger urban areas. For the prediction of the sound level due to all relevant roads, a hybrid approach of two types of models – an accurate model for the courtyard and a less accurate model covering the larger distance of sound propagation above roof level – is a plausible solution [157]. Such a method is the flat city method, where urban traffic noise sources and receiver positions are elevated to roof level and spherical spreading is considered. Correction factors for the underlying urban area and elevation of sources and receiver(s) are necessary in this method, which can be found by using an accurate model.

2.2. State of the art of prediction and modifying the acoustic soundscape in closed urban courtyards 15

Figure 2.2: Urban courtyards can create a soundscape of high quality [6].

2.2.2 Possible noise abatement schemes As the equivalent noise level in closed courtyards is influenced by both near and distant road traffic, in- and near-courtyard abatement schemes can be more efficient to reduce courtyard sound levels than abating at or close to all the relevant noise sources. We will here outline such possible noise abatement schemes, along with results from literature. Absorption To increase the absorption of building façades, suitable façade materials should be selected. However, a material that increases absorption for all relevant frequencies substantially does not exist. For the highest frequencies of our interest, granular plates, fractal walls [4] or façade vegetation [3] can be used. For the lower frequencies region, Helmholtz resonator based solutions such as microperforated panels and resonant systems such as thin plates on an air layer, can be used. Two studies were attributed to the effect of façade treatments in a shielded urban canyon. Ögren investigated the situation of a coherent line source outside a 11 m x 18 m single street canyon and took the averaged sound level at receiver points at a height of 1.5 m [181]. The reference case had rigid walls. Absorption at façades was found to be more effective than at the ground. Van Renterghem et al. investigated façade treatments at the shielded side in a configuration of two parallel canyons [167]. A coherent line source was located in one canyon and one receiver position in the other canyon. The canyons had the

16


dimensions 10 m x 10 m and were located 100 m apart. For a reference case with rigid façades and ground surface, a façade absorption coefficient of 0.33 (for normal wave incidence) led to a loss of 10 to 15 dB and a façade absorption coefficient of 0.55 gave 25 to 30 dB. These values are very high compared to the values from studies in the directly exposed street canyon; see e.g. [79, 90, 91]. The effect of a green flat roof compared with an acoustically rigid flat roof in between two urban canyons for the reduction in the non-exposed canyon was also studied by van Renterghem et al. [168]. A larger reduction was found for the higher frequencies, with a maximum reduction at the 1 kHz octave band of 10 dB. Diffusion Partly diffuse façade reflections to create a more diffuse sound field in the shielded canyon can be obtained by, for example, window depressions and balconies. Yet also street furniture leads to a more diffuse sound field. Ögren, in his single shielded canyon study, found that 0.3 m x 1.0 m niches in rigid façades led to a loss that increased with frequency, being around 7 dB at the 1 kHz 1/3-octave band. Van Renterghem et al. compared profiled façades with flat façades for the two parallel canyons geometry. The loss in the shielded canyon was around 10 dB. A case with balconies was also investigated in this study. Three balconies (with a width of 0.5 m and a height of 1.0 m) were applied to each façade. The reference case had flat façades with an absorption coefficient of 0.33. The balconies gave a loss of typically 10 dB, where however only one receiver point in the canyon was considered. Other effects The sound levels in shielded canyons also depend on atmospheric effects of mean and turbulent wind and temperature components. Van Renterghem et al. and Ögren show that the sound level at shielded sides is increased in the presence of strong atmospheric turbulence. Van Renterghem et al. also found that downwind could lead to higher sound level values at the shielded canyon [167]. Mixed effects A common conclusion of the studied influence of mixed façade treatments on the excess attenuation, is that the excess attenuation from two different treatments is not simply the addition of the two separate effects [167, 181]. Ögren found this for a combination of absorption and diffusely reflective boundaries and van Renterghem et al. showed it for a combination of balconies and absorption.

Chapter 3

Aim and strategy

3.1 Research needs applying the quiet side concept to courtyards In this work, the quiet side concept is embraced to create soundscapes of high quality in courtyards immersed in densely built urban environments. For this purpose, research needs from an acoustical point of view are derived from the previous Sections and are formulated as follows: 1. A prediction method for sound propagation to real-life courtyards is needed. Prediction of equivalent noise levels in major agglomerations is required by END. Available prediction software relies, however, on engineering calculation methods that underestimate noise levels in closed courtyards, leading to an overestimation of the amount of quiet urban areas. More accurate models are currently restricted in their application to real-life courtyards due to the demanding computation time (FDTD), or have been developed in two dimensions only (ESM). The needed prediction method can be accompanied by a long range engineering method (as the flat city model proposed before [157]) to include all contributing traffic noise sources. 2. Noise abatement schemes for real-life courtyards are needed. Quiet sides have been shown to have clear health benefits, and urban courtyards could qualify as quiet sides. Since indicative measurements, however, have shown that many courtyards exceed the quiet side equivalent noise level requirement, noise mitigation methods are needed as a tool to create courtyards 17

18

3. Aim and strategy

that fulfil this requirement. Current studied in-courtyard noise abatement schemes were evaluated using a 2-D model and used smooth façades in the reference geometries. A further investigation of noise abatement schemes for real-life courtyards is therefore needed1. 3. A method for predicting the transient sound field in courtyards is needed. The perceived soundscape is not assessed only by the equivalent sound level and its spectral content. The temporal variations of the acoustic soundscape are of importance as well. Also, to evaluate courtyards in a design or renovation stage, auralization of the transient acoustic soundscape would probably be helpful. A prediction method is therefore required, creating impulse responses. Of special interest is near road traffic, which determines the maximum noise levels in courtyards, and for which source identification is of relevance.

3.2 Thesis aim The aim of this thesis work is closely related to the acoustic research needs for urban courtyards as derived in Sec. 3.1. The aim of the thesis is to offer methods for prediction and modification of the soundscape in closed urban courtyards in order to qualify them as quiet sides2.

3.3 Strategy The strategy that has been followed to reach the thesis aim consists mainly of four aspects, which have a clear interaction. 1. A scale model study of urban canyons. This study has been carried out to investigate the sound field in a canyon directly exposed to road traffic noise and in the adjacent canyon or courtyard shielded from direct noise exposure. Sound propagates from the street canyon to the shielded side by propagation via the intermediate building block only. Also, the influence of façade absorption and façade irregularities on sound levels and the transient sound field has been studied. The results have been used to define requirements for the 1

The in- and near-courtyard noise abatement schemes as need to be proposed and evaluated have to be regarded as part of a holistic noise mitigation strategy as pointed out by the CALM paper [132], as they should be complemented by other means such as measures at-source and at-receiver. 2 The quiet side definition according to the soundscape support to health project of Chapter 2.1 is used here.

3.4. Thesis structure

19

prediction methods, to validate the prediction methods and to propose noise abatement schemes. The scale model study was chosen over field measurements because of the better control over parameters such as geometry, sound sources and background noise. 2. The development of a 2.5-dimensional (2.5-D) frequency-domain prediction method3 . This method allows predicting sound levels in the parallel canyons’ geometry from a point source or (finite) incoherent line source and evaluation of the proposed noise abatement schemes in the canyons’ geometry. The 2.5D method has been validated by results from the scale model study. 3. A parameter study of noise abatement schemes using the 2.5-D method. Noise abatement schemes in and near the closed courtyard are proposed. Courtyard geometries modelled from real-life courtyard cases have been used as reference situations to evaluate the effect of various abatement schemes. A finite incoherent line source outside the courtyard was modelled and the effect of noise abatement schemes was studied for various in-courtyard receiver positions. 4. The development of a three-dimensional (3-D) time-domain prediction method. This method calculates impulse responses and has been developed to predict the transient sound field and to investigate 3-D meteorological and geometrical effects in closed courtyards. The 2.5-D method and the scale model study results are used to validate this method.

3.4 Thesis structure The thesis is mainly based on work presented in the appended Papers. The contents of the Papers, their interaction and new aspects in them are briefly described in Sec. 3.5. The last part of Sec. 3.5 summarizes additional results that are contained in the thesis. The major part of the thesis, Part II, is devoted to the two developed prediction methods. It starts in Chapter 4.1 with the demands for the methods. Based on a literature overview on wave-based solution methods (Appendix A) that could be candidates for the methods intended, the two developed methods are discussed in Sec. 4.2. These methods, the 2.5-D ESM and the 3-D extended Fourier pseudospectral time-domain (PSTD) method, are described in Chapter 5 and 6. Part III deals with the abatement of traffic noise 3

A 2.5-D geometry is here defined as a 3-D geometry which is invariant in one direction (here the y-direction), yet where source and receiver positions can have arbitrary coordinates in the 3-D space; see e.g. Fig. 4.1(b).

20

3. Aim and strategy

Figure 3.1: Urban courtyards as a strategy when developing plans of new city areas, as here in Gothenburg [1].

in courtyards. The acoustic soundscape in courtyards is first characterized in Chapter 7 using results from real-life measurements. These characteristics are explained by transfer paths measured in the scale model study and calculated by the developed methods. Chapter 8 presents the reduction of noise by abatement schemes in courtyards as calculated by the 2.5-D ESM. The thesis is completed in Part IV by conclusions and recommendations for further work. Appendices B and C give further details on the scale model measurements and on pseudospectral methods, respectively.

3.5. Overview of appended Papers and additional thesis results

21

3.5 Overview of appended Papers and additional thesis results . Paper I. Sound propagation from a monopole sound source in a single urban street canyon and toward an adjacent shielded urban canyon or closed courtyard is studied from scale model measurements. The spatially and frequency dependent results are presented in decay times and sound levels and offer new insights into the acoustic soundscape of the studied geometry. The effect of façade diffusivity and façade absorption is studied. Differences between the acoustic soundscape in the 2.5-D canyon geometry and 3-D closed courtyard geometry are shown. Excess air absorption is corrected for by using a developed wavelet technique. . Paper II. The 2-D ESM method is combined with a technique to transform 2-D to 2.5-D results for the geometry of parallel canyons. Errors arising from numerically evaluating the transform are discussed, and results are shown to be consistent with reference solutions. The 2.5-D ESM now allows one to study sound propagation from point and incoherent line sources to shielded urban canyons, which has not been shown before4 . . Paper III. The developed 2.5-D ESM is used to evaluate proposed noise abatement schemes for urban courtyards, modelled by canyons. The 2.5-D ESM method is further validated by the scale model results. Noise abatement schemes are proposed with the aid of scale model results. Schemes with façade absorption, roof vegetation, a soft ground surface, horizontal façade screens (like walkways) and vertical roof screens have been investigated. The influence of a downward-refracting atmosphere is studied as well. An incoherent line source of finite length has been used to model a road traffic noise source and reference geometries from real-life courtyards have been taken. The study reports separate results for abatement schemes in the street canyon and in the shielded canyon.

4

This statement was valid at the time of the Paper submission. Later, Heimann also has performed calculations in similar geometries [73].

22

3. Aim and strategy

. Paper IV. For the 1-D situation of two discontinuous fluid media, a generalized eigenfunction expansion method (in further Papers and this thesis, called the extended Fourier PS method) is presented as a highly accurate calculation method for calculating the spatial derivative operator of the wave equation. The method extends the Fourier pseudospectral method to fluid media with discontinuous properties. Theory and numerical implementation are presented. Validation results are shown for the spatial derivative only and for time-domain results, where the combination with the k-space method for time iteration displays a dissipation- and dispersion-free scheme for piecewise homogeneous fluid media. The method keeps its accuracy up to almost two spatial points per shortest wavelength for mildly varying medium inhomogeneities. . Paper V. The method of Paper IV is extended to model 2D sound propagation through two fluid media with discontinuous properties. The theory of the method for the 2-D wave equation is presented and details on the numerical implementation are given. Two different time-iteration schemes are discussed. Successful validations are made for two fluid media with different density and sound speed. Results for a sound speed inhomogeneity on top of one of the media also led to good results. The method is applied to the water-air interface with a source close to the interface, showing enhanced transmission from water to air for the lower frequencies. A demonstration of the method for long-range propagation of the sound field transmitted from water to air in the presence of a realistic sound speed profile is given. . Paper VI. The extended Fourier PS method from Paper V is here applied to calculate the spatial derivative of the coupled first-order acoustic propagation equations. The solution is time-iterated by a Runge-Kutta method. The Paper presents the theory and contains several 2-D applications of outdoor sound propagation, ranging from propagation over a thin and thick screen, double screen and urban canyon. The ground surface is either rigid or has a finite impedance. Calculations are compared with analytical solutions and good agreement is found, both for a non-moving and moving atmosphere. A 3-D calculation of sound propagation over a barrier in a non-moving and moving atmosphere is also shown. The method is limited to boundaries modelled by a fluid with a different density. Only two spatial points per wavelength are needed in most cases, implying a lower computational cost than the currently used time-domain methods in outdoor acoustics with similar accuracy.

3.5. Overview of appended Papers and additional thesis results

23

. Additional thesis work The extended Fourier PSTD method, based on the extended Fourier PS method for evaluating spatial derivatives and a Runge-Kutta method for the time derivative, is further validated in the thesis for propagation to a 2.5-D canyon with façade absorption and façade irregularities and for the 3-D closed courtyard cases as studied in the scale model. Calculation results from the extended Fourier PSTD are used to study time-domain transfer paths.

24

3. Aim and strategy

Part II Numerical modelling of sound propagation to closed courtyards

25

Chapter 4

Motivation of selected modelling approaches

"...there is some historical precedent to believe that any scheme that incorporates some of the basic physical understanding of how waves propagate should have some intrinsic advantage over one based on brute force." (A.D. Pierce) [126]

4.1 Modelling demands A computational method is needed for accurate prediction of equivalent noise levels in closed courtyards. The necessary evaluation of noise abatement schemes will be done in terms of ∆LAeq and can therefore also be executed by the same method. Since road traffic noise entering courtyards through openings in façades and noise from local sources like fans can be abated relatively easy, sound propagation ’over’ the roof level to the courtyard is of interest. Only propagation to closed courtyards will therefore be studied here. In an urban environment, road traffic close to a courtyard is often located in an urban street canyon. Thus, to predict the equivalent sound level at the courtyard from nearby road traffic noise, the prediction method should be able to handle geometries as of Fig. 4.1(c), where dimensions of the street canyon and courtyard are typically 20 m for urban centres [153].

27

28

4. Motivation of selected modelling approaches

Figure 4.1: The 2-D, 2.5-D and 3-D situations of urban canyons and urban courtyard, with coordinate system and typical dimensions depicted in (c).

For the contribution of distant road traffic, intermediate effects between noise sources and receiver points in the courtyard – as meteorology, screening and multiple intermediate courtyards – could be taken care of by a simplified prediction method, such as the flat city model; see e.g. Schiff et al. [138]. To include distant traffic, only a single courtyard needs therefore to be modelled by the computational method. When approximating the geometry of Fig. 4.1(c) to be invariant in the y-direction, and modelling road traffic by a coherent line source, the situation is simplified to two dimensions. Such geometry of two urban canyons is sketched in Fig. 4.1(a) and has been studied before [164, 183]. However, using a coherent line source may lead to erroneous predictions, as pointed out in Fig. 12 of Paper II. For the evaluation of noise abatement schemes, the source modelling limitation needs therefore to be relaxed to an incoherent line source, which better represents road traffic. The 2.5-D canyons’ geometry of Fig. 4.1(b), which is invariant in the y-direction, yet where source and receiver positions can have arbitrary coordinates in the 3-D space, allows for using an incoherent line source as noise source. As was indicated by the scale model study results, see Paper I, predicted reduction in noise abatement schemes1 is expected to be larger for a 3-D courtyard environment than for a 2.5-D canyon geometry. A 2.5-D canyon geometry is however more efficient to compute for than a 3-D courtyard environment. 1

As pointed out in the research needs in Sec. 3.1, the noise abatement schemes considered in this thesis are in- and near-courtyard schemes.

4.2. Selected methods

29

Such a situation can therefore be used to study the effect of noise abatement schemes, keeping in mind that results represent a lower limit. With regard to the method needed for predicting the transient sound field in courtyards, near traffic is most important with respect to the variations of the transient acoustic soundscape. The typical geometry that needs to be modelled with this method is therefore similar to that for the sound level prediction method. However, modelling the full 3-D environment of the courtyard is necessary to evaluate the transient sound field. It is especially of relevance when using the model results for auralization purposes. As was found from the scale model study results, effects of the urban topology on the sound field are large – e.g. effects of façade reflections, effects of finite impedance surfaces and effects of façade irregularities such as depressed windows. Such effects should therefore be taken care of by both 2.5-D and 3-D prediction methods. Meteorological conditions influence urban sound propagation and should be calculated as well: sound waves are refracted by mean wind components and temperature gradients, scattered by atmospheric turbulence, and damped by atmospheric absorption. Especially the region above the buildings has meteorological conditions that are of importance for long-distance propagation to courtyards [78]. Although the frequency content of road traffic noise extends over 1000 Hz, 1/3octave bands up to 1000 Hz are most important to consider, for several reasons. The sound levels for higher frequencies are reduced more from geometrical shielding and larger air absorption. Also, the high-frequency content of noise is easier to mitigate by absorption material than the low-frequency part. Another aspect is that modelling the high-frequency range is more straightforward, as ray and diffusion methods become useful; see e.g. [91]. Finally, some practical aspects play a role. It is more difficult to accurately measure higher frequencies in a scale model, and the computation time of the wave-based numerical models increases with frequency.

4.2 Selected methods The overview of the numerical methods in Appendix A shows that a range of methods have been developed that provide more efficient prediction methods than the conventional finite element, finite difference and boundary element methods, by making use of the underlying wave phenomena in the solution method. Such methods are in Appendix A called wave-based solution meth-

30


ods. Not all wave-based solution methods, though, come without a price; e.g. the type of FEM that relies on a plane wave basis typically is ill-conditioned, which worsens with the increase of the number of plane waves involved [129]. Also, the necessary integration over the highly oscillating wave functions requires more attention than the conventional low-order polynomial functions do [26]. The method that has been developed for sound level prediction and evaluation of noise abatement schemes in courtyards is the frequency-domain equivalent sources method (ESM). Since only a limited number of frequencies per octave band are necessary for broadband predictions, a frequency-domain method such as the ESM may be more efficient than a time-domain model for noise level calculation. As pointed out in Sec. A.1.3, the ESM is a Trefftz-type method, since the spatially varying acoustic pressure is expanded in terms of wave functions exactly satisfying the Helmholtz equation. It has the advantage over the BEM that fewer boundaries need to be discretized, given that pressure and velocities are prescribed at some of the boundaries. The developed ESM extends the existing ESM from 2-D to 2.5-D, as required from Sec. 4.1. Since a large part of the costs in the ESM originates from the calculation of the Green’s functions for rectangular domains, these functions have here been improved for faster evaluation. The solved equation is the Helmholtz equation and a monopole with unity source strength and the ejωt convention have been used. The equation may be written as ∆ + (k − jα)2 p = −δ(x|xs ),

(4.1)

with k the wave number, p = p(x) the acoustic pressure, α = α(f ) is the frequency-dependent air absorption coefficient, δ(x|xs) = δ(x|xs )δ(y|ys)δ(z|zs ) and where the subscript s denotes the source position. The 2.5-D frequencydomain ESM excludes inhomogeneous and moving medium properties in the canyon. As described in Paper III, the region above the canyon can be modelled as inhomogeneous and moving using appropriate Green’s functions2 . The 2.5D ESM allows for a complex-valued locally reacting impedance, and boundary irregularities can be modelled. It is most convenient when the irregularities are staircase-shaped. For the transient sound field prediction method, a time-domain method has been developed. An alternative is to solve in the frequency domain and apply 2

Note that these Green’s functions are costly to evaluate for a complex moving or inhomogeneous medium.


31

an inverse Fourier transform over the spectrum to get the time-domain results. For the latter to be successful, the spectrum needs to represent a causal signal, the full frequency contents of the source need to be covered and it should be respected that the discrete frequency spacing corresponds to the signal length. Also, given the three-dimensionality of the problem, costs of solving the (implicit) equation system in a frequency-domain method are disadvantageous compared to costs of a (often explicit) time-domain method; see e.g. [127]. A timedomain method has therefore been chosen. The spatial domain is fully discretized to include spatially-dependent meteorological effects. As the calculation time is related to the number of spatial grid points, having an efficient method to compute spatial derivatives is of large importance for a 3-D geometry. The Fourier pseudospectral (PS) method to compute the spatial derivatives requires only two points per wavelength, is easy to implement and allows for a moving and weakly inhomogeneous atmosphere. The developed the 3-D extended Fourier pseudospectral time-domain (PSTD) method is an extension of this method to the case of discontinuous fluid media, i.e. allowing to include boundaries. The time-marching scheme is of less interest to elaborate on here, since the computational cost gain is only linearly decreasing with decreasing time step. Moreover, as spectral information on the modelled time-domain results is also of interest, imposing a certain sampling frequency requirement in time, a large time step is not relevant. The 3-D extended Fourier PSTD solves the Euler equations linearized with respect to the acoustic variables (LEE), where meteorological conditions can accurately be included: ∂u 1 + (u0 · ∇)u + (u · ∇)u0 + ∇p = 0, ∂t ρ0 ∂p + u0 · ∇p + ρ0 c2 ∇ · u = −sδ(x|xs ), ∂t

(4.2)

with c the adiabatic speed of sound and a sound source with strength s = s(t) is included on the right side of the second equation. The density ρa = ρa (x, t), pressure p = p(x, t) and velocity vector ua (x, t) = [ua , va , wa ]T are decomposed into their ambient values and acoustic fluctuations: ρa = ρ0 + ρ, pa = p0 + p and ua = u0 + u. Sound propagation is assumed to be adiabatic, ambient density and pressure variations are neglected. As can be seen from Eqs. (4.2), ambient velocity components influence sound propagation. These components can be found from the equations for the ambient flow only, as they are not assumed to be influenced by the acoustic fluctuations. The ambient components are assumed to be frozen with respect to the acoustic variables, i.e. they are time-independent. Ambient velocity components could act as aeroacous-

32


tic noise sources, but these are not considered here. Air absorption has so far been excluded in the extended Fourier PSTD method. The coupled first-order Eqs. (4.2) have the advantage over the convective wave equation, which only has one acoustic variable, that fewer approximations are made [122]. Also, for domain discretization methods such as the extended Fourier PSTD method, the perfectly matched layer to truncate open boundaries (see Sec. 6.2.4) can easily be included in the LEE [149]. Boundaries in the 3-D extended Fourier PSTD method are modelled as rigid or by a second fluid medium with a different density, which is shown to be a useful approximation for courtyards; see Chapter 6. As with the 2.5-D ESM, staircase-shaped boundary irregularities can be modelled. To summarize, Table 4.1 gives an overview of the features of the two developed methods.

2.5-D ESM

3-D extended Fourier PSTD

Description in

Chapter 5, Paper II and III

Chapter 6, Paper IV, V and VI

Purpose of method

Noise level prediction

Transient sound level prediction

Evaluation of noise abatement schemes

Investigation of 3-D geometrical effects

Equation solved

3-D Helmholtz equation, Eq. (4.1)

3-D linearized Euler equation (LEE), Eq. (4.2)

Geometry

2.5-D (Figure 4.1(b))

3-D (Figure 4.1(c))

Source

Point source / (Finite) incoherent line source

Point source / Finite incoherent line source

Boundaries

Locally reacting, complex valued impedance or rigid

Fluid medium with different density or rigid

Staircase shaped geometry

Staircase shaped geometry

Inhomogeneous and moving medium above

Inhomogeneous and moving medium

Meteorological conditions


Table 4.1: Developed prediction methods and their features.

roof level Discretization

Air absorption

No air absorption

Equivalent Sources (ES) at sub-domain interfaces

Total domain discretized

10 ES per λ

2 spatial points per λ

33

34


Chapter 5

The 2.5-D Equivalent Sources Method (ESM)

The idea behind the equivalent sources method (ESM) as it is applied here, is to divide the computational domain of interest into sub-domains and populate the sub-domain interfaces by equivalent sources. When the strength of these equivalent sources fulfil the governing equation, boundary conditions and subdomain interface conditions, the sound pressure at any point in the computational domain can be computed. The 2.5-D ESM is an extension of the 2-D ESM for the parallel canyons’ geometry as developed by Ögren and Kropp [183]. The 2.5-D ESM for parallel canyons is described in Sec. 5.1 and an extended description can be found in Paper II. A change of one of the Green’s functions has been made to speed up the ESM computation time, and is presented in Sec. 5.2. As discussed in Sec. 5.3, the developed method allows for modelling an incoherent line source of infinite and finite length. The accuracy of the 2.5-D ESM is discussed in Sec. 5.4, where results from the 2.5-D ESM are compared with results from the scale model study.

5.1 The 2.5-D ESM Figure 4.1(b) shows the situation and Eq. (4.1) the governing equation that the ESM aims to solve in Cartesian coordinates. Equation (4.1) is first transformed from the (x, y, z)-domain to the (x, ky , z)-domain and solved in that domain.

35

36

5. The 2.5-D Equivalent Sources Method (ESM)

Figure 5.1: (a) A simplified street canyon cross-section with two façade impedance surfaces and a balcony; (b) ESM representation of the situation of (a), where the domain is divided in sub-domains by placing equivalent sources; (c) Extended Fourier PSTD representation of the situation of (a), where the total domain is discretized.

5.1. The 2.5-D ESM

37

The 2.5-D results are then obtained by an inverse transform. We therefore first multiply Eq. (4.1) by ejky y and integrate over y, which gives 2 ∂2 ∂ 2 2 + P + (k − jα) − k P = −ejky ys δ(x|xs )δ(z|zs ), (5.1) y ∂x2 ∂z 2 with Z∞ P (x, ky , z) = p(x, y, z)ejky y dy, −∞

q k = kx2 + ky2 + kz2 ,

where α = α(f ) is the frequency-dependent air absorption coefficient. From this equation, we find the following equivalent 2-D Helmholtz equation: 2 ∂ ∂2 2 (5.2) + + (K − jα2D ) q = −δ(x|xs )δ(z|zs ), ∂x2 ∂z 2 with q q x, K = k 2 − ky2 , z = P (x, ky , z)e−jky ys , q q 2 2 2 2 α2D (f, ky ) = j (k − jα) − ky − k − ky .

This equation is solved by the 2-D ESM. A rigorous description of the ESM is given e.g. by Bérillon and Kropp [23]. We will here illustrate the ESM methodology for a typical cross-section of a canyon with impedance patches and a balcony; see Fig. 5.1(a). The discretization of this cross-section by equivalent sources in the ESM is shown in Fig. 5.1(b). The domain is split into sub-domains for which Green’s functions, fulfilling Eq. (5.2) and boundary conditions, are known. The interfaces of the sub-domains are populated by equivalent sources. In the case of the geometry of Fig. 5.1(a), several sub-domains are discerned: rectangular cavities, impedance patches and a free space above the canyon. Opposite to the physical situation, the sub-domain interfaces are rigid to get useful Green’s functions. The primary source with unity source strength and equivalent sources are coherent line sources in the 2-D geometry. The solution is simplified, compared to solving the 3-D case since the equivalent sources cover lines in 2-D instead of surfaces in 3-D. The crux of the ESM is to find the equivalent sources strengths. These strengths are expressed in v, the volume velocity per unit length, and are equal to the normal surface velocities in this case [23]. They are found by imposing continuity of the pressure and normal velocity across the sub-domain interfaces. The normal velocity condition is fulfilled by setting the signs of the equivalent sources strengths opposite for the two bordering subdomains. To express the pressure at both sides of the interface for position r in

38


Fig. 5.1(b), we use the Kirchhoff-Helmholtz integral equation, see e.g. [23]:

−

Zx2 0

Zx1 0

+

(5.3)

v(x, z4 )G1 (xr |x, zr |z4 ) dx + G1 (xr |xs , zr |zs ) = v(x, z4 )G3 (xr |x, zr |z4 ) dx +

Zz1 0

Zz3

z1

v(x1 , z)G3 (xr |x1 , zr |z) dz +

v(0, z)G3 (xr |0, zr |z) dz

Zz4

z2

v(x1 , z)G3 (xr |x1 , zr |z) dz,

with G1 the semi-free field Green’s function, G3 the Green’s function inside a rectangular cavity, and s the index of the primary source. Equation (5.3) fulfils the governing equation, the boundary conditions and the conditions at the interface position r. Numerically, the integrals become discrete sums over equivalent sources. When writing down this equation for all equivalent source positions N, a matrix system can be constructed: (5.4)

Av = b,

where A is a square N × N matrix containing the Green’s functions, v a column vector with length N representing the unknown source strengths v and b a column vector with length N containing the primary source contributions. The system is solved using LU decomposition. For the case of Fig. 5.1(b), the complex pressure q at a receiver point r3 inside the canyon is now obtained by an integration over the equivalent sources bordering the canyon: x Z1 Zz3 q(xr3 , zr3 ) = jωρ0  v(x, z4 )G3 (xr3 |x, zr3 |z4 ) dx + v(0, z)G3 (xr3 |0, zr3 |z) dz z1

0

+

Zz1 0

v(x1 , z)G3 (xr3 |x1 , zr3 |z) dz +

Zz4

z2



v(x1 , z)G3 (xr3 |x1 , zr3 |z) dz  .

(5.5)

The pressure at a point r1 above roof level is calculated by a summation of contributions of the primary source and the equivalent sources at roof level:  x Z2 q(xr1 , zr1 ) = −jωρ0  v(x, z4 )G1 (xr1 |x, zr1 |z4 ) dx − G1 (xr3 |xs , zr3 |zs ) . (5.6) 0

5.1. The 2.5-D ESM

39

The 2.5-D solution is obtained by applying an inverse Fourier transform over solutions q. We retrieve the 2.5-D solution by: 1 p(x, y, z) = 2π

Z∞ q q x, K = k 2 − ky2 , z ejky (ys −y) dky .

(5.7)

−∞

The pressure p can thus be calculated by an integral over 2-D solutions with p wave numbers K = k 2 − ky2 . For equivalence of a locally reacting boundary condition, we write (see Paper II): Zs,2D (K, f ) =

Zs (f )K , k

(5.8)

with Zs the normalized surface impedance and Zs,2D the normalized surface impedance of the 2-D problem governed by Eq. (5.2). Physically, this condition can be explained as follows. A locally reacting boundary condition as used in the ESM (see Sec. 5.2), means that sound waves travel into the boundary medium perpendicular to the interface. Equation (5.7) represents a decomposition of the pressure field in waves with different wave numbers ky representing different propagation angles. Each decomposition entails a projected wave number K in the 2-D domain. For every different ky , the wave propagates in the boundary medium with wave number k ccb , where cb is the (complex) sound speed of the boundary medium. However, the use of Zs would imply a propagating wave number in the boundary of K ccb . Use of Eq. (5.8) is therefore necessary. When the boundary impedance is finite, a set of 2-D calculations over q should therefore be done for every frequency f of p, since Zs,2D depends on ky and f 1 . This does increase the calculation time. The integral of Eq. (5.7) has singularities for ky = −k and ky = k which are treated by a change of variables, similar to what has been done before by Salomons [136], and details can be found in Paper II. For ky > |k|, K becomes imaginary, and q corresponds to a near-field solution. An important numerical issue when the integration is approximated by a summation is whether solutions with imaginary frequencies are necessary to include in the integral in order to find an accurate 2.5-D solution (since solutions for imaginary frequencies are not straightforward to calculate for all cases). Omitting the imaginary frequencies mostly affects the lowest frequencies and smaller source-to-receiver distances due to the fact that the near-field solution decays more rapidly with the increasing frequency and source-to-receiver distance. The largest contribution to the integral of Eq. (5.7) 1

This also holds if air absorption is included, since α2D in Eq. (5.2) is a function of ky and f as well.

40


Table 5.1: Required frequency discretization ∆f of q in the integral of Eq. (5.7) for obtaining the 2.5-D ESM solution with a tolerance of 0.5 dB. Geometry of Fig. 4.1(b) with point source coordinates (9,0,0) and incoherent line source coordinates (9,y,0). For the incoherent line source calculations, ∆f = 0.2 Hz up to 50 Hz. Diffusion patches refer to the applied façade diffusion patches as in the scale model study (see Fig. 1(b) of Paper I).

Receiver coordinates Rigid façades Diffusion patches

Point source f = 100 Hz, y = 0 m (0,0,15) (60,0,15) 0.4 0.1 0.4 0.2

Incoherent line source f = 1000 Hz (49,0,0) 0.4 1.6

lies around ky = k sin θ, with θ as of Fig. 1 in Paper II. To prevent the imaginary frequencies from making a substantial contribution to the integral, sin θ should be much smaller than 1. Another numerical issue is how fine the frequency discretization of the integral of Eq. (5.7) should be. For a constant frequency discretization of q and in free field, the solution q increases to oscillate with increasing r. The solution therefore breaks down for a certain source-receiver distance. Table 5.1 shows the required frequency discretization for calculations in the parallel canyons geometry of Fig. 4.1(b). Since higher order façade reflections have a larger contribution to the sound level in the shielded canyon than in the directly exposed canyon, see also Sec. 7.2, the required frequency discretization is smaller in the shielded side.

5.2 Green’s functions The ESM relies on Green’s functions in the several sub-domains, and calculating the Green’s functions between the equivalent sources causes the major part of the ESM calculation time. There are three types of domains involved, each with a different Green’s function. G1 : The Green’s function above roof level The Green’s function above roof level is a semi-free field Green’s function for the 2-D Helmholtz of Eq. (5.2): j (2) (5.9) G1 (x|xs ) = − H0 (K|x|xs|) , 2 p (2) with |x|xs| = (x − xs )2 + (z − zs )2 and with H0 the Hankel function of order zero and second kind. For an inhomogeneous or moving medium, a different Green’s function might be used. Also, Paper III uses the Hadden-Pierce diffraction expression to calculate approximate Green’s functions in the presence of a

5.2. Green’s functions

41

wedge. G2 : The Green’s function at an absorption surface When assuming a locally reacting impedance surface, the Green’s function at the surface is: (5.10)

G2 (x|xs ) = Zs,2D (x),

where x = xs and Zs,2D as from Eq. (5.8). The pressure at the impedance surface is found from q(x|xs) = G2 (x|xs )vn (x|xs), with vn the volume velocity per unit length of the equivalent source at the impedance surface. G3 : The Green’s function in a rectangular domain with rigid boundaries Two ways to calculate this Green’s function are by a modal summation G3,modal and by a summation of the image sources G3,IS : G3,modal (x|xs ) ≈ N M 4 X X cos (mπx/lx ) cos (mπxs /lx ) cos (nπz/lz ) cos (nπzs /lz ) , lx lz n=0 m=0 m2 π 2 /lx2 + n2 π 2 /lz2 − K002

with:  0.25  = 0.5   1

m, n = 0 m = 0, n > 0 or m > 0, n > 0,

(5.11)

n = 0, m > 0

q N M j XX (2) 2 2 0 H0 K (2nlz − |zs + z|) + (2mlx − |xs + x|) G3,IS (x|xs) ≈ − 4 n=0 m=0 q (2) 2 2 0 + H0 K (2nlz − |zs + z|) + (2mlx − |xs − x|) q (2) 2 2 0 + H0 K (2nlz − |zs − z|) + (2mlx − |xs + x|) q (2) 2 2 0 + H0 K (2nlz − |zs − z|) + (2mlx − |xs − x|) , (5.12) where K 0 = (K − jα2D ), with N, M the highest mode numbers in Eq. (5.11) and the highest reflection order in Eq. (5.12) and lx and lz the dimensions of the rectangle. Both methods have a double summand and the disadvantage that they converge slowly with increasing N and M. G3,modal has been used before in the 2-D ESM [181]. Since G3 takes the largest part of the ESM computation time due to the summation, it is worth searching for a faster way to calculate G3 . Several methods were developed to accelerate the Green’s function of the 2-D Helmholtz equation for periodic domains. Linton gives an overview of ana-

42


lytical techniques [105]. From these techniques, the lattice sum method has the advantage that the major part of its summand is independent of the positions of the source and the receiver. Having calculated its summand for a certain geometry, one can efficiently calculate for multiple source and receiver positions. Although this principle is advantageous here, the method does not converge for the typical source-receiver distances of the canyon geometry. Ewald’s method is another attractive method, yet only for very short distances from the source [105]. Kummer’s method, which relies on the Kummer transformation, improves the convergence of Eqs. (5.11) and (5.12) [105]. It has been applied to acoustics by Zinoviev [180] and relies on subtracting and adding back a series which has the same asymptotic behaviour as the summations. These are thereby converted into two series which both converge faster. The transform can be applied to the modal as well as to the image sources approach and for both methods some kind of modal term is retrieved. It is here found that, for the same accuracy, a factor 3.3 in time gain can be achieved in 1-D, or a factor 11 in 2-D. Horoshenkov and Chandler-Wilde used an integral expression to obtain a more efficient Green’s function calculation in an acoustical application with an impedance boundary [80]. Their method is, however, not expected to be faster than Kummer’s method in our frequency range of interest. An even more attractive method than Kummer’s method is the use of a Green’s function, which is a combination of a modal and a wave approach. In one direction (here the x-direction), a single-term expression based on the wave approach is used and in the z-direction, the modal summation is left. The derivation of this method was based on the expression of the Green’s function in a rectangular domain with soft boundaries [47]. Below, the derivation is done for rigid boundaries. We assume that the Green’s function solution to Eq. (5.2) has the form: G3 (x|xs) ≈ with( =

0.5 1

N X

Gn (x|xs ) cos (nπz/lz ) ,

(5.13)

n=0

n=0 n > 0.

Inserting this Green’s function in Eq. (5.2), multiplying by 2 cos (nπz/lz ) /lz and integrating over z, we obtain: 2 ∂ 2 2 − κn Gn = − cos (nπzs /lz ) δ(x − xs ), (5.14) 2 ∂x lz


Amplitude (Pa)

4

−2.1

1.4 1.3

3

1.2 2

−2.2

1.1

1 0

(c)

(b)

(a) 100 N (−)

200

2 Amplitude (Pa)

43

1 0

100 N (−)

1

−2.3 200 0

100 N (−)

200

100 N (−)

200

−1.7

0.9 1.5

0.8 −1.8

0.7

(e)

(d) 1 0

100 N (−)

200

0.6 0

(f) 100 N (−)

200

0

Figure 5.2: Values of Green’s functions calculated for various source-receiver combinations (see sketch at the right) as a function of the number of modes N. The frequency is 50 Hz. Thin: Green’s function calculated using Eq. (5.11) with M = N ; Thick: Green’s function calculated using Eqs. (5.18).

where κ2n is equal to (nπ/lz )2 − K 02 . The wave solution to this equation is ( 0 ≤ x ≤ xs 1 An eikn x + e−ikn x (5.15) Gn (x|xs ) = ik (l −x) −ik (l −x) n x n x 2 Bn e +e xs ≤ x ≤ lx ,

where An and Bn are yet unknown coefficients and kn2 = −κ2n . These solutions satisfy Eq. (5.14) and the boundary conditions, and can physically be seen as two travelling waves in opposite directions, multiplied by a yet unknown coefficient. The coefficients An and Bn are obtained from the conditions of pressure and normal velocity across the source position: An cos (kn xs ) = Bn cos (kn (lx − xs )) , kn Bn sin (kn (lx − xs )) + kn An sin (kn xs ) = −

(5.16) 2 cos (nπzs /lz ) . lz

We now find for An and Bn : 2 cos (nπzs /lz ) cos (kn (lx − xs )) , lz kn sin (kn lx ) 2 cos (nπzs /lz ) cos (kn xs )) . Bn = − lz kn sin (kn lx )

An = −

(5.17)

This gives the Green’s function G3 after insertion Eqs. (5.17) in Eq. (5.15) and the resulting Eq. (5.13):

44

5. The 2.5-D Equivalent Sources Method (ESM) G (x|xs) ≈ 3 N P  x)) cos(nπzs /lz ) cos(nπz/lz )  cos(kn (lx −xs )) cos(kknn sin(k  − l2z n lx )    − l2z

n=0 N P n=0

s )) cos(nπzs /lz ) cos(nπz/lz ) cos(kn (lx −x)) cos(kknnxsin(k n lx )

(5.18) 0 ≤ x ≤ xs xs ≤ x ≤ lx .

When assuming a similar convergence rate for N and M for the modal summation of Eq. (5.11), the computation time gain is quadratic. In fact, this new Green’s function is even more efficient, since its solution is exact in one direction. Figure 5.2 shows the convergence with N for a geometry with typical dimensions of a canyon, calculated with the modal summation of Eq. (5.11) and the modal-wave approach of Eqs. (5.18). In the modal summation, both sums have a maximum number equal to N. For a receiver position equal to the source position, no convergent results have been found for N up to 200. For all other receiver positions and x 6= xs , the modal-wave approach converges faster with N than the modal summation does; see Figs. 5.2(b) to (f).

5.3 Incoherent line sources To calculate the equivalent noise level from a dense traffic flow, it can be modelled by a row of uncorrelated point sources (an incoherent line source). From the point source solution of Sec. 5.1, it is straightforward to calculate the incoherent line source solution. The incoherent line source solution level (re 1 Pa) may be written as:  ∞  Z LILS (x, z, f ) = 10 log  |p(x, y, z, f )|2 dy  , (5.19) −∞

with p(x, y, z, f ) the solution calculated by Eq. (5.7). By the Parseval theorem, we may write: Z∞

−∞

1 |p(x, y, z, f )| dy = 2π 2

Z∞

−∞

|q(x, K, z)|2 dky .

(5.20)

The incoherent line source solution can thus also be calculated by integrating over the 2-D solutions |q(x, K, z)|. Since |q(x, K, z)| is less fluctuating over the wave number K then the complex-valued q(x, K, z) from Eq. (5.7), the integral on the right side of Eq. (5.20) can be evaluated more efficiently compared to evaluation of the integral of Eq. (5.7). Calculations with an incoherent line source in the canyons’ geometry of Fig. 4.1(b) by evaluating the right side of

5.4. Validation

45

Eq. (5.20) have been done in Paper II, and Table 5.1 reports on the required frequency discretization of q. A solution with an incoherent line source of finite length can be obtained using the 2.5-D ESM as well by limiting the integration limits:   kZy,max q 2 1  LFILS (x, z, f ) = 10 log  (5.21) q x, K = k 2 − ky2 , z dky  2π 

= 10 log 

−ky,max

1 2π

Zθmax

−θmax



|q(x, k cos(θ), z)|2 cos(θ)k dθ ,

where θmax is the maximum angle between receiver position and (image) source. The appendix of Paper III shows the details of this expression. The finite incoherent line source has in that Paper been used to evaluate the effect of noise abatement schemes.

5.4 Validation The 2-D ESM was successfully validated against the boundary element method for low frequencies and rigid boundaries [181]. The 2.5-D ESM is in Paper II validated for infinitely high façades, and convergence is shown in the shielded canyon. A comparison of 2.5-D ESM results with scale model study results has been done in Paper III to show its capabilities for the canyons’ geometry. The error made in the 2.5-D ESM is due to the following aspects: ◦ ESM approach. The semi-free field Green’s function is known analytically and its error in a computational environment is restricted to the computer round-off error. A locally reacting impedance is assumed for non-rigid boundaries. For most outdoor porous materials, this is a good approximation. For a more rigid material, this approximation is less good. The rectangular domain Green’s function converges with an increasing number of terms included. The number of terms is chosen high to obtain convergent results. Small numerical damping, included in the air absorption coefficient, is indispensable to prevent modes having too large amplitudes. The equivalent sources’ strengths are determined by imposing conditions consisting of pressure and normal velocity across the sub-domain interfaces. The pressure and normal velocity fluctuate over the length of the interface on a scale decreasing with increasing frequency. These fluctuations have to be captured by the number of equivalent sources applied.

46


The combined error of the approximated Green’s functions and the number of equivalent sources is difficult to predict, mainly due to the modal behaviour of the sound field in the canyon. A small error in one of the two sources of error could lead to a large error for a certain receiver point. The error generally decreases with an increasing number of equivalent sources. ◦ Transform from 2-D to 2.5-D. The limited frequency discretization in the 2-D to 2.5-D transform means that an approximate solution is obtained as printed in Table 5.1. When omitting the imaginary frequencies, a small error (< 1 dB) in the directly exposed street canyon could also be present in the lowest 1/3-octave bands considered, i.e. up to 100 Hz. ◦ The number of frequencies per 1/3-octave band. For comparison with the scale model study results and presentation of the noise abatement schemes, results are presented in 1/3-octave bands. To save calculation time, the number of calculated frequencies per 1/3-octave band has been reduced to a number that is estimated to be sufficient for convergence. This number depends on the boundary conditions: for rigid boundaries, the solution is more fluctuating over frequency than for cases with finite impedance boundaries or with diffuse boundary reflections. A number of 30 frequencies per 1/3-octave band has been used throughout. For the validation presented here in Fig. 5.3, results from the scale model study have been used. A more detailed comparison has been done in Paper III. A comparison of sound levels at directly exposed and shielded canyon receiver positions with a height of 5 m in the outer façade is shown. An average over receiver positions for y ∈ [0 m, 40 m] has been taken. The agreement is rather good for all situations.

5.4. Validation

47

(III)

Lre free (dB)

30

10

(dB) L

re free

−10 −20

re free

−10 −20

10

125 250 500 1000 Frequency (Hz)

−20

10 (dB)

0

−10

−30

125 250 500 1000 Frequency (Hz)

L

(dB) (dB)

20

0

−30

125 250 500 1000 Frequency (Hz)

(dB)

0

L

10

0

10

re free

20

10

−30

125 250 500 1000 Frequency (Hz)

re free

Lre free (dB)

30

L

10 0

(II)

re free

20

10

125 250 500 1000 Frequency (Hz)

0 −10 −20 −30

125 250 500 1000 Frequency (Hz)

0 3−D extended FPSTD Scale model 2.5−D ESM

−10 −20

L

(I)

Lre free (dB)

30

−30 0

125 250 500 1000 Frequency (Hz)

−40

125 250 500 1000 Frequency (Hz)

Figure 5.3: Comparison of scale model measurement results and results from calculations with the 2.5-D ESM and 3-D extended Fourier PSTD method. Results in 1/3-octave bands. (a) Source at (9,0,0) and receivers at (0,y,5); (b) Source at (51,0,0) and receivers in the shielded canyon at (0,y,5); (c) Source at (9,0,0) and receiver at (49,0,0). (I) Rigid façades; (II) Horizontally oriented diffusion patches; (III) Horizontally oriented absorption patches. Results have been averaged over the y-position of the receivers in (a) and (b).

48


Chapter 6

The 3-D extended Fourier pseudospectral time-domain (PSTD) method

Pseudospectral (PS) methods have become popular to solve various kinds of time-dependent physical problems, as testified by the amount of books written on this topic; see [74] for an overview. A major advance in spectral methods, to which PS methods belong, was made in 1972 by Kreiss and Oliger [99] and Orszag [120] who used trigonometric functions and Chebyshev polynomials to compute spatial derivatives. Early applications of PS methods concerned mainly elliptic and parabolic equation problems [67], as fluid dynamics and weather prediction [32, 66]. Since hyperbolic problems are more sensitive to instability than elliptic and parabolic problems are, PS methods have only been applied to hyperbolic problems such as gas dynamics, electromagnetics and acoustics from the late 1980s and early 1990s [67]. Another reason for the slow introduction of PS methods for hyperbolic problems is the appearance of the Gibbs phenomenon for discontinuous media problems. Since use of Chebyshev polynomials could solve these problems, see Sec. 6.1.2, this problem could be overcome [67]. With regard to acoustics, the use of PS methods is currently still not numerous, with applications such as photo-acoustics [42], non-linear elastic wave propagation [68], ultrasonics [110, 149], aeroacoustics [82] and atmospheric wave propagation [169]. To introduce the developed extended Fourier PSTD method, the two most widely used PS methods, the Fourier PS method for a periodic domain problem and 49

50

6. The 3-D extended Fourier pseudospectral time-domain (PSTD) method

the Chebyshev PS method for non-periodic problems, are described in Sec. 6.1. They are used to calculate the spatial derivative operator of the one-dimensional wave equation. Then, in Sec. 6.1.3, the extended Fourier PS method is presented as a method for periodic problems with discontinuous media. The extended Fourier PS method is used to calculate the spatial derivative operator of a time-dependent problem, with the time-marching scheme operated by another method. Such a combination is known as a pseudospectral timedomain (PSTD) method. Section 6.2 is devoted to the extended Fourier PSTD method to model sound propagation to a 3-D courtyard, where the linearized Euler equations (LEE) are solved. For a homogeneous non-moving medium, the theory and numerical implementation to calculate the spatial derivatives are presented in Sec. 6.2.1, and in Sec. 6.2.2, the treatment of media inhomogeneities and a moving medium is discussed. The time-marching scheme used is shown in Sec. 6.2.3. The treatment of open boundaries is discussed in Sec. 6.2.4 followed by a demonstration of the implementation of the method. The Section concludes with estimates on the numerical costs. In Sec. 6.3 finally, calculation results from the method in 2-D and 3-D are validated.

6.1 PS methods to calculate spatial derivatives In PS methods, the residual of the governing equation is forced to hold at a set of discrete spatial points, collocation points. Other spectral methods are the Galerkin and tau method, which require that the projection of the residual on some polynomial vanishes [74]. The PS method thus decomposes the physical domain by a number of discrete points. For the Fourier PS, Chebyshev PS and extended Fourier PS methods, we consider the 1-D wave equation: 2 ∂ 1 ∂2 − p(x, t) = 0. (6.1) ∂x2 c2 ∂t2 Different boundary conditions apply for the three PS methods, as will be described in the following Sections and Appendix C.2, where the treatment of boundary conditions in the Chebyshev PS method is presented in detail. The PS ∂2 methods approximate the spatial derivative operator L = ∂x 2 , which is treated independently of the method approximating the time derivative operator. We thus write Eq. (6.1) as: ∂2 p(x, t) = Lp(x, t), ∂t2

(6.2)

6.1. PS methods to calculate spatial derivatives

51

where the right side is computed by the PS methods. In a time iteration scheme, the operator Lp(x, t) needs to be evaluated for every time step, where p(x, t) is known from the former time step. The core of all PS methods for transient problems such as Eq. (6.2) is that the function p(x, t) is projected onto a set of orthogonal basis functions ψn (x) [74]: p(x, t) =

X

Pn (t)ψn (x),

(6.3)

|n| |N/2|, this will lead to an aliasing error. The error in p(x) is thus composed of the truncation error from Eq. (6.7) and the aliasing error from Eq. (6.8). Both errors decrease when the function p(x) is smoother, i.e. when the values of |Pn | for large n are suppressed. The maximum resolved wave number is therefore determined by the chosen spatial discretization and corresponds to two spatial points per wavelength. If we increase the signal period interval from [0, 2π] to (−∞, ∞), the discrete projection of Eq. (6.5) becomes an integral transform, and Eqs. (6.5) and (6.6) become the Fourier transform pair: 1 p(x) = 2π

Z∞

P (k)ejkx dk,

(6.9)

−∞

with P (k) =

Z∞

p(x)e−jkx dx,

(6.10)

−∞

which holds for finite energy signals p(x). These integrals are also approximated by numerical quadrature of discrete sums: 1 p(l∆x) ≈ ∆xN

n=N/2−1

X

n=−N/2

ln

P (n∆k)ej2π N ,

(6.11)


53

with N/2−1

P (n∆k) ≈ ∆x

X

ln

p(l∆x)e−j2π N ,

(6.12)

l=−N/2

where discretization of both spatial and wave number domain again leads to periodicity in both domains and to aliasing and truncation issues. The discrete Fourier transform can thus be applied to a finite domain on which p(x, t) is periodic2 . Using the Fourier-transform pair, the spatial derivative operator L can now be calculated by applying the derivative operator to the Fourier kernel: 1 Lp(x) = 2π =

=

1 2π 1 2π

Z∞

−∞ Z∞ −∞ Z∞ −∞

P (k)Lejkx dk

P (k)

(6.13)

∂ 2 jkx e dk ∂x2

−k 2 P (k)ejkx dk.

To calculate Lp(x), we thus need to calculate P (k), as approximated by Eq. (6.12), multiply by −k 2 ejkx and integrate numerically over the wave number k, as in Eq. (6.11). The summations are evaluated by FFTs, which require N log2 N calculations as compared to N 2 for a discrete Fourier transform (i.e. direct evaluation of the summations of Eqs. (6.11) and (6.12). The Fourier PS method is global since all spatial points are involved to compute a spatial derivative. This is in contrast to the finite-difference method where a limited number of spatial points is involved to compute a spatial derivative. In Appendix C.1, it is shown that the Fourier PS method is equivalent to a finite-difference method wit an infinite order of accuracy when |P (k)| = 0 for k > π/∆x. The stability of an equation (system) of the form of Eq. (6.2) depends on the eigenvalues of the matrix operator L. For the Fourier PS method, the matrix operator is normal, i.e. LLT = LT L and stability of the numerical scheme can be determined from the maximum eigenvalue of L [74]. The maximum eigen√ Dπc value of L using an equidistant grid spacing ∆x is max|| = ∆x , with D the dimensionality of the problem and c the adiabatic speed of sound. The largest discrete time step ∆t for stability, can be found by the maximum eigenvalue and 2

Periodicity can most conveniently be obtained by adding a numerical absorption layer at the domain ends such that p(x, t) has compact support on the domain. This implies that p(x, t) is periodic on the domain and moreover prevents wrap-around effects.

54


the stability region of the time iteration scheme used. This ∆t is proportional to the chosen discretization step ∆x. The above Fourier basis functions fail for a signal on a spatial domain of interest [a, b] which is, together with its periodic extensions, not smooth. The reason is that P (k) will have substantial components for k > π/∆x, causing truncation and aliasing errors. Such an error is introduced when the domain [a, b] consists of media with discontinuous properties. This error is known as the Gibbs phenomenon and can make the numerical solution of Eq. (6.2) unstable, especially since the wave equation is a hyperbolic equation without dissipation. Increasing the grid density can reduce the error linked with the Gibbs phenomenon. This will make the error in the approximation more local, yet will not remove it completely. The medium properties can also be low-pass filtered spatially for a reduced effect of the Gibbs phenomenon, although leading to a lower accuracy for high wave numbers. Also, a post-processing method can be applied, yet is computationally inefficient [109]. Another way to reduce the Gibbs phenomenon is the mapped Fourier PS method proposed by Lu, where the grid points are redistributed by a pre-determined mapping curve [109].

6.1.2 Chebyshev PS method A way to completely remove the Gibbs phenomenon as it arises when using the Fourier PS method for media with discontinuous properties is by using a different set of basis functions in the expansion of Eq. (6.3) [74]. PS methods for spatially bounded domain problems, where the periodic extension of p(x) and its derivatives are not required to be continuous, have therefore been developed with polynomials that form a non-periodic basis. The polynomials commonly used are eigensolutions to Sturm-Liouville problems, since this class of polynomials has extensively been studied and has nice convergence properties. The Sturm-Liouville problem can be formulated as: d d Lψn (x) = − u(x) + q(x) ψn (x) = n w(x)ψn (x) x ∈ [−1, 1], (6.14) dx dx dψn (−1) =0 dx dψn (1) =0 α+ ψn (1) + β+ dx

α− ψn (−1) + β−

2 α− + β−2 6= 0

α− β− ≤ 0,

2 α+ + β+2 6= 0

α+ β+ ≥ 0,


55

where u(x) ∈ C 1 [−1, 1]3 and positive in (−1, 1), q(x) ∈ C 0 [−1, 1] and nonnegative and w(x) ∈ C 0 [−1, 1] and non-negative. The eigenfunctions ψn (x) and eigenvalues n form a L2 [−1, 1]-complete basis [74]. Since the operator L is self-adjoint4 , the eigenfunctions ψn (x) are orthogonal, and we can use them to expand p(x): p(x) =

Pn =

∞ X

n=0 Z1

1 cn

Pn ψn (x),

(6.15)

p(x)ψn (x)w(x) dx,

−1

with cn the normalization coefficient and where the overbar denotes the complex conjugate. Eigenfunctions that are solutions to Sturm-Liouville problems with u(−1) = u(1) = 0 are special cases of Jacobi polynomials. Two orthogonal Jacobi polynomials are often used: Legendre polynomials and Chebyshev polynomials. Legendre polynomials are the best with regard to minimizing the L2 error. The Chebyshev polynomials have simple expressions and are easy to compute. The Chebyshev polynomials Tn (x) are solutions to the Sturm√ √ Liouville problem with u(x) = 1 − x2 , q(x) = 0 and w(x) = 1/ 1 − x2 such that Eq. (6.14) becomes: n d √ d 2 +√ Tn (x) = 0, (6.16) 1−x dx dx 1 − x2 with the solution Tn (x) = cos(n arccos(x)). When changing variables according to x = cos(ξ), Eq. (6.16) reduces to: 2 d + n Tn (ξ) = 0, (6.17) dξ 2 with solution Tn (ξ) = cos(nξ). The discrete grid points xl can be chosen differently. A uniformly chosen grid point distribution does not lead to convergent solutions near the boundaries, this being known as the Runge phenomenon. An often made choice is to make use of the Gauss-Lobatto distribution, xl = − cos(πl/N), l ∈ [0, ..., N], which is close to optimally distributed [159]; see Fig. 6.1.

3

u(x) ∈ C n [−1, 1] means that function u(x) and its n-order derivatives are continuous in domain [−1, 1]. 4 L is a self-adjoint operator if hLf, gi = hf, Lgi, with f, g ∈ C ∞ [−1, 1].

56


−1

0 x−position (−)

1

Figure 6.1: Location of discrete grid points xl on a 1-D domain with x ∈ [−1, 1] according to the Gauss-Lobatto distribution in the Chebyshev PS method, N = 16. This distribution is also efficient from a computational efficiency point of view, since the FFT can now be used. The Chebyshev method with Gauss-Lobatto points reads: p(ξl ) ≈ Pn ≈

γn =

(

Pn cos(πln/N),

n=0 N X

1 γn

with ( wl =

N X

(6.18)

p(ξl ) cos(−πln/N)wl ,

l=0

π n = 0, N π/2 n ∈ [1, ..., N − 1], π/(2N) l = 0, N π/N l ∈ [1, ..., N − 1],

and values at xl are found from xl = cos(ξl ). For geometries with different dimensions than the standard [-1,1], mappings can project these geometries on the standard [74]. The derivative operator using the Chebyshev polynomial expansion can be calculated as [159]: Lp(xl ) =

∞ X

n=0 ∞ X

Pn L cos(nξ)

(6.19)

∂ 2 cos(nξ) ∂x2 n=0 ∞ X ∂ ∂ξ ∂ cos(nξ) = Pn ∂x ∂x ∂ξ n=0 ∞ X ∂ 1 √ = nPn sin(nξ) 2 ∂x 1 − x n=0 ∞ ∞ X 1 X 2 x nPn sin(nξ) − n Pn cos(nξ). = (1 − x2 )3/2 n=0 1 − x2 n=0

=

Pn

To calculate Lp(xl ), both Pn in Eq. (6.18) as the summations in Eq. (6.19) can be calculated using FFTs.


57

In the Chebyshev method with the Gauss-Lobatto distribution, the ratio between the maximal spatial discretization distance and the average spatial discretization distance is ∆xmax /∆xavg = π/2, resulting in a minimum number of π points per wavelength. As we also have seen in Sec. 6.1.1, explicit timemarching schemes have a stability criterion for the choice of ∆t related to the smallest spatial discretization ∆xmin . Since ∆xmin /∆xavg decreases linearly with domain size N, ∆t decreases with increasing N. The total domain needs therefore to be subdivided to relax ∆t. However, since the accuracy of the method decreases with N, the choice of N is a trade-off [67]. This choice is further influenced by the computational efficiency, since the number of operations needed per FFT is N log2 N, and sub-domain calculations can be parallelized. As shown for the Fourier PS method in Appendix C.1, the Chebyshev PS method can also be written as an interpolation of grid point values at the bounded domain. Adopting that approach, the derivative operator becomes a matrix-vector multiplication. For domains where N < 64, this interpolation method is recommended over the approach of Eq. (6.19) from an efficiency point of view [74]. As no boundary conditions are applied in the calculation in Lp(x) of Eq. (6.19), the results are accurate for the internal domain points only. In Appendix C.2, the treatment of the sub-domain boundary conditions and outer boundary conditions, which also influence the stability properties, are discussed.

6.1.3 Extended Fourier PS method Like the Chebyshev PS method, the extended Fourier PS method uses a different set of basis functions to avoid Gibbs phenomenon arising in the Fourier PS method for discontinuities in the media properties. In accordance with the Fourier PS method and in contrast to the Chebyshev method, the basis functions in the extended Fourier PS method account for the whole domain. As an example here, we consider a two-media problem, with density ρ1 and sound speed c1 for x ∈ (−∞, 0] and ρ2 , c2 for x ∈ [0, ∞). The sought basis functions of Eq. (6.5) are now solutions to the following eigenvalue problem: d 1 d − Rj ψ(, x) = 0 x ∈ (−∞, ∞), (6.20) dx ρj dx d 1 d with L = dx a self-adjoint operator5 , ψ(, x) the orthogonal eigenfuncρj dx

58


tions, Rj = ρj1c2 and = −kj2 c2j the eigenvalues. Because of the infinite spatial j domain, ψ(, x) and are both generalized, i.e. continuous functions. The eigenfunctions can be written as:  √ √  α e c1 x + β e− c1 x x ≤ 0 1 1 √ (6.21) ψ+ (, x) = N+ ()  e c2 x x ≥ 0,  √  e− c1 x x≤0 √ √ ψ− (, x) = N− ()  α e− c2 x + β e c2 x x ≥ 0, 2

2

where ∈ (−∞, 0]. The coefficients α1 , β1 , α2 and β2 can be calculated by the continuity of pressure and normal velocity at x = 0. The eigenfunction can be seen as a plane wave decomposition of the wave field. The coefficients N+ () and N− () in Eq. (6.21) are normalization constants chosen so that the following orthogonality condition is satisfied: Z∞

−∞

ψ± (, x)ψ± (0 , x)Rj dx = δ( − 0 ),

(6.22)

where the overbar denotes the complex conjugate. We can now expand the function p(x) with the orthogonal eigenfunctions as a continuum basis: 0 XZ

p(x) =

± −∞

P± ()ψ± (, x) d,

(6.23)

where P± () =

Z∞

p(x)ψ± (, x)Rj dx.

(6.24)

−∞

Applying operator L to the left-hand side of Eq. (6.23) and making use of Eq. (6.20), i.e. Lψ = Rj ψ, we find: Lp(x) =

0 XZ

± −∞

5

Rj P± ()ψ± (, x) d.

(6.25)

In the (extended) Fourier PS method, L is a self-adjoint operator if hLf, gi = hf, Lgi, with f, g ∈ C ∞ (−∞, ∞), and f, g → 0 as x → ±∞. Numerically, the spatial domain is truncated on domain [a, b], requiring f, g ∈ Cp∞ [a, b].

6.2. The extended Fourier PSTD method

59

Inserting the eigenfunctions and making use of = −kj2 c2j , we can calculate Lp(x) by:  R∞ k2 β1  1 1 1  P (k ) + P (k ) ejk1 x dk1 x ≤ 0 − P (k ) + 1 1  2π ρ1 α1 1 1 α2 2 1 −∞ Lp(x) = R∞ k22 0  β2 1 1  P2 (k1 ) + α2 P2 (k1 ) + α1 P1 (k1 ) ejk1 x dk1 x0 ≥ 0,  − 2π ρ2 −∞

with P1 (k1 ) =

R0

(6.26)

p(x)e−jk1 x dx, P2 (k1 ) =

R∞ 0

−∞

0

p(x0 cc12 )e−jk1 x dx0

and where x0 = x cc12 . For a

single homogeneous medium, these equations return to the Fourier transform pair of Eqs. (6.9) and (6.10). Numerical evaluation of Eqs. (C.29) is treated in Sec. 6.2.1 and is similar to the Fourier PS method. For discontinuous properties, the calculation of Lp(x) requires one additional forward and inverse FFT compared to the Fourier PS method. The extended Fourier PS has the same two points per wavelength and stability criteria as the Fourier PS method. Results of calculations with this method for the 1-D wave equation are shown in Paper IV.

6.2 The extended Fourier PSTD method 6.2.1 Computing the spatial derivatives of the linearized Euler equations To model sound propagation to closed urban courtyards in the time domain, the LEE of Eqs. (4.2) are solved by the extended Fourier PSTD method. As for the 1-D wave equation in Sec. 6.1.3, the orthogonal basis functions to expand the spatial dependent acoustic variables of the LEE are again found from an eigenvalue equation. We first consider the two-media problem of Fig. 6.2, and assume the media to be non-moving and piecewise homogeneous with c1 ≤ c2 . The non-moving linearized Euler equations for media with piecewise constant properties in a Cartesian coordinate system and matrix-vector notation can be written as:    

0 0 0

0 0 0

0 0 0

∂ ∂x ∂ ∂y ∂ ∂z

∂ ∂x

∂ ∂y

∂ ∂z

0





   q(x, t) =   

∂ −ρj ∂t 0 0 0

0 ∂ −ρj ∂t 0 0

0 0 ∂ −ρj ∂t 0

0 0 0 −1 ∂ ρj c2j ∂t



  q(x, t), (6.27) 

60


Figure 6.2: The 3-D problem of two semi-infinite fluid media, medium 1 and medium 2, with the interface at (0, y, z). Directions of eigenfunctions ψ− and ψ+ according to Eq. (6.30) are shown.

with q = [u, v, w, p]T a vector containing the acoustic variables, x = [x, y, z] the three-dimensional position vector and j denotes the medium. The orthogonal basis of eigenfunctions are found from the following eigenvalue problem:     |

0 0 0

0 0 0

0 0 0

∂ j ∂x

∂ j ∂y

∂ j ∂z

∂ j ∂x ∂ j ∂y ∂ j ∂z 0

{z





   ψ(, φ, θ, x) =    }

L

ρj 0 0 0

|

0 ρj 0 0

0 0 ρj 0

{z Rj

0 0 0 1 ρj c2j



  ψ(, φ, θ, x),  }

(6.28)

which can be written in a more compact form as: [L − Rj ] ψ± (, φ, θ, x) = 0,

(6.29)

with L a self-adjoint operator, Rj a matrix with medium-dependent constants ρj and cj . The eigenfunctions can be seen as a set of plane wave solutions with different angles of incidence, depicted in Fig. 6.2. For access to the use of FFTs, the eigenfunctions are written as a function of the wave number vector6 k1 = [kx1 , ky , kz ]. The orthogonal generalized eigenfunctions that are solutions to Eq. (6.29) are:

6

Note that kx2 =

s

k12

2 c1 c2

2 ,k − 1 + kx1 y2 = ky1 = ky and kz2 = kz1 = kz .


 

        ψ− (k1 , x) = N− (k1 )        

     

− k1kρx11 c1 α1 e−jk1 ·x − β1 e−jk1 ·x ky −jk1 ·x −jk1 ·x − k1 ρ1 c1 α1 e + β1 e − k1kρz1 c1 α1 e−jk1·x + β1 e−jk1 ·x 1 ·x α1 e−jk1 ·x + β1 e−jk − k2kρx22 c2 e−jk2 ·x    − k kρy c e−jk2 ·x  2 2 2    − kz e−jk2 ·x    k2 ρ2 c2 −jk2 ·x e





        ψ+ (k1 , x) = N+ (k1 )        

      

61

kx2 k2 ρ2 c2 ky k2 ρ2 c2 kz k2 ρ2 c2

    

kx1 ejk1 ·x k1 ρ1 c1 ky ejk1 ·x k1 ρ1 c1 kz ejk1 ·x k1 ρ1 c1 jk1 ·x

e

α2 ejk2 ·x − β2 ejk2 ·x α2 ejk2 ·x + β2 ejk2 ·x jk2 ·x jk2 ·x α2 e + β2 e

α2 ejk2 ·x + β2 ejk2 ·x

    x≤0   x ≥ 0,

     



x≤0 

    x ≥ 0,  

(6.30)

with k1 · x = kx1 x + ky y + kz z, k1 · x = −kx1 x + ky y + kz z, k1 = |k1 |, kx1 ∈ [0, ∞) and ky , kz ∈ (−∞, ∞). The normalization coefficients N± (k1 ) can be found by the orthogonality condition and the coefficients α1 , α2 , β1 , β2 are found from the physical interface conditions at x = 0 as follows: α1 = α2 =

ρ2 kx1 + ρ1 kx2 2 ρρ21 kkx1 x2 ρ1 kx2 + ρ2 kx1 2 ρρ12 kkx2 x1

1 1

,

β1 =

,

β2 =

ρ2 kx1 − ρ1 kx2 x1 2 ρρ12 kkx2 ρ1 kx2 − ρ2 kx1 x2 2 ρρ21 kkx1

1 1

,

(6.31)

.

We can now decompose the variable vector q(x, t), the solution of Eq. (6.27) at a certain time, onto the orthogonal eigenfunctions ψ± (k1 , x) by: Q± (k1 , t) =

Z∞ Z∞ Z∞

−∞ −∞ −∞

q(x, t)ψ± (k1 , x)Rj dx,

(6.32)

62


where q(x, t) =

∞ ∞ ∞ XZ Z Z ± −∞ −∞ 0

Q± (k1 , t)ψ± (k1 , x) dk1 .

(6.33)

The spatial derivative operator Lq(x, t) then follows from Eq. (6.29) as: Lq(x, t) =

∞ ∞ ∞ XZ Z Z ± −∞ −∞ 0

Rj Q± (k1 , t)ψ± (k1 , x) dk1.

(6.34)

However, for our geometry or interest, i.e. Fig. 4.1(c), the eigenfunctions ψ± (k1 , x) cannot be found in an exact analytical expression as in Eq. (6.30). Yet, for fluid media with equal sound speeds but different densities, physical reflection coefficients and the normalization coefficients become angular-independent and the spatial derivative operator L can be decomposed in uncoupled one-dimensional transforms; see also Paper VI. This means that the spatial derivatives of the LEE are calculated separately on a 1-D basis. This approach is followed here. Several spatial derivative cases are considered in Eqs. (6.35) to (6.39). These cases represent derivatives in the geometry of Fig. 4.1(c), where non-rigid boundaries are modelled by a fluid medium with a different density. The spatial derivatives are presented in x-direction for the pressure and the horizontal velocity component in domains [−x1 , x1 ] and [−x1 , 2x1 ]7 , and derivatives in the y- and z-direction can be found similarly. The domains are discretized in equidistantly spaced points, and the pressure and horizontal velocity are staggered in space by ∆x . According to the LEE of Eqs. (6.27), the spatial derivatives are evaluated 2 at the staggered positions. In Eqs. (6.35) to (6.39), the variables Ri,j and Ti,j are the physical reflection and transmission coefficients from medium i to j. We further have used m∆x+ to denote m(∆x + ∆x ), Fx1 and Fx−1 are the forward and inverse Fourier transform 2 over and to the x-variable and sgn(x) is the signum function. For a rigid medium 1 as of Eq. (6.37), we only need to solve for medium 2. Some directions in Fig. 4.1(c) cross three media, e.g. the horizontal direction in the courtyard. An eigenfunction expansion for the three-media case has been developed similarly to the two-media case. The eigenfunctions are built up by a summation of terms, analogous to multiple reflections, and are truncated at some point. The eigenfunctions can be found in the appendix of Paper VI. 7

The derivative of the other acoustic velocity components v and w are corresponding to Eq. (6.27) only taken in the y- and z-directions respectively.


∂p ∂x l∆x+

p1 ∂u ∂x l∆x u1

∂p ∂x l∆x+

p1 p2

∂u ∂x l∆x

u1

u2

∂p ∂x l∆x+


63

∆x = Fx−1 jkx e−jkx 2 Fx1 [p1 ] −l1 ≤ l ≤ l1 − 1, = p(m∆x) −l1 ≤ m ≤ l1 − 1,

(6.35)

p1 −jkx ∆x 1 2 F = jkx e x p2 p(m∆x) = R1,1 p(−m∆x) + T2,1 p(m∆x) R2,2 p(−m∆x) + T1,2 p(m∆x) = p(m∆x)

(6.36)

= Fx−1 jkx ejkx = u(m∆x+ )

∆x 2

Fx1 [u1 ]

−l1 ≤ l ≤ l1 − 1, −l1 ≤ m ≤ l1 − 1.

Fx−1

jkx ∆x 2

u1 u2

= Fx−1 jkx e Fx1 u(m∆x+ ) = + + −R1,1 u(−m∆x+ ) + T1,2 u(m∆x+ ) −R2,2 u(−m∆x ) + T2,1 u(m∆x ) = u(m∆x+ )

−l1 ≤ l ≤ −1 0 ≤ l ≤ l1 − 1, −l1 ≤ m ≤ −1 0 ≤ m ≤ l1 − 1, −l1 ≤ m ≤ −1 0 ≤ m ≤ l1 − 1, −l1 ≤ l ≤ −1 0 ≤ l ≤ l1 − 1, −l1 ≤ m ≤ −1 0 ≤ m ≤ l1 − 1, −l1 ≤ m ≤ −1 0 ≤ m ≤ l1 − 1.

∆x = Fx−1 jkx e−jkx 2 Fx1 [p2 ] 0 ≤ l ≤ l1 − 1, = p(|m∆x|) −l1 ≤ m ≤ l1 − 1, ∆x

= Fx−1 jkx ejkx 2 Fx1 [u2 ] = sgn(m)u(|m∆x+ |)

0 ≤ l ≤ l1 , −l1 ≤ m ≤ l1 − 1.

(6.37)

64


∂p

∂x l∆x+

p3

∂x l∆x

=

u3

∂p ∂x l∆x+


 p1 Fx1  p2  p3

Fx−1



jkx ejkx ∆x 2



 u1 Fx1  u2  u3

u(m∆x+ ) + +  −R1,1 u(−m∆x +) + T1,2 u(m∆x +)  −R2,2 u(−m∆x ) + T2,1 u(m∆x ) = u(m∆x+ )  + + −R2,2 u(2x1 − m∆x+ ) + T2,3 u(m∆x+ ) −R3,3 u(2x1 − m∆x ) + T3,2 u(m∆x ) = u(m∆x+ )

u1 = u2

∆x Fx−1 jkx e−jkx 2



p(m∆x)  R1,1 p(−m∆x) + T2,1 p(m∆x)  R2,2 p(−m∆x) + T1,2 p(m∆x) p(m∆x) =  R2,2 p(2x1 − m∆x) + T3,2 p(m∆x) R3,3 p(2x1 − m∆x) + T2,3 p(m∆x) = p(m∆x)

p1 = p2

∂u

=



Fx−1

−jkx ∆x 2

= jkx e = p(|m∆x|) Fx−1

jkx ∆x 2

Fx1

Fx1 +

= jkx e = sgn(m)u(|m∆x |)

[p2 ]

[u2 ]

−l1 ≤ l ≤ −1 0 ≤ l ≤ l1 − 1 l1 ≤ l ≤ 2l1 − 1, −l1 ≤ m ≤ −1 0 ≤ m ≤ l1 − 1, −l1 ≤ m ≤ −1 0 ≤ m ≤ l1 − 1 l1 ≤ m ≤ 2l1 − 1, 0 ≤ m ≤ l1 − 1 l1 ≤ m ≤ 2l1 − 1,

(6.38)

−l1 ≤ l ≤ −1 0 ≤ l ≤ l1 − 1 l1 ≤ l ≤ 2l1 − 1, −l1 ≤ m ≤ −1 0 ≤ m ≤ l1 − 1, −l1 ≤ m ≤ −1 0 ≤ m ≤ l1 − 1 l1 ≤ m ≤ 2l1 − 1, 0 ≤ m ≤ l1 − 1 l1 ≤ m ≤ 2l1 − 1.

0 ≤ l ≤ l1 − 1, −l1 ≤ m ≤ l1 − 1,

0 ≤ l ≤ l1 , −l1 ≤ m ≤ l1 − 1.

(6.39)


65

For media 1 and 3 being infinitely rigid, i.e. ρ1 = ρ3 = ∞, we only solve for medium 2. Periodicity leads to the same equations as for the two fluid case with a rigid medium 1 as of Eq. (6.37). The Eqs. (6.35) to (6.39) constitute all types of spatial derivative operators used in the extended Fourier PSTD method to solve LEE for the courtyard problem. FFTs are used to evaluate the above spatial derivative operators numerically. We illustrate this for the 1-D two-fluid problem of Eqs. (6.36), and compute the spatial derivative of p(x) in the media. The domain is discretized with N = 2l1 equidistantly spaced points, ranging from −x1 to x1 − ∆x and with p(−x1 ) = p(x1 ). The wave number domain is also discretized by N discrete points. The spatial discretization leads to a periodicity in the wave number 2π domain with period 2kx,max = ∆x and the wave number domain discretization 2π . The former leads to a periodicity in the spatial domain with period 2x1 = ∆k 8 leads to the requirement of two points per wavelength , and the latter requires the acoustic variable to be periodic on the domain. We therefore assume that p(x) has compact support on the domain, which can be enforced by applying a numerical absorption layer at the domain ends (such as the PML used here; see Sec. 6.2.4). The forward Fourier transform from Eqs. (6.36) is calculated as: " # " # p (l∆x) P (n∆k) 1 1 Fx1 = (6.40) p2 (l∆x) P2 (n∆k) " # N/2−1 X p1 (l∆x) ln ≈ ∆x e−j2π N . p2 (l∆x) l=−N/2 A problem that arises when calculating the spatial derivative is the derivative at the highest wave number N/2, which returns a purely imaginary derivative component; see also [186]. However, by staggering the velocity and pressure in space as has been done here, the derivative component for staggered positions becomes purely real for the highest wave number component. The spatial derivative of the pressure of Eqs. (6.36) is calculated as: " # N/2−1 X −l1 ≤ l ≤ −1 ln P1 (n∆k) ∆x 1 ∂p ≈ j(n∆k) e−j(n∆k) 2 ej2π N ∂x l∆x+ N∆x 0 ≤ l ≤ l1 − 1. P2 (n∆k) n=−N/2

(6.41)

The transforms for the horizontal velocity derivative are found similarly. 8

This follows from the maximum resolved wave number kx,max = resolved wave number.

2π λmin ,

with λmin the smallest

66


6.2.2 Moving inhomogeneous medium

The expressions to compute spatial derivatives in Sec. 6.2.1 are derived for a homogeneous non-moving medium. Assuming that the acoustic variables and their spatial derivatives only have wave number components below kmax = π/∆x9 , the extended Fourier PS method of Sec. 6.2.1 can be applied to the spatial derivatives in the case of moving inhomogeneous media as well. High values or large gradients of atmospheric wind components that lead to spectral components larger than kmax will cause the solution to suffer from aliasing and truncation errors which can be remedied by using more points per wavelength. The sound speed c may vary spatially in the atmosphere, due for example to temperature differences. Again, if this leads to spectral components of the acoustic variables and their first derivatives above kmax , aliasing and truncation-related errors are introduced. For a moving medium, the linearized Euler equations can be written from Eq. (4.2) as: ∂ 1 ∂p ∂ ∂ ∂ ∂ ∂ ∂u u0 + u0 u=− + u +v +w + v0 + w0 , ∂t ∂x ∂y ∂z ∂x ∂y ∂z ρj ∂x ∂v ∂ ∂ ∂ ∂ ∂ ∂ 1 ∂p + u +v +w + v0 + w0 , v0 + u0 v=− ∂t ∂x ∂y ∂z ∂x ∂y ∂z ρj ∂y ∂w ∂ ∂ ∂ ∂ ∂ ∂ 1 ∂p + u +v +w + v0 + w0 , w0 + u 0 w=− ∂t ∂x ∂y ∂z ∂x ∂y ∂z ρj ∂z ∂p ∂u ∂v ∂w ∂ ∂ ∂ 2 p = −c ρj − sδ(x|xs), + u0 + v0 + w0 + + ∂t ∂x ∂y ∂z ∂x ∂y ∂z (6.42) with u, v and w the acoustic velocity components, u0 , v0 and w0 the atmospheric velocity components, p the acoustic pressure, ρj the medium dependent density and c the adiabatic speed of sound. Compared to the non-moving medium case, we have additional spatial derivative terms. As we assume a steady flow with respect to the acoustic time scales, the derivatives acting on the atmospheric velocity components only have to be calculated once and will not increase the computation time substantially. We also find terms with cross-derivatives as ∂u ∂z that were not encountered before. However, the extended Fourier PS method as developed in Sec. 6.2.1 is based on the derivative in the velocity variable

9

for an equidistant grid with ∆x = ∆y = ∆z.


67

direction. We therefore rewrite Eq. (6.42) as: ∂u ∂(u0 u) ∂(v0 u) ∂(w0 u) ∂ ∂ ∂ 1 ∂p + u +v +w + + , u0 + =− ∂t ∂x ∂y ∂z ∂x ∂y ∂z ρj ∂x ∂ ∂ ∂ ∂v 1 ∂p ∂(u0 v) ∂(v0 v) ∂(w0 v) + u +v +w + + , v0 + =− ∂t ∂x ∂y ∂z ∂x ∂y ∂z ρj ∂y ∂(u0 w) ∂(v0 w) ∂(w0 w) 1 ∂p ∂ ∂ ∂ ∂w w0 + =− + u +v +w + + , ∂t ∂x ∂y ∂z ∂x ∂y ∂z ρj ∂z ∂p ∂u ∂v ∂w ∂(u0 p) ∂(v0 p) ∂(w0 p) 2 = −c ρj − sδ(x|xs ), + + + + + ∂t ∂x ∂y ∂z ∂x ∂y ∂z (6.43) where we have made use of incompressibility of the medium10 . The Eqs. (6.43) now contain derivatives where products of atmospheric and acoustic velocity components arise, such as ∂(w∂z0 u) . We make use of the no-slip boundary condition for the atmospheric velocities, i.e. they have zero values at the boundaries. Spatial derivatives of these variables within the atmospheric domain can consequently be evaluated using the Fourier PS method. The Fourier PS method has successfully been applied to calculate spatial derivatives for weakly inhomogeneous media; e.g. [110]. In Papers IV, V and VI, it is shown that the extended Fourier PS also returns accurate results for a weakly inhomogeneous sound speed and a moving medium with typical values for the lower part of the atmosphere.

6.2.3 Time iteration Runge-Kutta scheme To solve a time-domain problem as the LEE, a PS method need to be accompanied by a scheme for time iteration, which together is generally called a PSTD method. Several different time-marching schemes have been used in combination with a PS method, such as the acoustic wave propagator [109], the AdamsBashforth method [56], the Adams-Moulton method [172], the k-space method [110], the symplectic operator [140], the leap-frog method [42], the alternating direction implicit (ADI) method and the split-step (SS) scheme [107], and the Runge-Kutta scheme [177]11 . The k-space method has been shown to be attractive in combination with the Fourier PS method [110]. This method has successfully been applied in Paper IV to the 1-D two-fluid problem and in Paper V to the 2-D two-fluid problem, and is described in Appendix C.3. The k-space 10

∂v0 ∂w0 0 In particular, the relation ∂u ∂x + ∂y + ∂z = 0 has been utilized. 11 The number of references per time-iteration scheme is here limited to one.

68


method requires access to the full wave number-time (k-time) domain. For a 3-D courtyard geometry, spatial derivatives are calculated by using 1-D transforms with evaluation of spatial derivatives in the kx -time, ky -time and kz -time domains. As there is no access to the k-time domain, the k-space method is not applicable and a Runge-Kutta (RK) scheme for the time marching is used instead. RK schemes are single time-step schemes (only information from the former time step is needed), similar to a high-order Taylor series approach. Higherorder derivatives are however not involved in the RK schemes. Instead, several intermediate stages within one time step are taken. High-order RK schemes are more stable than low-order schemes but require more calculation operations and memory storage [74]. High-order yet low-storage RK schemes have also been developed. Such a recently developed low-storage optimized 6-stage Runge-Kutta method by Bogey and Bailly, RKo6s, where only storage of the variables at two time stages is needed, has here been used [27]. This algorithm has optimized the dispersion and dissipation error for frequencies corresponding to 4 points per wavelength. To apply the RKo6s scheme, we first write Eqs. (6.43) in compact notation: ∂q(x, t) = −Lq(x, t) − s(t)δ(x|xs), ∂t

(6.44)

with matrix L containing all spatial derivative terms of Eqs. (6.43). The RKo6s solution of this expression is written as: (6.45)

q(x, t0) = q(x, t), q(x, ti) ≈ q(x, t0 ) − γi ∆t(Lq(x, ti−1 ) + s(ti−1 )δ(x|xs )) for

i = 1...6,

q(x, t + ∆t) ≈ q(x, t6 ),

where the spatial derivative operator L and source function s are evaluated at time stage i − 1 and with coefficients γi real numbers between 0 and 1 as can be found in [27]. The time step ∆t is chosen to be ∆t = ∆x , since the highest 2c resolved spatial wave number then corresponds to 4 points per wavelength of the temporal resolved frequency, for which the RKo6s method was optimized. This time step has been found to give stable results for the calculated cases in this thesis and Paper VI. Source function To initiate a calculation, zero acoustic variable values and the following waveletshaped source function have been used: 2

s(t) = A sin(2πfc t)e−a(t−tc ) ,

(6.46)


69

with amplitude A, fc and tc the central frequency and time of the wavelet, and a a constant determining the bandwidth. In the calculations, the values A = 1 c0 c0 2 −2 3 Hz, a = 16 Nm/s, fc = 8∆x s and tc = 0.01 s have been used. The draw∆x back of this source function is that it changes the pressure value after every time stage at a single point in space. This leads to a spatial discontinuity, limiting the maximum discrete time step using a PS method. It is shown in Paper VI that for the chosen ∆t in the extended Fourier PSTD method with the RKo6s method, this error can be neglected.

6.2.4 Perfectly matched layer (PML) The computational domain in the extended Fourier PSTD method needs to be truncated. For open boundaries (in the extended Fourier PSTD method, these are all boundaries apart from acoustically rigid boundaries), a reflection-free termination and avoidance of the wrap-around effect inherent to using discrete Fourier transforms are desired. Since reflection-free terminations are a recurrent problem in every domain decomposition method for exterior acoustics, many approaches have been proposed; see e.g. [111] for an overview. A widely adopted method to model such a boundary is the perfectly matched layer (PML), as first presented by Berenger [19] for electromagnetic problems. The idea behind the PML is to include an absorption layer at the edge of the computational domain acting on the normal components of the acoustic velocity and pressure, which will create a layer without reflections for all incoming propagation angles. Several types of PMLs have been presented where both the pressure and velocities are split into components [81], only the pressure is split [19] or none of the acoustic variables is split [53]. We have here followed the approach of Berenger and split the pressure into components parallel and perpendicular to the PML. The linearized Euler equations including PML then read:

70


∂eσx t u 1 ∂p = −u · ∇u0 − ∇ · (u0 u) − , ∂t ρj ∂x ∂eσy t v 1 ∂p e−σy t = −u · ∇v0 − ∇ · (u0 v) − , ∂t ρj ∂y 1 ∂p ∂eσz t w = −u · ∇w0 − ∇ · (u0 w) − , e−σz t ∂t ρj ∂z σx t ∂u sδ(x|xs ) px −σx t ∂e e = −ρ0 c2 − ∇ · (u0 p) − , ∂t ∂x D ∂eσy t py ∂v sδ(x|xs) e−σy t = −ρ0 c2 − ∇ · (u0 p) − , ∂t ∂y D ∂eσz t pz ∂w sδ(x|xs) e−σz t = −ρ0 c2 − ∇ · (u0 p) − , ∂t ∂z D e−σx t

(6.47)

with p = px + py + pz and D the dimensionality of the problem. The PML coefficients σ = [σx , σy , σz ]T are zero for internal points and zero or positive in the PML and are written following Berenger as for σx : m x − xPML , (6.48) σx = β D with x − xPML the position of the grid point in the PML layer, D the thickness of the layer, xPML ≤ x ≤ xPML + D, β the maximum amplitude of σx and with m an exponent. The values m = 4, β = 1e4 s−1 and D = 20∆x m have been found to return a reflection coefficient of R = 0.003 (corresponding to an attenuation of -50 dB) for most frequencies and propagation angles. Paper VI gives some more details on the implementation of the PML and its performance.

6.2.5 Implementation demonstration of a 2-D canyon As for the ESM, we illustrate the use of the extended Fourier PSTD to the 2-D canyon situation of Fig. 5.1(a). In the example here, the medium is considered to be moving. Equations (6.47) with v = v0 = 0 are the governing equations to be solved by the extended Fourier PSTD method for this problem. Figure 5.1(c) shows a representation of Fig. 5.1(a), discretized for being solved by the extended Fourier PSTD method. Pressures and velocity components are staggered in space. Boundaries that form a horizontal interface coincide with grid points of pressure and vertical velocity components and boundaries that form a vertical interface coincide with grid points of the horizontal velocity component. For non-rigid terminations, a PML is included. The solution is now calculated as follows:


71

1. All acoustic variables have zero initial values. The atmospheric velocities and their spatial derivatives are known.

2. The RKo6s time-iteration is taken from the initial value at t0 to t1 = t0 + γ6 ∆t by six times iterating the following equations: t ∂u0 ∂u0 ∂(u0 u) ∂(w0 u) i−1 1 ∂p ti t0 +u +w + + ux+ ,z = ux+ ,z − γi ∆t + ρj ∂x ∂x ∂z ∂x ∂z x ,z ti−1 ∂w0 ∂w0 ∂(u0 w) ∂(w0 w) 1 ∂p ti t0 +u +w + + = wx,z wx,z + + − γi ∆t + ρj ∂z ∂x ∂z ∂x ∂z x,z ti−1 ∂u ∂(u0 px ) ∂(w0 px ) s(ti−1 )δ(x|xs ) 0 i px |tx,z = px |tx,z − γi ∆t c2 ρj + + + ∂x ∂x ∂z 2 x,z ti−1 ∂w ∂(u0 pz ) ∂(w0 pz ) s(ti−1 )δ(x|xs) 0 i + + + pz |tx,z = pz |tx,z − γi ∆t c2 ρj ∂z ∂x ∂z 2 x,z i i ptx,z = (px + pz )|tx,z .

(6.49)

where x+ = x + ∆x , z + = z + ∆z , ti = t0 + γi ∆t and the variables are written 2 2 i as the numerical approximate values; e.g. ptx,z ≈ p(x, z, ti ). 3. The spatial derivatives on the right side are calculated by the extended Fourier PS method of Sec. 6.2.1. The derivatives of products of atmospheric and acoustic variables are only non-zero in the atmosphere and are calculated using the Fourier PS method (Eqs. (6.35)). The other spatial derivatives are calculated using the Eqs. (6.36) to (6.39) depending on whether they cross two or three media. Pressures and velocity components are staggered in space which should be taken into account when computing some of the spatial derivatives (see Sec. 6.2.1). 4. After the 6th stage, i.e. one time step, the variables are updated by the PML coefficients: utx+ ,z = t wx,z = +

px |tx,z = pz |tx,z =

t6 e−∆tσx u x+ ,z t6 e−∆tσz w x,z + t6 e−∆tσx px x,z t6 e−∆tσz pz x,z

6 ptx,z = (px + pz )|tx,z .

(6.50)

72


5. The acoustic variables are updated as: utx0+ ,z = utx+ ,z

(6.51)

t0 t wx,z = wx,z + + 0 px |tx,z = px |tx,z

0 pz |tx,z = pz |tx,z .

and are again iterated six times again according to step 2.

6.2.6 Numerical efficiency The numerical costs of the extended Fourier PSTD method are compared to the current state-of-the-art time-domain method for sound propagation to courtyards, the second-order-accurate FDTD method. The numerical costs can be divided in storage capacity and calculation time. We consider the case of a single 20 m x 20 m x 20 m courtyard situation and a frequency range up to 1000 Hz. Assuming that the boundaries are explicitly modelled in both methods so that the same computational domain is modelled, we can compute a storage capaNFDTD city factor of MD spat , with Mspat = NPS the ratio of the number of discrete points required for similar accuracy in both methods in one spatial dimension, which can be estimated to range from 4 to 8, and D the dimensionality of the problem. The storage capacity factor for D = 3 then ranges from 64 to 512 in favour of the extended Fourier PSTD method. The necessary computation time is related to the number of computational operations. To estimate the ratio of the calculation time between FDTD and the extended Fourier PSTD method for the 3-D courtyard problem, we make use of previous estimates for the Fourier PSTD method from Liu [106]. For a single time step in the Fourier PSTD method, the number of operations amounts to D log2 (NPS ) [106], with KPS a constant. The number of operations in the KPS NPS FDTD method amounts to KFDTD (2NFDTD )D . Using Mspat = 4 − 8, Liu estimated the Fourier PS method to be 4D − 8D times as efficient as the FDTD method for NPS ≤ 64 [106]. In the extended Fourier PSTD method, derivatives for a single medium amount to the same computational costs as in the Fourier PSTD method, and take approximately 2 and 3 times as many operations for a derivative across 2 and 3 media respectively. For a derivative between two rigid terminations finally, this factor is 2 (see Sec. 6.2). For the single-courtyard situation, approximately twice as many calculations are therefore needed with the

6.3. Validation

73

extended Fourier PSTD method compared to having (erroneously) modelled the problem by the Fourier PSTD method. Furthermore, as the operations for a single FFT scale with NPS log2 (NPS ), the courtyard problem with NPS = 256 requires more computational effort per grid point than the reference case of Liu with NPS = 64. Taking these two factors into account, a computational efficiency factor of 24−192 of the extended Fourier PSTD in favour over the FDTD method is then found. Since the time steps in FDTD calculations are often smaller than in the extended Fourier PSTD method, an even larger factor will be obtained.

6.3 Validation The extended Fourier PSTD method has subsequently been implemented and validated in 1-D, 2-D and 3-D. These implementations are presented in Papers IV, V and VI. For most cases, highly accurate results were found with a requirement of two points per wavelength, which needed to be slightly increased for the inhomogeneous and moving media cases. The error made in the extended Fourier PSTD method can be attributed to the following aspects: ◦ Extended Fourier PS method. The error introduced by the extended Fourier PS method to compute the spatial derivatives of an acoustic variable is related to the truncation and aliasing error. The method has spectral accuracy, i.e. is exact when the spectral content of the acoustic variable is limited up to half the spatial sampling frequency. The numerical error is related to the spectral content at half the spatial sampling frequency and bounded by the computer round-off error. As the extended Fourier PS method computes 1-D derivatives separately, 2-D and 3-D effects geometrical effects as edges and corners are only resolved approximately. The introduced error varies among the various cases, see Paper VI. Boundaries are treated as being either acoustically rigid or are modelled by a fluid with a different density. The applicability of these treatments should be evaluated for the modelled situation of interest. ◦ PML. The implemented layer to obtain reflection free boundaries, the PML, showed to give a higher reflection coefficient for higher frequencies, see Paper VI. Although the PML reflection coefficient is low even for the highest resolved frequency, these reflections could influence cases with highly shielded receiver positions. ◦ RKo6s method. The method implemented for time iteration, the optimized low-storage six-stage explicit Runge-Kutta method (RKo6s), was optimized

74


by Bogey and Bailly for amplitude and phase errors for a frequency corresponding to 1/4 of the sampling frequency [27]. The selected discrete time step in the calculations has been chosen such that 1/4 of the temporal sampling frequency corresponds to the maximum resolved frequency from the spatial derivative operator. The error introduced by the RKo6s method is therefore low, and can be found in [27]. In addition to the cases studied in the appended Papers, the extended Fourier PSTD method has here been validated for a shielded canyon where both absorptive and diffusive façade reflections occur; see Fig. 6.3. The 2-D ESM is used for comparison. The agreement between the methods is very good. Since boundary materials are modelled as locally reacting in the ESM in contrast to being modelled by a second fluid with a different density in the Fourier PSTD method, the calculations show that the latter implementation is useful for the courtyard situation. Also, 3-D calculations in the canyons’ and canyon-courtyard geometry of the scale model study have been performed. The canyon results are shown in Figs. 5.3(a) and (b), along with the 2.5-D ESM and scale model results. Details of the geometry and used façade absorption can be found in Appendix A of Paper III12. The results are calculated with a spatial sampling of ∆x = 1/3 m, corresponding to a maximum resolved frequency of 510 Hz. This upper frequency is bounded by the computational resources. Calculations have been done using a Matlab implementation on a computer with 8 GB memory and a 2.13 GHz processor, without having attempted to parallelize the code or having made use of a more efficient programming language. For the courtyard geometry with horizontally oriented diffusion patches (where all façades are modelled by a boundary medium to include some façade absorption), the implemented method required 82 s per time step. The extended Fourier PSTD canyon results show good agreement with the scale model study results and a very good agreement with the ESM results. For the courtyard cases with hard boundaries and diffusion patches at the façades, results are shown in Figs. 5.3(c) along with the scale model results and display good agreement.

12

The façades used in the scale model study are modelled to have a finite impedance.

6.3. Validation

75

10 Extended Fourier PSTD 2−D ESM

−10

L

re free

(dB)

0

−20

(a) −30

63

125

250

500 1000 2000

Frequency (Hz) 10

−10

L

re free

(dB)

0

−20

(b) −30

63

125

250

500 1000 2000

Frequency (Hz)

Figure 6.3: One-third octave band sound levels relative to free field calculated with the extended Fourier PSTD method and the ESM. Spatial sampling frequency in the extended Fourier PSTD method equal to 4000 Hz. Two-dimesional geometry of a shielded canyon. (a) Smooth façades with rigid and finite impedance (Zn = 9) parts; (b) Façades with depressed rigid and finite impedance (Zn = 9) parts.

76


Part III Abatement of road traffic noise in courtyards

77

Chapter 7

The acoustic soundscape in closed courtyards

The features of the acoustic soundscape in urban courtyards are briefly touched upon in Sec. 2.1. The acoustic soundscape will here further be characterized in Sec. 7.1, as described by its equivalent sound level, time variation, spatial dependence and spectral contents. This description is based on results from real-life measurements. As the acoustic soundscape is a composition of many road traffic source contributions, its features can be understood from the physical aspects of point-to-point transfer paths, i.e. a noise source position outside the courtyard and a receiver position inside the courtyard. Such transfer paths have been measured in the scale model study and calculated using the 2.5-D ESM and 3-D extended PSTD method. The transfer paths are considered in the time and frequency domain in Sec. 7.2, and dependence of the source position on these transfer paths is shown. Also, the influence of façade surface properties such as absorption and irregularities is discussed. Transfer paths to receiver positions in the directly exposed street canyon are used for comparison.

79

80

7. The acoustic soundscape in closed courtyards

7.1 Signatures of the acoustic soundscape Equivalent sound levels For equivalent sound levels in courtyards, we consider the histogram of Fig. 2.1. The figure is based on 687 indicative equivalent sound level measurements at courtyards in the city of Gothenburg with a typical measurement time of 15 minutes [7]. As already mentioned in Sec. 2.1, the equivalent noise level criterion for quiet sides of 45 dB(A) is exceeded in 60 % of the measured situations. The results show further that more than 75 % of the courtyards have equivalent sound levels between 40 and 50 dB(A) with a mean of 47.2 dB(A) and a standard deviation in LAeq of 5.7. This rather small spread of the equivalent noise levels for the different courtyards is caused by many road traffic noise source contributions for every single courtyard, as already found by Ögren and Kropp [183]. In the soundscape support to health (SSH) project, many LAeq measurements in courtyards of several Swedish cities were executed as well. These results also showed LAeq > 45 dB(A) for the major part of the measurements, with an even lower spread than from Fig. 2.1 [181]. Time variations The transient variation of the acoustic soundscape can be studied in several ways. One way is to construct histograms from short-time equivalent sound levels. Figure 7.2(a) shows a typical histogram from a 20-minute daytime LAeq measurement at a courtyard in Gothenburg, with a time averaging of 0.19 s per sample [171]. The figure also shows the distribution as measured at the directly exposed side of the same building. The distribution of the sound levels can be expressed by a standard deviation which is lower for the courtyard than for the directly exposed receiver position, a result which has been found repeatedly when performing such measurements in courtyard situations [40, 59, 102]. It means that the transient variation of the sound level is smaller in the yard than at the directly exposed positions. On an octave band basis, it was found that the standard deviation of the transient sound level in the courtyard is higher for the lower frequencies [60, 40] and outliers of high sound levels can be attributed to local traffic occurrences [61]. Spatial variations Based on measurements from the SSH programme, it was concluded by Ögren that the sound field in courtyards can be considered as diffuse, i.e. sound levels are rather even throughout the courtyard [181]. Several measurement campaigns to investigate the difference between the acoustic soundscape at the directly exposed side and at the courtyards of apartment buildings in

7.1. Signatures of the acoustic soundscape

81

Figure 7.1: Pictures from a measured situation in Gothenburg. Left: View in the street canyon. Right: View in the courtyard adjacent to the street canyon. Pictures taken from the same apartment [171].

0.5

80

Shielded side Exposed side

70

0.4

eq

L (dB)

PDF (−)

60 0.3 0.2

50 40

0.1

30

(a) 0 40

(b) 50

60 L

Aeq

70 (dB(A))

80

90

20

63 125 250 500 1000 2000 Frequency (Hz)

Figure 7.2: Results from a 20-minute measurement in the courtyard and adjacent street canyon of Figs. 7.1 [171]. (a) Histogram; (b) Spectrum. Gothenburg consisted of measurement points located at ground level and outside the uppermost apartments [171, 40]. The results indicated higher equivalent sound levels at the highest apartments compared to the levels at the ground floor.

82

7. The acoustic soundscape in closed courtyards t t 0

Shielded canyon Exposed canyon

1

Lre max (dB)

0 −20 −40 −60

1

2 Time (s)

3

4

Figure 7.3: Energy-time curves for receiver positions in the directly exposed and shielded canyon measured in the scale model study. 250 Hz 1/3-octave band. To construct the energytime curves from the impulse responses, an integration time of 0.04 s has been used. Thick: Receiver position (60,0,5); Thin: Receiver position (0,0,5). −8

Amplitude (Pa)

4 2 0 −2 −4

Amplitude (Pa)

1

0 −9 x 10

1

2 Time (s)

3

4

0 −9 x 10

1

2 Time (s)

3

4

0

1

2 Time (s)

3

4

0.5 0 −0.5 −1 1

Amplitude (Pa)

x 10

0.5 0 −0.5 −1

Figure 7.4: Calculated 250 Hz octave band impulse responses with the extended Fourier PSTD method with the source at (9,40,0) and smooth façades with small absorption (α = 5 %). (a) Canyons’ situation, receiver position (0,0,5); (b) Canyons’ situation, receiver position (49,0,0); (c) Courtyard situation, receiver position (49,0,0).

7.2. Fundamentals of the acoustic soundscape

83

Spectrum Figure 7.2(b) shows unweighted spectra from a 20-minute measurement. The measured situation is the same as for the shown histograms in Fig. 7.2(a), and spectra are typical for directly exposed and shielded urban areas [171]. The spectrum at the directly exposed side displays the well-understood signature of the road traffic noise spectrum, with a large low-frequency content from the power train noise and the large contribution around 1 kHz due to tyre-road noise. The spectrum at the courtyard mildly resembles this signature, yet levels are clearly reduced more for the higher frequencies.

7.2 Fundamentals of the acoustic soundscape Impulse responses Impulse responses for a point source in one canyon and a receiver position in the parallel shielded canyon or adjacent courtyard have been measured in the scale model study and calculated by the extended Fourier PSTD method. An important concept for the shielded receiver positions that can be derived from the impulse responses is that of the rise time. Figure 7.3 shows measured energy-time curves for a receiver position in the directly exposed canyon and shielded canyon. Whereas the sound level drops after the direct contribution at the directly exposed side, the sound level at the shielded side first increases after the first contribution t0 until a time t1 , when the sound level starts to drop. The rise time ts is equal to t1 − t0 and indicates the importance of façade reflections at the shielded side: higher-order reflections affect the sound level to a larger extent than the first contributions. The rise time can be understood from a decreasing diffraction angle for an increasing order of reflection (see also Paper I). From the scale model results, the rise time was found to increase with frequency and has been shown to be proportionally to the steady-state sound level. It can further be noticed that ts is rather constant for increased transverse source-to-receiver distances (the y-distance). To quantify the reverberation effect, the decay time T10 has been calculated from the scale model measurements using the impulse response1 . The measured T10 results in the shielded canyon are calculated starting from t1 and exceed the values of the directly exposed street canyon, meaning that the sound energy dies out slower at the shielded side. This can be explained by the larger num1

The decay time T10 uses the part of the energy-time curve, which is obtained after turning off a sound source that has generated a steady state sound field, where the energy drops from -5 to -15 dB relative to the maximum value. In Paper I, the calculation of T10 is discussed.


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ber of reflection contributions and smaller spherical divergence difference for subsequent waves at the shielded side compared to the directly exposed side. Also, there is a decreased geometrical shielding for higher-order reflection contributions at the shielded side. As the ts , the T10 is rather constant over the transverse source-to-receiver distance. In the shielded closed courtyard situation, the decay times T10 and rise times ts are even larger than in the shielded canyon due to the contribution of reflections in two horizontal dimensions (see Paper I). This is further illustrated by impulse response calculations from the 3-D extended Fourier PSTD method in Figs. 7.4(a-c). The calculated impulse responses are for the 250 Hz octave band and the façades are rigid. Figure 7.4(b) and (c) have the same source and receiver position, yet for the canyons’ and canyon-courtyard geometry respectively. The impulse response is more dense if Fig. 7.4(c) due to the higher number of reflections in the courtyard. The


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larger importance of higher-order reflections at the shielded side compared to the exposed side means a smaller dependence on the source-to-receiver distance, explaining that distant sources become important. It further explains the narrow spread of sound levels measured in individual courtyards and among courtyards, as mentioned in Sec. 7.1. The low standard deviation of the time variations can be understood from this similarly. Frequency responses The frequency-dependent transfer functions are obtained by Fourier-transforming the impulse responses from the scale model measurements and extended PSTD method calculations. Figure 7.5 shows the sound level relative to the free field level at 1 m from the source, Lre free,1m over the frequency and transverse source-to-receiver distance. The level is shown in (a) for source and receiver in the same canyon, in (b) for the receiver in the shielded canyon and in (c) for the receiver in the shielded courtyard. For all cases, rigid façades are considered. In the street canyon case (a), the level is rather constant over frequency and decreases with the transverse source-to-receiver distance. In contrast to the street canyon results of (a), the levels are rather constant over the transverse source-toshielded-receiver distance for both canyon and courtyard, with higher levels in the courtyard case. For both shielded cases, the level decreases with frequency due to the decreasing edge diffraction coefficient with frequency. The increased shielding with frequency corresponds with the results from real-life measurements; see Fig. 7.2(b). The higher sound levels at higher courtyard positions from the signatures of Sec. 7.1 has also been found in the scale model measurements, where results have shown that the sound levels for lower receiver positions in the shielded canyon are lower than for higher positions in the canyon (see Fig. 8 of Paper I). It can also be found in Paper III as a result from a finite incoherent line source. The level difference can be attributed to the larger geometrical shielding for lower positions. It is opposite to the directly exposed street canyon levels, where higher receiver positions have lower sound levels. Façade absorption and diffusion Absorption and diffusion are two fundamental types of façade surface properties affecting the sound field. Where absorption merely damps (and disperses) the reflected sound waves, applied diffusion patches change the type of reflection from specular to partly diffuse. In Figs. 7.6 and 7.7, snapshots of a calculation with smooth façades and with irregularly shaped façades are shown. The results clearly illustrate the influence of irregularly shaped façades.

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Figure 7.6: Snapshots of the sound field in a 2-D canyon calculated with the extended Fourier PSTD method for smooth rigid façades. A sound source has generated a short wavelet pulse at time t = 0.01 s and is located at (-6 m, 18 m), spatial sampling corresponding to fs = 4000 Hz. Time instances are subsequently 0.05 s, 0.10 s, 0.15 s, 0.20 s, 0.25 s and 0.30 s.


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Figure 7.8: Excess attenuation for additional façade absorption and diffusion patches (see text). Results calculated with the extended Fourier PSTD method with frequencies up to 500 Hz and a flat spectrum over the 1/3-octave bands is used. Thin: Source at (9,y,0) and receiver at (0,0,5); Thick: Source at (9,y,0) and receiver in shielded canyon at (49,0,0); Dashed: Source at (9,y,0) and receiver in closed courtyard at (49,0,0).


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Figure 7.8 shows level differences in dB(A) for façade absorption and façade diffusion patches in the directly exposed canyon, shielded canyon and shielded courtyard compared to the case with façades with an absorption coefficient of 0.05. Results are calculated with the extended Fourier PSTD method and are presented in A-weighted results for frequencies up to 500 Hz. For the case of diffusion patches in Fig. 7.8(a), the scenario of horizontally oriented patches from the scale model study was studied. In the absorption case, Fig. 7.8(b), the absorption coefficient of all façades was set to 0.5. The difference between results for the shielded receiver positions and the exposed receiver positions is clearly visible. Since façade reflections have a larger importance at the shielded side, excess attenuation from façade absorption and diffuse reflections are larger at the shielded side than at the directly exposed side. The diffuse reflections cause a decrease of the sound level despite their small dimensions compared to the wavelengths. This is due to the effect of multiple reflections: the diffuse part of a multiple reflected sound wave is dominating over the specular reflected part even if the diffuse reflection part for a single reflection is small. Scale model study results have further shown that horizontally orientated diffusion elements in the canyons yield a lower sound level than vertically orientated elements. The reason for this is that in the x-z plane, the sound field with horizontally oriented diffusion elements is more diffuse than for the vertically oriented elements. Results in Fig. 7.8(b) show that the effect of absorption in the courtyard case is larger for larger transverse source-to-receiver distances than in the canyon case. This can be understood from the higher number of reflection contributions in the courtyard case; see Figs. 7.4(b,c). The difference is less obvious for the diffusion case in Fig. 7.8(c). Finally, Figs. 7.9 show impulse responses calculated for the three different façade scenarios with a source in the street canyon and a receiver position in the adjacent closed courtyard. The reduction of the magnitude of the higherorder façade reflections for the cases with façade diffusion and absorption is very obvious. Type of source Since the multiple road traffic noise sources that compose the total acoustic soundscape in closed urban courtyards are uncorrelated, the 2.5-D ESM has been developed to calculate the effect of noise abatement schemes using an incoherent line source of finite length. Using a coherent line source, as from a 2-D method, would not respect the incoherence of the multiple sources and

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Figure 7.9: Calculated 250 Hz octave band impulse responses with the extended Fourier PSTD method with source at (9,40,0) and with receiver at position (49,0,0) for the courtyard situation and three different façade scenarios: (a) Smooth façade surfaces with a small façade absorption coefficient of α = 5%; (b) Façade diffusion according to the scale model study (see Fig. 1(b) of Paper I); (c) Façade absorption α = 50%.


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would display different results, which could be more dominated by interference effects; see e.g. Fig. 9 of Paper II. However, the resulting effect of the noise abatement schemes, which are expressed in a sound level difference from a reference situation, showed that coherent line source results are very close to the finite incoherent line source results (with a maximum deviation of 0.3 dB(A); see Paper III). The reason is that coherent effects are cancelled out by façade diffusion, as already present in the reference situation, and presentation of the results on a 1/3-octave band basis. The abatement scheme results for an incoherent line source of infinite length showed a larger deviation from the finite incoherent line source results. These deviations can be attributed to the effects of near grazing incident waves from distant sources of the infinite incoherent line source. Since these effect are not likely to occur in real-life situations, the incoherent line source of infinite length is less suitable for noise prediction in a shielded courtyard.

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Chapter 8

Noise abatement schemes for closed courtyards

To fulfil the definition of a quiet side, one requirement for closed urban courtyards is to meet the LAeq < 45 dB(A) criterion. To be able to adapt courtyards that do not meet this requirement, noise abatement schemes for closed courtyards have been proposed and evaluated. Results from the scale model study and extended Fourier PSTD method in Sec. 7.2 and Paper I show a large effect of façade properties on the sound level for receiver positions in a closed courtyard, indicating that façade treatments could be efficient1 . For most noise abatement schemes, we therefore have studied in-courtyard façade treatments. Some treatments at the roof of the nearest building have also been considered. The 2.5-D ESM has been used to evaluate the noise abatement schemes2 . In Sec. 7.2 and in previous studied [167, 181], façade effects are compared to a situation with smooth façades. However, as we are interested in noise abatement schemes for real-life shielded courtyards where façades already have finite impedance surfaces and yield partly diffuse reflections, a comparison of such treatments with smooth façades is not very realistic. The effects of noise abatement schemes has therefore been related to existing façade situations, with geometries taken from real-life courtyards in Gothenburg as shown in Figs. 7.2. 1

This was also indicated by previous studies for a 2-D geometry [167, 181]. The scale model and extended Fourier PSTD results show that façade effects are equal or larger for the 3-D closed courtyard than for a 2.5-D canyon geometry. Since calculating the effects of noise abatement schemes using a 2.5-D canyon geometry is preferable from a computational time point of view, the 2.5-D ESM has been utilized for evaluating them. The presented results thus have to be seen as a lower limit of the actual effects expected. 2

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Figure 8.1: Façade pictures and aerial views over the two real-life scenarios in Gothenburg taken for the noise abatement scheme study. Courtyards are marked by a white dot, adjacent street canyons by a white line [8]. Furthermore, it is of importance to model road traffic noise from a realistic source type. A finite incoherent line source has therefore been implemented to represent a traffic flow. Table 8.1 gives excess attenuation results in dB(A) assuming a typical traffic noise source spectrum for a finite incoherent line source outside the courtyard at roof level with 11 to 24 receivers in the courtyard, and for two finite incoherent line sources in a canyon and a receiver at roof level outside the canyon. For absorption treatments, only the non-window parts were covered, resulting in a 40% average coverage in the upper scheme of Table 8.1. Results are averaged over all receiver positions of the two geometries with two height-to-width ratios each. It is important to repeat that the noise abatement schemes should be seen as a complement to other road traffic noise reduction measures. More details of the calculations and results can be found in Paper III.

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Table 8.1: Averaged effects of the investigated noise abatement schemes in dB(A). ∆Lp (dB(A)) Receivers Receiver in shielded outside canyon source canyon Scheme

Description Façade absorption α = 0.4-0.8

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Part IV Conclusions and Further work

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Chapter 9

Conclusions

Previous research shows that quiet sides are essential urban environments; they reduce adverse health effects from road traffic noise, and their soundscapes are perceived positively. Closed urban courtyards can be such quiet sides. The acoustic soundscape in a closed urban courtyard is characterized by its uniform noise level over space and time, and a low variation in noise levels across different courtyards. In many cases, measured equivalent noise levels in urban courtyards exceed the 45 dB(A) limit for a quiet side, feeding the need for noise abatement measures for courtyards. In order to assess the perceived soundscape quality in courtyards, the ability to predict the transient acoustic soundscape is important as well. In this thesis, tools have been developed to be able to create courtyards that qualify as quiet sides. A scale model study of parallel urban canyons was carried out, and two numerical methods were further developed for sound propagation to closed courtyards. One of these methods has also been used to evaluate noise abatement schemes for closed courtyards.

9.1 Scale model study A 1 to 40 scale model consisted of two parallel urban canyons with a source in one of the canyons and receiver positions both in the canyon directly exposed to noise and in the canyon shielded from direct noise. A geometry with a closed courtyard adjacent to the street canyon was also investigated. Noise from the 99

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street canyon could only reach the shielded side by propagating over the intermediate building block. As the highest measured frequency in the scale model was 45 kHz, excess air absorption occurs. For the measured impulse responses, air absorption enters as a function of frequency and time. A wavelet-based method, with time- and frequency-dependent basis functions, has successfully been developed to correct for this excess air absorption. From the measurements, several characteristics of the sound field in the shielded canyon and courtyard were found with a point source located in the adjacent street canyon: ◦ The sound level and decay times are rather constant over the length of the shielded canyon (at least up to the measured transverse source-to-receiver distance of 40 m), whereas they respectively decrease and increase with the transverse source-to-receiver distance in the directly exposed canyon. ◦ The sound level decreases with frequency in the shielded canyon and courtyard due to diffraction shielding. ◦ The impulse response in the shielded canyon is characterized by a rise time, which can be related to frequency and sound level. ◦ Higher-order façade reflections are of larger significance in the shielded canyon and courtyard than in the directly exposed canyon, explaining the larger excess effect of absorptive and diffusive façade properties. ◦ Horizontally oriented façade diffusion elements are more effective to reduce the sound level in the shielded canyon than vertically oriented façade elements. ◦ For the shielded courtyard, results suggest that the effects of façade properties are of even higher significance than for the shielded canyon. The importance of higher-order façade reflections at the shielded side explains the significance of distant sources to the sound level in shielded urban courtyards, as well as the observed low noise level variations in space and time.

9.2 The 2.5-D ESM The two-dimensional frequency-domain equivalent sources method (2-D ESM) solves the Helmholtz equation in the geometry of parallel urban canyons. The method relies on subdivision of the geometrical domain by locating equivalent sources at sub-domain interfaces. This method is extended to a 2.5-D ESM. This allows one to calculate the response of a point source or (finite) incoherent line source for the 3-D geometry of urban canyons, which is geometrically invariant

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in the direction along the canyons. In the 2.5-D ESM, the 3-D Helmholtz equation is first transformed from the (x, y, z)-domain to the (x, ky , z)-domain, where ky is the wave number for the y-direction. The 2.5-D solution is then found by an inverse transform over solutions in the (x, ky , z)-domain, which are obtained from the 2-D ESM. Numerical issues that arise in the evaluation of the inverse transform, the necessity of including imaginary frequency solutions and the discretization of the transform, are investigated for the parallel canyons geometry. The solution for an incoherent line source of finite or infinite length can numerically be found more efficiently than a point source solution. The ESM relies on the use of Green’s functions for the various sub-domains. The efficiency of the ESM was improved by using a more time-efficient Green’s function for the rectangular sub-domains – a modal-wave function instead of a complete modal function. For the parallel canyons situation with rigid façades as well as for façades with absorption and diffusion properties, 2.5-D ESM results are in good agreement with results from the scale model study.

9.3 Noise abatement schemes The 2.5-D ESM has been developed for sound level prediction in a closed courtyard (approximated by a canyon) and investigation of in- and near-courtyard noise abatement schemes for courtyards. As scale model study results indicated that the simplification of a courtyard by a canyon could lead to an underestimation of 2.5-D noise abatement schemes, results from the noise abatement scheme study must be regarded as lower limit values. The large influence of façade properties on the sound field as found from scale model results has motivated the choice of several façade treatments as noise abatement schemes. From reallife urban courtyard cases, reference situations were taken for investigation of these schemes, and a finite incoherent line source outside the courtyard was modelled. From the studied cases, the following conclusions can be drawn for treatments in or near the courtyard: ◦ Façade treatments lead to a larger noise level reduction for observer positions in the lower part of the courtyard and for narrower courtyards. ◦ At-roof treatments lead to a constant reduction of noise throughout the courtyard, being 2 dB(A) for a 15 cm grass (saddle) roof and 4 dB(A) for a 1 m tall noise screen. ◦ An increase of the façade absorption coefficient to 0.8 (for the non-window parts) leads to a reduction of 4 dB(A) for most observer positions in the courtyard.

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◦ An increase of the façade absorption coefficient in the upper part of the façades was found to be most favourable for noise reduction. ◦ Horizontal screens with absorption underneath, like walkways, lead to an average reduction of 2.5 dB(A) in the courtyard. ◦ A downward-refracting atmosphere can reduce the excess effect of additional façade absorption. ◦ For observer positions in the closed courtyard, noise abatement schemes applied to the source street canyon lead to larger reductions than the same schemes applied in the shielded courtyard. However, for an urban environment this means that many street canyons must be treated to obtain these reductions. Calculations using a coherent line source were found to give nearly the same results with differences up to 0.3 dB(A).

9.4 The 3-D extended Fourier PSTD method The 3-D extended Fourier pseudospectral time-domain (PSTD) method numerically solves the linearized Euler equations (LEE) and allows one to model sound propagation to a 3-D closed courtyard in a moving and weakly inhomogeneous atmosphere in the time domain. The method relies on the extended Fourier pseudospectral (PS) method to compute the spatial derivative operator of the LEE and uses a low-storage optimized 6-stage Runge-Kutta (RKo6s) method for time-iteration. The extended Fourier PS method extends the existing Fourier PS method by allowing for discontinuous media properties. To compute a spatial derivative of an acoustic variable with the extended Fourier PS method, the variable is transformed through orthogonal basis functions. The spatial derivative operator on the acoustic variable is then applied to these basis functions, which derivatives are known analytically. These basis functions are generalized eigenfunctions and can physically be seen as plane wave solutions to the discontinuous media problem. An inverse transform returns the spatial derivative of the variable. In the case of a single medium, the basis functions reduce to the trigonometric functions of the Fourier transform. Fast Fourier transforms are involved to operate the forward and inverse transforms. The extended Fourier PS method is accurate for wave number components obeying the Nyquist criterion, i.e. two spatial points per wavelength. The chosen discrete time step is in accordance with the optimized range of the RKo6s method, i.e. four points per wavelength. To obtain reflection-free boundaries at open domain

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ends of the discretized domain, a perfectly matched layer (PML) is included in the method, also when a moving grid was implemented. The extended Fourier PSTD method is more time-efficient than the state-of-the-art time-domain methods in outdoor acoustics with similar accuracy. The method has subsequently been validated for 1-D, 2-D and 3-D situations, showing a low dissipation and dispersion error up to two spatial points per shortest wavelength for most cases. In 2-D, it has been applied to transmission through the water-air interface, with continued propagation through a moving atmosphere. It has further been validated for several typical outdoor acoustic situations, such as sound propagation over a rigid ground surface with a thin noise screen with or without the presence of a wind velocity profile. As the method is limited to modelling boundaries as being rigid or by a fluid medium with a different density, it is best suited to situations dominated by meteorological conditions and complex geometries. However, application of sound propagation to the shielded canyon showed a very good agreement with results from the 2.5-D ESM, where impedance surfaces are modelled as locally reacting. Comparison for the 3-D courtyard situation with results from the scale model study showed good agreement. The method has been developed to study the transient sound field in courtyards and for investigation of 3-D effects. Calculations of an impulsive wave propagating into the courtyard clearly display the mechanics of multiple (diffuse) reflection and diffraction processes, and obtained impulse responses show the effect of a closed courtyard as compared to a shielded canyon.

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Chapter 10

Further work

In this thesis work, various developments have been made and investigations undertaken for modelling and modifying the acoustic soundscape in closed urban courtyards. The derived thesis material could be a departure of interesting further work, which is listed here. An application using the 2.5-D ESM ◦ The flat city method is a simple model to compute sound levels in closed courtyards and relies on elevating sources and receiver to the urban roof height. The 2.5-ESM can be used to estimate coefficients in the flat city method related to the elevation. Numerical issues of the extended Fourier PSTD method ◦ The effect of air absorption can be included in the linearized Euler equations, which are solved by the extended Fourier PSTD method. ◦ The implemented PML to obtain reflection-free boundaries displayed a lower performance for the higher frequencies, which could be improved. ◦ The stability of the method has not been studied. By computing the stability region of the RKo6s scheme and the maximum eigenvalue of the spatial derivative operator, one could find the maximum discrete time step regarding stability. ◦ A numerically more efficient implementation should be feasible. To accomplish this, a more appropriate programming language than Matlab can be used and the implementation can be parallelised. This would allow to model larger geometries and higher frequencies. 105

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Applications of the extended PSTD ◦ Existing software systems for modelling room acoustics can be used for auralization. These methods are ray approaches, and studies have indicated that auralization (in comparison with auralization of a real room) is difficult: the methods underestimate the speech-intelligibility in rooms with low reverberation; see e.g. [76]. Wave-equation-based methods, where higher-order diffraction effects are modelled more accurately, as in the extended Fourier PSTD method, could provide better tools for auralization. ◦ The currently presented extended Fourier PSTD method is able to model atmospheric sound propagation including 3-D meteorological and topographical effects such as crosswind over a noise barrier. It is of interest to further investigate these 3-D effects. ◦ A further investigation of 3-D urban environments with shielded receiver positions and various façade properties is of interest. Further work related to using the Chebyshev PSTD method ◦ The Chebyshev PSTD method can be developed for solving the linearized Euler equations including general impedance boundaries, something which the current extended Fourier PSTD method is unable to model. ◦ A hybrid Chebyshev vs. extended Fourier PSTD method could be developed, making use of the (more costly) Chebyshev PS method to evaluated spatial derivatives in the presence of impedance boundaries and the extended Fourier PS method for evaluation of spatial derivatives in other directions.

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10. Further work

Appendix A

State of the art of wave-based solution methods

For an accurate prediction of equivalent noise levels from road traffic in real-life closed courtyards and computation of impulse responses, sophisticated numerical methods, i.e. wave-equation-based methods, are necessary. As reported in Sec. 2.2.1, current accurate prediction methods developed for this purpose either are computationally heavily demanding in three dimensions (FDTD) or have appeared in two dimensions only (ESM). The studied problem in this thesis constitutes a domain which typically contains ∼ 102 wavelengths per dimension for the highest frequency of interest. Generally, accurate simulation of such short wave acoustical problems is still considered to be an unsolved problem for most engineering and physical problems, as reported in [119, 178]. For the low-frequency range, finite-difference (FDM) and finite-element (FEM) methods are still the most widely used methods for geometries that do not have analytical solutions. Such generalized domain decomposition methods do not make use of the underlying wave phenomena, and therefore require a typical number of 10 spatial discretization points per wavelength for acceptable accuracy. The development of numerical prediction methods for sound propagation to closed courtyards has in this thesis been devoted to methods being accurate and cost-efficient1 . In order to extend wave-equation-based methods to higher 1

Cost-efficient with respect to computational time and storage capacity compared to conventional methods.

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frequencies, several cost-efficient methods have been proposed in recent years. The aim of this Chapter is to give an overview of such cost-efficient computational methods that derive their efficiency from including a priori knowledge of the exact solution in their solution method. These solution methods are therefore either based, or enriched by, wave-based functions and are further denoted as wave-based solution methods. These methods are classified in frequency- and time-domain methods in Sec. A.1 and A.2. Apart from wave-based solution methods, other ways exist to improve the efficiency of computational sound propagation methods, which are only mentioned here: ◦ Domain decomposition can provide an effective way to solve a problem, especially if computations can be parallelized. ◦ In most frequency-domain methods, a matrix needs to be inverted. Since direct inversion is computationally costly, iterative solvers are often used. The mathematical community continuously improves these iterative solvers and makes them generally applicable. In order to improve the convergence rate of iterative solvers, preconditioners are being developed. ◦ Hybrid modelling, which can be accomplished e.g. in space or in frequency.

◦ Simplification of the geometry.

◦ Parallel computing, such as for multi-domains or multi-frequencies. ◦ Use of high-end computational environments and languages.

A.1 Frequency-domain methods Regarding the numerical techniques for time-harmonic acoustics, it is worth referring to some valuable recent review papers, which have a wider scope than the current section [71, 129, 154]. The frequency-domain wave-based solution techniques are here classified in several methods: boundary element, finite element and Trefftz-based methods.

A.1.1 Boundary element methods For problems where Green’s functions (for parts) of the domain are known, the governing acoustical equation in its integral form can be solved. Since these fundamental solutions (i.e. the Green’s functions) are included in the solution method, such a method is a type of wave-based solution method. Using the free-space Green’s function of the homogeneous medium problem, the boundary element method (BEM) only requires discretization of the boundaries and

A.1. Frequency-domain methods

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reduces the dimensionality of the problem as compared to the traditional FDM and FEM, see e.g. [39]. In BEM, all boundaries of a homogeneous medium are populated by monopole and dipole sources, which together fulfil the governing equation and the boundary conditions and satisfy the required Sommerfeld radiation condition at infinity. Another boundary discretization method is the equivalent sources methods (ESM), which is here classified as a Trefftzbased method (see Sec. A.1.3). BEM is limited mainly to a non-moving and homogeneous propagation medium, since Green’s functions for a moving or inhomogeneous medium are less straightforward to compute; see e.g. Zampolli et al. for layered media [176], Li et al. and Uscinski for a medium with a linear sound speed profile [104, 162] and Suzuki and Lele for an arbitrarily moving medium [148]. Another disadvantage of BEM is the associated fully populated non-symmetric system matrix. Most of the computational efforts in BEM are needed to construct the matrix and solve the equation system. Developments in BEM therefore aim at decreasing the density and dimension of the matrix. Wave BEM Perrey-Debain et al. [123, 124] developed an extension to the conventional BEM which is called the wave BEM. Using ideas from the partition-of-unity FEM, see Sec. A.1.2, the underlying wave behaviour of the solution is incorporated into the formulation of the boundary elements. The basic idea is that the conventional polynomial BE shape functions are multiplied by plane wave functions. The number of plane waves and their direction may be chosen freely. Evenly distributed directions are most commonly applied, but they may be irregularly distributed if some knowledge about the prevailing wave direction is available. As a result, the boundary elements are larger than in the conventional BEM, reducing the size of the system matrix rather than attempting to introduce sparsity. The presence of a large number of plane waves causes the conditioning of the system to degrade. Wave-number-independent BEM A recently developed type of BEM is the wave-number-independent method, see e.g. [10, 37]. The idea is to enrich the BE shape functions by incorporating the known asymptotic behaviour of the solution at high frequencies. The integrands involved with the new BE approaches may be decomposed into a highly oscillating factor and a smooth factor. The method aims for a prediction accuracy with a given discretization, which is frequency-independent. Fast multipole method The idea of the fast multipole BEM (FMM) is to use two formulations for the

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fundamental Green’s functions, one in the far field and one in the near field. Sources far away from a receiver point are clustered in a multipole expansion, whereas standard functions are used for nearby sources. The FMM makes the BEM matrix sparser. The method has also been extended to a multilevel FMM. Tong et al. have listed references on the (multilevel) FMM [158].

A.1.2 Finite element methods In FEM, the domain is discretized in arbitrarily shaped elements. Compared to FDM, FEM has the advantage that the spatial domain can be discretized more arbitrarily and in comparison with BEM, media inhomogeneities can be modelled more easily. The accuracy of the conventional FEM approximation, which is characterized by dispersion errors, can traditionally be controlled by two factors. The element size can be made smaller to improve accuracy, e.g. [84], and the polynomial degree of the basis functions can be increased, e.g. [85]. Since higher-order elements generally provide greater computational efficiency, fewer degrees of freedom are generally needed to achieve a given discretization error for oscillatory wave solutions. The following overview discusses wavebased solution methods within FEM. Wave envelope approach Wave shapes were introduced into large finite elements by Astley and were termed wave envelope finite elements [11]. In application to a realistic threedimensional problem, this wave envelope scheme has been shown to offer significant computational efficiencies when compared to an equivalent BEM. Bettess and Chadwick developed the wave envelope approach for bounded problems [25]. Instead of modelling the highly oscillating complex pressures, the wave envelope and the phase are modelled in the method, which vary much more gradually over the domain. An iterative procedure is used whereby an estimate of the phase is first given and from the standard finite element calculation for the wave envelope, a better estimate for the phase is obtained. Even if a very poor estimate for the initial phase is given, the iterated values for the phase and wave envelope converge to the exact values, but very slowly [35]. Stabilized methods Stabilized methods reduce artificial dispersion by including additional residual terms in the FEM formulation. For example, the Galerkin least-squares (GLS) method includes the residual on the Helmholtz equation in this way, and the Galerkin gradient least-squares method the gradient of this residual, see [129] for references. However, the GLS can only eliminate the dispersion error in a


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defined direction: the application of this type of stabilization scheme has less influence on the overall accuracy if the solution does not include large contributions of wave components in this direction. The quasi-stabilized method, as presented by Babu˘ska et al. [13], minimizes the dispersion error over all directions. In the residual-based finite element method of Oberai and Pinsky, the residuals are computed in element interiors and on inter-element boundaries [118]. This is in contrast to the GLS, where the added term contains residuals in the element interiors. Generalized methods These methods aim at reducing the artificial dispersion by using known fundamental solutions, such as plane waves, to define or enrich the FEM approximate solution space. Many generalized methods are based on the partition-of-unity FEM (PUFEM), as first proposed by Babu˘ska and Melenk [15]. In the PUFEM, a priori knowledge is incorporated by replacing the local approximation space with a linear combination of free-space solutions (plane waves) multiplied by the conventional polynomial basis functions. As such, many wavelengths can be included within a single element, leading to ultra-sparse meshes. The application of free-space solutions shows an improved computational efficiency compared with the stabilized methods. However, the boundary conditions and the numerical integration require special attention. Furthermore, the set of algebraic equations becomes ill-conditioned for a large number of free-space solutions. These computational issues hold generally for all PUFEMs. When the medium is moving, plane waves are no longer an exact solution and hence the mesh also has to be refined to obtain a convergent solution with wave enrichment based on plane waves. The PUFEM for moving inhomogeneous media has been developed as well with the same benefits as the PUFEM for non-moving homogeneous media [63]. The basis functions are then local solutions of the governing equations taking into account the convective effect of the flow. Another member of this class is the element-free Galerkin method (EFGM) which requires nodal data only and no element connectivity; e.g. [18]. It is based on the moving least-squares method. A weighted sum of a number of free-space solutions approximates the pressure in the neighbourhood of a node. Numerical experiments show that the level of artificial dispersion obtained with the element-free Galerkin method is comparable with the levels obtained with the quasi-stabilized method [129]. Lacroix et al. [103] proposed an improved EFGM. Trigonometric functions have been chosen for the local basis of the element-free

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Galerkin method with a spatially dependent phase function. Since this phase is a priori unknown, it is first computed, for instance, with an EFGM with a polynomial basis. Based on this initially obtained wave phase, the solution is iteratively computed using the local basis of trigonometric functions. With this meshless basis method, very accurate results are demonstrated on academic as well as real-life 3-D problems within a large frequency range. Similar to the EFGM are the Green’s function discretization (GFD) [33] and wave expansion method (WEM) [34], where the pressure can be approximated by the superposition of wave functions with unknown strength. The wave functions are plane waves in WEM and 2-D or 3-D Green’s functions in the GFD. A type of WEM has also been developed for an inhomogeneous medium [17]. The generalized FEM (GFEM) is a combination of the classical FEM and PUFEM. The construction of the discrete space for the GFEM is similar to that for the PUM. In addition to the local enrichment functions, a standard polynomial field is included in the local approximation spaces [147]. Multiscale methods Multiscale methods follow an additive approach rather than the multiplicative approach from the generalized methods. The basic assumption of the multiscale method is that the solution of the acoustic problem is given by an addition of the coarse-scale solution, for which the standard polynomial FE solution method is chosen, and the fine-scale solution. The fine-scale solution method is the linear combination of free-space functions of the Helmholtz equation. It is defined for element interiors only, i.e. it vanishes on the element boundaries. Consequently, the direct sum of the FE approximation and the fine-scale solution may violate the pressure continuity across element boundaries. The application of the Lagrange multiplier technique enforces the pressure continuity weakly. The method is known as the discontinuous enrichment method, see [54]. Numerical experiments show that the discontinuous enrichment methods suffer less from pollution errors than the classical FEM. Furthermore, the involved models are not as ill-conditioned as PUFEM of comparable numerical accuracy. Omitting the coarse scale finite element basis, the method reduces to the discontinuous Galerkin method (DGM) as derived in [55].

A.1.3 Trefftz-based methods In 1926, Trefftz proposes the use of a priori knowledge in the definition of approximate solutions [160]. His approximate solutions satisfy the governing do-


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main equations but violate the boundary conditions. The Trefftz-based methods have made their entry in the field of computational acoustics in the mid-1990s. In the most frequently used indirect formulation of the Trefftz approach, the approximate solution is selected from a complete solution, which is an infinite sum of waves. The sum is truncated in the approximate solution. The approximated pressure satisfies the homogeneous Helmholtz equation and the Sommerfeld radiation condition a priori. The violation of the boundary conditions needs to be enforced to zero in an integral sense by application of the weighted residual approach. In the indirect Trefftz approach, the unknown degrees of freedom (weights of the waves) do not represent physical field quantities. The Trefftzbased method is a boundary integral method similar to BEM, although with the advantage that it does not require the evaluation of singular integrals. On the other hand, it is based on a weighted residual formulation with test functions similar to FEM. Various derived Trefftz-based methods can be classified, with the discontinuous Galerkin method derived by Farhat et al. [55] belonging to the hybrid Trefftz finite element methods. A frameless Trefftz method is the Wave Based Method (WBM) [45], where a combination of propagating and evanescent waves is applied. The WBM adopts a weighted residual approach to enforce the boundary and continuity residuals to zero. The method has been applied successfully for many time-harmonic interior acoustic problems [128]. It is shown that the WBM exhibits an enhanced computational efficiency as compared to the conventional FEM for problems with a relatively moderate geometrical complexity [129]. The ultra weak variational formulation constitutes another class. It is based on a discretization of the problem domain into elements. A boundary variable is introduced at the element boundaries satisfying an alternative variational formulation, namely the ultra weak variational formulation. It is shown that the associated field variable satisfies the Helmholtz equation if the boundary variable satisfies the ultra weak variational formulation. The field variable is approximated by a set of plane wave functions. Although the ultra weak variational formulation may solve wave problems at reduced computational cost, it may suffer from numerical instability. Huttunen et al. [83] have applied the ultra weak formulation to problems in which the wave speeds are constant within each element, but are allowed to vary from element to element. Ortiz has extended the approach to the case where the wave velocity may vary within each element and refraction can occur [121].

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Source simulation techniques constitute a class of Trefftz-based methods as well. Although various names have appeared for this class of methods – e.g. superposition method, source simulation technique, equivalent source method, full-field method, null-field method, equivalent sphere method, energy source simulation method – they all apply the same modelling principle [129]. An interior/exterior acoustic boundary value problem is solved by positioning a number of sources exterior/interior to the problem boundary. The equivalent sources method applied to street canyons, where the domain is decomposed in sub-domains with monopole sources at the sub-domain interfaces, can be classified as a Trefftz-based method. In contrast to BEM, where most commonly freefield Green’s functions are used, the Green’s functions in the ESM hold for the separate sub-domains. The advantage of the ESM is that the number of sources could be smaller due to prescribed values of normal velocity and pressure at boundaries and the system matrix is sparser due to the multiple sub-domains. The method of fundamental solutions is comparable to the equivalent source methods, since also sources are positioned outside the problem domain with the source strengths determined by enforcing the boundary conditions at the boundary in a collocational way, see e.g. [65].

A.2 Time-domain methods Since time-domain investigation provides insightful understanding of the governing physical phenomena, various numerical time-domain schemes have been developed to obtain the transient response of an acoustic source at a receiver position. The developments of wave-based solution methods in the time domain are here classified into techniques applied to the spatial derivative operator, the temporal derivative operator and direct time-domain formulations. The first two classes can be seen as separate parts of the solution method, and the third class consists of methods that solve the time-domain formulation completely. In contrast to the popularity of BEM and FEM among the frequency-domain methods, finite-difference methods (FDM) are often used as time-domain solution methods (called FDTD). The FDTD is considered attractive due to its simplicity and straightforward way to implement arbitrarily shaped geometries. A cornerstone for the FDTD methods is the Yee algorithm, a second-orderaccurate FDTD method [174]. A second-order-accurate FDTD method is still state of the art within the outdoor sound propagation community; e.g. [73]. Besides the FDTD method as mentioned in Chapter 2.2.1, the transmission line model as in [100, 75, 69] and the equivalent discrete Huygens model, see e.g.

A.2. Time-domain methods

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[94], have also appeared. These methods, however, require a computational effort similar to the second-order-accurate FDTD methods. Many developments to reduce costs of processing time and memory in the broader wave propagation community have been devoted to improving the FDTD algorithm of Yee; see e.g. [179, 27, 139] for an overview of methods and references. These improvements are not listed here as wave-based solution methods, since they do not capture a priori information on the fundamental wave-solution into the method. Nevertheless, these methods have been shown to improve the standard FDTD method substantially, regarding both accuracy and stability. Improvements for spatial derivative operators include: ◦ Higher-order methods, where more discrete points in the finite-difference stencil are involved for calculating derivatives. ◦ Compact implicit schemes, where coefficients in the finite-difference scheme are matched to fulfil a certain order of accuracy. ◦ Optimized schemes, where coefficients in the finite-difference scheme are modified to produce reduced errors over a certain range of wave numbers. For the temporal derivative operator (i.e. time-marching methods), some popular developed schemes are the Adams-Bashforth, Adams-Moulton, RungeKutta and MacCormack schemes; see e.g. [27]. Also these methods have appeared in higher order, optimized for a reduced error over a range of wave numbers and developed as low-storage algorithms. Furthermore, unconditionally stable methods have been presented, aiming to relax the small time-step size that is imposed by the stability criterion of conditionally stable methods. Such implicit methods are the alternating direction implicit (ADI) method and split-step (SS) schemes. The choice of the time-step in these methods is determined on the basis of the desired accuracy. While much effort has been given to improving the FDTD algorithm of Yee by non-wave-based solution methods, wave-based solution methods in the time domain have been developed to a smaller extent than in the frequency domain.

A.2.1 Spatial derivative operator methods Spectral methods Spectral methods can offer an efficient way to solve the spatial differential derivative operator. Spectral FEMs have been applied to time-domain problems, typically require 5 discrete points per spatial wavelength, and are able to model arbitrary geometries and medium inhomogeneities; see e.g. [97]. In pseudospectral

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(PS) methods, the governing equation is forced to hold at a discrete set of grid points; collocation points [159]. Although frequency-domain PS methods have been developed as well, e.g. [30], most literature is devoted to PS methods for transient problems. PS methods calculate spatial derivatives by decomposing the spatial variables at a certain time through a set of basis functions. The basis functions are global functions as compared to the local functions in finitedifference methods (FDM). In contrast to FDM, PS methods have no phase error. For a homogeneous medium, spatial derivatives for an acoustic variable with compact support can be calculated with spectral accuracy (accurate up to two spatial points per wavelength) by the Fourier PS method by first applying a Fourier transform to the wave number domain. The Fourier transform has basis functions that can be seen as a basis of plane waves. The spatial derivative operator is then applied in the wave number domain (i.e. multiplication by ±jk for a first derivative) and an inverse Fourier transform returns to the spatial domain. The use of FFTs makes this method more efficient than applying finite differences, especially for higher dimensions. The Fourier PS method is still accurate for a weakly inhomogeneous medium as shown in [110]. However, spatial derivatives are known not to be continuous for a discontinuity in the medium properties, and therefore cannot be represented well by the Fourier PS method. PS methods for spatially bounded domain problems, where the periodic extension of spatial derivative of the acoustic variable is not required to be continuous, have been developed with polynomials that form a non-periodic basis, such as Chebyshev polynomials. This will however require a higher spatial resolution than in the Fourier PS method (π points per wavelength), a more stringent stability criterion and a multiple subdivision of the spatial domain. Band-limited functions, prolate spheroidal wave functions (PSWF), have recently been proposed as another basis in PS methods; see e.g. Chen et al. [38]. Such a basis was demonstrated, under certain conditions, to be more accurate than the Chebyshev basis and allow for a larger time step regarding stability. Discontinuous Galerkin FEM A frequency-domain FEM development that has also appeared for time-domain problems is the discontinuous Galerkin method (DGM). As in the case of conventional finite element methods, the computational domain is divided into a set of finite elements. Basis functions, such as e.g. plane waves in [62], are then defined for each element which hold for the element interior. In contrast to the finite-element methods, the discontinuities between two elements on every interface in the DGM must be handled by using an approximate numerical flux, whose choice is not unambiguous. In general, one advantage of the DGM is the


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possibility of computing solutions separately for each element, indicating that the DGM is easy to parallelize [24]. Wavelet-based methods A wavelet-based method for approximation of the spatial derivative operator has been presented by Hong and Kennett [77]. A review of applications of wavelets to solve partial differential equations is given there and e.g. in [112]. Wavelets have compact support in time and space, and the developed wavelet method is shown to give accurate and stable results for wave propagation problems through highly inhomogeneous media. However, the method is not faster than the FDTD method, due to the time-consuming calculation of the differentiation in the wavelet-based method.

A.2.2 Temporal derivative operator methods Exponential operator methods When studying wave propagation departing from the first-order coupled equations for non-moving media, a first step in the solution method can be to write the solution for free space in its exact form: q(t) = eLt q(t0 ), with t0 = 0, q(t) = [u, v, w, p]T the acoustic variable vector and:   1 ∂ 0 0 0 ρ0 ∂x  1 ∂    0 0 0 ρ 0 ∂y  . L=  1 ∂  0 0 0   ρ0 ∂z ∂ ∂ ∂ ρ0 c2 ∂x ρ0 c2 ∂y ρ0 c2 ∂z 0

(A.1)

(A.2)

This starting point has often been used in electromagnetics (with slightly different equations), and an overview of methods to evaluate the exponential operator eLt of Eq. (A.1), of which Taylor series and Padé series are some, can be found in [16]. The most accurate and stable method is the Chebyshev polynomial expansion scheme, pioneered by quantum chemists Tal-Ezer and Kosloff [150]. It allows very long time steps. Pan and Wang introduced this method in acoustics, called the acoustic wave propagator (AWP). Boundaries and spatial medium variations can be included in the system operator L. The operator eLt is expanded in Chebyshev polynomials, with zero- and first-order Chebyshev polynomials defined and higher-order polynomials calculable by recursive relations. The number of spatial derivative operators that have to be applied for a single time step in the AWP is equal to the order of the polynomial expansion.

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Another method to evaluate the exponential operator is the symplectic integrator method, which factors the exponential in two exponential terms. A single time step is split into a number of stages. Since variables at each stage are directly updated, no additional storage is required (in contrast to standard multistep schemes such as the Runge-Kutta method). The symplectic integrator method is non-dissipative and stable, and accurate calculations with moderately lower memory usage than standard schemes such as the 4th-order 5-stage explicit Runge-Kutta have been performed [140]. The symplectic operator method has been applied to the Maxwell equations mostly, yet applications to the wave equation also appeared; see e.g. [173]. Time-domain beam propagation method (TD-BPM) The idea of the time-domain beam propagation method (TD-BPM) is that for a band-limited function, a highly fluctuating carrier wave in time is removed and the remaining variable is solved for, which is slowly fluctuating in time. Further, the slowly varying envelope approximation (SVEA) is applied to the wave equation by neglecting the second-order time derivative. The TD-BPM method is the counterpart of the well-established frequency-domain parabolic equation (PE) method in acoustics; see e.g. [137]. The SVEA allows one to use an implicit scheme with a subsequent larger time step than that in the FDTD method. To solve the TD-BPM equation, Padé schemes or an alternating-direction implicit (ADI) scheme have been used, as in [141]. Using the TD-BPM, calculation times of 30% of conventional FDTD methods were reported. One drawback of the method is that only band-limited frequency contents are considered, whereby several runs are needed for broadband calculation. Recently, an improved technique, a wide-band approach, was proposed [133].

A.2.3 Direct methods BEM and ESM The boundary element method has been formulated in the time domain (TDBEM), see e.g. [72], where Green’s functions become impulse responses as a function of time and space and the source strength of the sources at the boundary elements become functions of time. Some work on accelerating the TDBEM has been made, such as the plane wave time-domain (PWTD) BEM in [51, 175]. This method can be considered the time-domain counterpart of the frequencydomain fast multipole method. However, developing stable TDBEM methods seems currently to be a problem of larger relevance than accelerating the method. The ESM has also been presented in the time domain [101]. As in the


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frequency domain, ESM could be faster than BEM. The k-space method In the k-space method, as first presented by Bojarski for electromagnetics problems [28], Eq. (A.1) is transformed to the wave number-time (k-time) domain and spatial derivatives are calculated in this domain (according to the Fourier PS method). Since the exponential operator is analytical in the k-time domain, the solution at any time can be calculated. This approach, see Appendix C.3 for more details, is advantageous over the second-order-accurate FDTD for a mildly inhomogeneous moving medium, where this k-space method, without extra computational complexity, allows for a larger time step than the FDTD method.

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Appendix B

Aspects considered for a successful scale model study

In the scale model study as reported in Paper I, a geometry of two parallel canyons was investigated. One canyon represents a street canyon directly exposed to road traffic noise whereas the other canyon represents a relatively quiet side, shielded from direct noise. The results from the scale model study have been used to define requirements for the developed prediction methods in Chapter 4.1, to characterize the sound field at the shielded side in Sec. 7.2 and Paper I, and to validate the prediction methods in Chapter 5 and Paper III. The purpose of this Appendix is to discuss the various aspects that have been taken care of to make the scale model study successful. Details of the executed measurements can be found at the end of this Appendix. In Fig. 1 of Paper I, cross-sections and plans of the scale model study setup are shown. The chosen scale is 1 to 40 and full-scale distances which are representative for European city centres [153] are shown in the figure. The model was placed in the anechoic room of the Division of Applied Acoustics with a ground construction of 3 m x 3 m, consisting out of 9 elements. The elements were placed on 145 mm high wooden beams, which made it possible to place the microphone with preamplifier below the floor and with the microphone membrane level with the floor. The microphone, when mounted below the ground, was placed in a patch that could be moved in one direction. The sound source was also placed below the ground floor. To prevent diffusion effects due to the small gaps between the floor elements, the gaps were covered by electricity tape. The building blocks consisted of cubic units made of 5 mm thick Plexiglas 123

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B. Aspects considered for a successful scale model study

(250 mm × 250 mm × 250 mm). These units gave the freedom to change the geometry easily. The monopole source is located in the left street canyon; see Fig. 1(a) of Paper I. Its position was not altered during the measurements. The receiver positions are also shown in Figs.1(a,b,h,g) of Paper I and vary over y and z in the façades of the canyons. In the shielded canyon, the microphone was also positioned at the canyon ground floor. Table B.1 gives an overview of the receiver positions during the measurement campaign. All receiver positions were repeated for four cases: ◦ Acoustically hard façades.

◦ Horizontally oriented façade absorption patches. ◦ Horizontally oriented façade diffusion patches. ◦ Vertically oriented façade diffusion patches.

The absorption patches consist of velvety felt and the diffusion patches of Plexiglas. Mixed cases, like façade absorption patches combined with façade diffusion patches, were not considered. A closed courtyard situation has also been investigated. When the courtyard was investigated at the shielded side, the directly exposed side was kept as a canyon. Figure 1(h) of Paper I shows a situation with a courtyard at the shielded side and vertically applied diffusion elements. The closed-courtyard situation at the directly exposed side is symmetric to the closed-courtyard at the shielded side as shown in Fig. 1(h) of Paper 1, with (30,y,z) as symmetry plane. Acoustical scale model study measurement results have a certain degree of accuracy. This accuracy depends on many factors, of which the chosen scale is probably the main factor. In general, one can say that with a larger scale, the expected geometrical accuracy increases and the negative effects of increased air attenuation at higher frequencies can be remedied more easily. The drawbacks of a large scale are the limitations of the measurement space available and practical matters of constructing and adapting the physical model. The chosen scale here is 1 to 40 with a 1/3-octave band frequency range from 4 to 40 kHz (100 to 1000 Hz at full scale). In the preparation for the scale model measurements, several choices had to be made. The several aspects that have been considered here for a successful scale model study are now discussed. Measurement room An anechoic termination of the measurement enclosure as well as a low background noise level were desired. The anechoic room of the Division of Applied Acoustics was used for the measurements. The size of the room with effective

125 dimensions of 8 m x 8 m x 6 m, limited the dimensions of the scale model. The anechoic room was designed for the frequency range 75 Hz to 10 kHz. Room reflections, due to for example metal tubes or lamp bulbs, were detected in the measured signal for some receiver positions. The reflections were suppressed by mounting mineral wool between the sound source and the reflecting objects. Acquisition system Output. The measurement system MLSSA was chosen [113]. Impulse responses could be measured with this system, yielding both time and (after a Fourier transform) spectral information on the sound field in the investigated geometries. Signal to noise ratio (SNR). An important criterion for the choice of the acquisition system is the SNR. From the MLSSA system, a known source signal is emitted to the sound source used. The receiver signal is then correlated with this known signal and the impulse response is calculated. The technique to accomplish this in MLSSA is the Maximum-Length Sequence (MLS) technique [113]. The SNR was improved by averaging subsequent measurements (as built into the software). Frequency range. The acquisition system should be capable of sampling the measured signal with a sufficiently high sample frequency. For the equivalent 1 kHz 1/3-octave band, a sample frequency of around fs = 135 kHz was required in MLSSA since the MLS power falls by 1.6 dB at one third of the sample frequency. Therefore, the highest possible fs in MLSSA, 133 kHz, was used. Source Frequency spectrum. The source should have a sufficiently high amplitude in the frequency range of interest to measure a sound pressure that is signal strong enough, considering the needed signal to noise ratio at shielded receiver positions. Since measurement results are sensitive to the reference signal being resonant, a smooth frequency spectrum was desired as well. Impulse response. The impulse response of the source itself should be short. A long impulse response of the source affects the impulse response of the measured system (the canyons). The source impulse response could be removed from the system impulse response by deconvolution. This is however a sensitive process since the source impulse response is not exactly omnidirectional. With a short source impulse response, the system impulse response is not much affected and a deconvolution is not necessary.

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Repeatibility. As measured signals are related to a reference signal, a high repeatability of the source was desired. A tweeter source with a smooth frequency response was chosen as the sound source. The repeatability of this source is high compared with other candidate sources such as a spark source. The source also has sufficient power in the frequency range of interest and a short impulse response. Directionality. The source should preferably be omnidirectional, since that facilitates the comparison with a calculation model. To accomplish an omnidirectional field one should fulfil ka < 1, where k is the wave number and a the radius of the source [125]. The radiation efficiency however decreases with a decreasing ka number. To obtain omnidirectionality, the source was placed in a cavity below a Plexiglas plate with a small hole (see Fig. B.1). The hole was located at the ground level and acted now as the source. The cavity was filled with felt material to suppress resonances. A 10 mm circular hole was used for the lower frequency region up to 16 kHz and a 3 mm circular hole for the upper frequency region. As the output power drops with the decreasing hole diameter, the hole-size choice is a trade-off. The impulse and frequency responses of the source mounted in the cavity are shown in Fig. B.2. Both source and receiver (when mounted in the shielded canyon ground) were placed off-centre in the canyons. This was done to prevent exciting or measuring one of the first modes at their maximum or minimum pressure level. When measurements were done in a courtyard situation, the source was placed off-centre in both horizontal directions. Receiver Frequency response. A microphone smaller than usual for the audible acoustics range was necessary to measure frequencies up to 45 kHz. A 1/8-inch condenser microphone was therefore used. Directionality. The directionality of the receiver at a certain frequency is related to its membrane dimension relative to the wavelength, being more directional for higher frequencies. The 1/8-inch microphone has an angular response difference of at most 6 dB at 45 kHz according to the factory data, with a maximum for normal incidence. Since sound waves reach the microphone under various angles in the canyons’ geometry, improving the directionality was desirable. This was done by placing the microphone in a hole of a Plexiglas box, with its membrane placed in the plane of the Plexiglas. Sound waves could now only reach the microphone in a hemisphere and diffraction effects around the microphone were reduced. To keep the same microphone position relative to the box

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Figure B.1: Cross-section sketch of the tweeter source mounted below the floor plane of the scale model. Dimensions of the scale model are shown.

0 −20 1

1.5

(a)

2 2.5 3 −3 Time (ms) x 10

Magnitude (dB)

Amplitude (Pa)

Amplitude (Pa)

50 20

20 0 −20 1

1.5

(b)

2 2.5 3 −3 Time (ms) x 10

40 30 20 10 0

0

1

2

3

4

5

(c) Frequency (Hz) x 104

Figure B.2: (a) Impulse response of the source in the cavity of Fig. B.1 in the low frequency region of 3.5 kHz to 16 kHz; (b) Impulse response of the source in the cavity in the high-frequency region of 16 kHz to 45 kHz; (c) Frequency response of the source in the cavity. Vertical lines indicate the used frequency ranges. The shown time and frequency are in the scale model units.

during a measurement sequence, the box with the microphone was moved to every receiver position and to the reference positions as well. The same procedure was followed for receiver positions in the ground surface with the microphone fixed in a ground surface patch. A second advantage of the chosen microphone position is that the microphone and connected pre-amplifier were not present in the canyon or courtyard and did not influence the sound fields. Disturbing transfer paths Measurements with high insertion losses, such as measurements at the shielded receiver positions, are sensitive to disturbing acoustical transfer paths. Such encountered transfer paths are listed here with the measures taken to suppress them:

128


◦ Transmission through the Plexiglas boxes due to excitation of modes in the boxes – Insertion of mineral wool. ◦ Transmission through the gaps between the Plexiglas boxes – Application of electricity tape covering the gaps between the Plexiglas boxes. ◦ Waves diffracted around the Plexiglas boxes at the canyons’ ends – Anechoic termination by absorption patches at the ends of the canyons. ◦ Sound radiation from the Plexiglas boxes – Attachment of 5 mm thick material with a high internal loss factor inside the most relevant Plexiglas boxes. ◦ Room reflections – Mounting mineral wool between source and reflecting objects. Non-linear scaling effects All relevant scaling laws have to be respected in a physical scale model. In fluid dynamics, several non-dimensional parameters have been defined that should be equal at all scales, such as the Reynolds and Strouhal numbers. These numbers are not relevant for sound propagation through a non-moving medium. The most obvious scaling law concerns the frequency that scales up by scale factor 40 and the dimensions that scale down with the same factor. Apart from these linear scaling effects, there are some aspects that scale non-linearly with frequency. Acoustic impedance of materials. Materials in outdoor acoustics can often be modelled as porous media. The impedance of these materials is non-linearly related to the frequency. The use of the same material at full scale and in the scale model would lead to different impedances, whereas the same impedance is desired. For a field problem built at scale, it is necessary to find a material with properties that correspond with the real scale material properties at the chosen scale. The current experimental measurement results have been compared with model calculations and the impedances of the chosen materials were measured and their values have been used in the calculation model. Acoustic boundary layer. A second effect is due to the acoustic boundary layer. In a thin layer close to a flat boundary, called the acoustic boundary layer, viscosity and heat conduction cause a loss of acoustic energy. The loss is relative to the square root of the frequency and increases with the angle of incidence relative to the normal to the surface. The boundary layer effect is most pronounced in cases of sound propagation over rigid surfaces, since the admittance due to the boundary layer effect is small compared to the admittance of materials having a finite impedance [125]. Due to the multiple reflections in the source and shielded canyon, the effect of the acoustic boundary layer is of relevance for the

129 admittance of the ground surface. Its effect has been taken into account in the calculation models (see Paper I for a further investigation). Excess air attenuation. Damping of sound waves by air is mainly caused by two effects [137]. Temperature and velocity gradients in a sound wave are reduced by heat and momentum transfer, which depend on the thermal conductivity and viscosity of air, this being known as classical absorption. A sound wave loses energy due to these processes, which is converted into heat. The attenuation loss for this effect is proportional to the frequency squared and hard to avoid. The second effect is due to the presence of oxygen and nitrogen molecules in air. Relaxation and compression of these molecules caused by the acoustic pressure fluctuations result in a loss of energy, again converted into heat. Water vapour acts as a catalyst to this process. This type of loss is also non-linearly dependent on frequency and can be influenced by changing the consistency of the propagation medium. The facilities to achieve this were however not available. Since the effect of absorption is different in the scale model than in reality, a correction is necessary. A wavelet-based method to correct the measurements for excess air attenuation has successfully been applied here (see Paper I).

130


Table B.1: Microphone positions in the scale model study for all four façade types. All microphone positions are given in x and y coordinate with the coordinate system as in Fig. 1a of Paper I. (The notation (0,-40:5:40) means that the microphone is at x = 0 m and y varies form -40 m to 40 m in steps of 5 m). The source position was fixed at (9,0,0).

z-coordinate microphone z=0m z=2m z=5m z=8m z = 12 m z = 15 m z = 18 m

Directly exposed street canyon Canyon Courtyard (0,0) (0,0) (0,-40:5:40) (0,0) (0,0) (0,0) (0,0) (0,0) (0,-40:5:40) (0,0) (0,0) (0,0)

Shielded canyon Canyon Courtyard (49,-40:5:40) (49,0) (60,0) (60,0) (60,-40:5:40) (60,0) (60,0) (60,0) (60,0) (60,0) (60,-40:5:40) (60,0) (60,0) (60,0)

Table B.2: Equipment list of the scale model study. Device type Acquisition system Source Power amplifier Frequency filter Watt meter Pre-amplifier Microphone pre amplifier 1/8 inch condenser microphone

Specification MLSSA Version 10WI-4 Ceramic driver tweeter Thiel and Partner, Accuton C 2 12-6 [5] NAD 3020 Krohn-Hite 3202 filter Feedback electronic wattmeter EW 604 Larson-Davis type 2200 C Larson-Davis type 910 B B&K type 4138

Table B.3: Details of the scale model study. Sample frequency Measurement frequency range

133 kHz low: 2 kHz - 20 kHz high: 10 kHz - 52 kHz

Number of samples of the impulse responses MLS order Averaging in MLSSA Air absorption correction Ground material Absorption material Building blocks material Measurement chamber

65535 16 16 x pre-averaging (raw signal treatment) 1 to 3 averages of the obtained impulse response Wavelet-based method (see Appendix Paper I) Chipboard plate with a melamine cover Velvety felt (see Appendix Paper I) 250 mm × 250 mm × 250 mm boxes of 5 mm thick Plexiglas Anechoic room (Chalmers, Gothenburg)

Appendix C

Aspects of PS methods

C.1 Fourier PS methods: finite differences with an infinite order of accuracy The (extended) Fourier pseudospectral (PS) method is global, i.e. computation of the spatial derivative of an acoustic variable at a discrete point depends on values of all points in the entire domain1 . Finite-difference methods, in contrast, are local since only a part of the domain is involved in the calculation of a spatial derivative. In a finite-difference method, a variable is approximated by a local polynomial interpolant. The derivative of the variable is then obtained by differentiating the polynomial. The first-order spatial derivative at position xk calculated by central finite-differences of accuracy order 2m (with m an integer) reads: m X ∂p(x) p(xk + l∆x) − p(xk − l∆x) , (C.1) ≈ αlm dx xk 2l∆x l=1

where αlm are the weights. Higher-order accurate approximations involve more spatial points and are less local. Consider the following case where the order of 1

For the calculation of a spatial derivative, the entire domain refers here to the single dimension of that derivative.

131

132

C. Aspects of PS methods

the approximation of periodic function p(x) is infinite: ∞ X p(xk + l∆x) − p(xk − l∆x) ∂p(x) = αl∞ , ∂x xk 2l∆x l=1

with

p(x) = ejnxl ,

xl =

2πl N +1

(C.2)

l ∈ [0, ..., N].

From Eq. (C.2), it can be shown that αl∞ = 2(−1)l+1 for l ≥ 1, and using periodicity of the function p(x), it can be shown that [74]: N X dp 1 = (−1)l+k ∂x xk 2 sin l=0

p(xl ) . π (k − l) N +1

(C.3)

Thus, for a periodic function, infinite order of accuracy is obtained by making use of all points within one period. We now return to the numerical implementation of the Fourier PS method for expanding a periodic function as in Sec 6.1: ! n=N/2 N X 1 X p(x) ≈ p(xl )e−jnxl ejnx (C.4) N +1 l=0 n=−N/2   n=N/2 N X X 1 ejn(x−xl )  p(xl )  = N + 1 l=0 n=−N/2

=

N X

p(xl )hl (x),

l=0

where the expansion has been rewritten, such that p(x) is obtained by an interpolation of the grid point values p(xl ) with interpolant hl (x), and xl = N2πl , l∈ +1 [0, ..., N]. The interpolant is the Lagrange polynomial: n=N/2 X 1 hl (xk ) = ejn(xk −xl) N +1

(C.5)

n=−N/2

1 sin N 2+1 (xk − xl ) , = N + 1 sin 12 (xk − xl )

which has the following derivative: ∂hl (−1)l+k 1 . = π ∂x xk 2 sin N +1 (k − l)

(C.6)

C.2. Boundary conditions in the Chebyshev PS method

133

Notice now the two different yet equivalent ways to compute spatial derivatives, using the expansion method as used in Chapter 6 and using the interpolation method. The second is known as the matrix method, since the calculation involves a matrix-vector multiplication. Calculating the derivative in the Fourier PS method with the interpolation method can now be found as: N X ∂p(x) dhl (xk ) ≈ p(xl ) ∂x xk ∂x xk l=0 =

N X (−1)l+k l=0

2

sin

(C.7) 1

π (k N +1

− l)

p(xl ).

When comparing the matrix method with the expansion method, the former requires O(N 2 ) and the latter O(N log2 N) number of operations. The matrix method can be more effective to use for smaller domains. Equation (C.7) is equal to what we found in Eq. (C.3), showing that the Fourier PS method for a periodic signal is equivalent to the finite-difference method with an infinite order of accuracy. From Eq. (C.4), we notice that the highest wave number n that can be captured is N/2, which implies a minimum of two points per wavelength.

C.2 Boundary conditions in the Chebyshev PS method The Chebyshev pseudospectral (PS) method to calculate spatial derivatives for finite domain problems is briefly described in Sec. 6.1.2. As the method is constructed for finite domains, spatial derivatives for the inner grid points can be calculated. Calculation of these derivatives at the boundaries requires boundary conditions. Different ways to fulfil the boundary conditions have been proposed. Strongly imposed boundary conditions imply that the variables or their derivative operator are replaced by their corrected values after each time step. Updating the variables is referred to as the correctional form, and updating the derivative operator as the differential form. Strongly imposed boundary conditions may cause instability [74]. Another way to impose boundary conditions is the weakly imposed method, where both the equation and the boundary conditions hold at the boundary points. This method is more stable than the strongly imposed conditions. The collocation formulation for the one-way wave equa-

134


tion then reads as follows [74]: ∂p(x, t) ∂p(x, t) +c = ∂t ∂x −τ−1 Q− (x)c[p(−1, t) − h(−1, t)] − τ1 Q+ (x)c[p(1, t) − h(1, t)]

(C.8)

x ∈ [−1, 1],

where c is the adiabatic speed of sound, Q± (x) a polynomial that vanishes at all grid points except for x = ±1, τ−1 and τ1 parameters to adjust stability and h(−1, t), h(1, t) the boundary conditions. The method is consistent since it holds for p(±1, t) = h(±1, t). When τ±1 = ∞, the boundary conditions are enforced strongly, as in the correctional method. When τ±1 = 0 on the other hand, only the equation is fulfilled but not the boundary condition. For all other values of τ±1 , both the boundary condition and the equation are fulfilled. The idea behind the weakly imposed boundary conditions is that it is only necessary to satisfy the boundary conditions consistent with the order of the scheme, i.e. imposing them exactly (i.e. strongly) is not necessary if the solution is not exact [74]. To fulfil boundary conditions, an often used approach in the Chebyshev method is to utilize the method of characteristic variables. We consider the wave equation in three dimensions and a problem of two sub-domains (1 and 2), with the sub-domain interface at (a, y, z):  2  ∂ p21 − c2 ∂ 2 p21 + ∂ 2 p21 + ∂ 2 p21 = 0 (x, y, z) ∈ ([−1, a], [−1, 1], [−1, 1]) ∂t ∂y ∂z ∂x  ∂ 2 p22 − c2 ∂ 2 p22 + ∂ 2 p22 + ∂ 2 p22 = 0 (x, y, z) ∈ ([a, 1], [−1, 1], [−1, 1]). ∂t ∂x ∂y ∂z (C.9)

To find the characteristic variables at the sub-domain interface, we first write the equations as a system of coupled first-order equations: ∂qi,j ∂qi,j ∂qi,j ∂qi,j +A +B +C = 0, ∂t ∂x ∂y ∂z

(C.10)

where qi,j = [ρj cj uj , ρj cj vj , ρj cj wj , pj ]T , with j the medium index and 

  A = c 

0 0 0 1

0 0 0 0

0 0 0 0

1 0 0 0





    ,B = c  

0 0 0 0

0 0 0 1

0 0 0 0

0 1 0 0





    ,C = c  

0 0 0 0

0 0 0 0

0 0 0 1

0 0 1 0



  , 

(C.11)

C.2. Boundary conditions in the Chebyshev PS method

135

where all matrices are symmetric and have four real eigenvalues, where = [0, 0, c, −c]T . The characteristic variables in the x-direction can be found by diagonalizing the matrix A, such that: P T AP = Λ = diag(i )4i=1 ,

(C.12)

with orthogonal matrix P: 

1   P =√  2

0 0 √ − 2 0 √ 0 − 2 0 0

1 1 0 0 0 0 1 −1



  . 

(C.13)

0 We introduce the characteristic variables qi,j = P T qi,j and substitute variables in Eq. (C.10):

P

0 0 0 0 ∂qi,j ∂qi,j ∂qi,j ∂qi,j + AP + BP + CP = 0, ∂t ∂x ∂y ∂z

(C.14)

where

0 qi,j



  =  

−ρj cj vj −ρj cj wj 1 √ (ρj cj uj + pj ) 2 √1 2

(ρj cj uj − pj )



  ,  

(C.15)

which holds since P is independent of t, x, y and z. We then multiply by P T : 0 0 0 0 ∂qi,j ∂qi,j ∂qi,j ∂qi,j + P T AP + P T BP + P T CP ∂t ∂x ∂y ∂z 0 0 0 0 ∂qi,j ∂qi,j ∂q ∂q i,j i,j = +Λ + P T BP + P T CP ∂t ∂x ∂y ∂z ∂q0  ∂q0 0 ∂q4,j 1,j 3,j c  + 2 − ∂y + ∂y = 0  ∂t   0 0 0  ∂q4,j ∂q  2,j + c − ∂q3,j + =0 ∂t 2 ∂z ∂z = 0 0 0 ∂q ∂q ∂q 0 ∂q 3,j   + c ∂x3,j = − √c2 ∂y1,j − √c2 ∂z2,j  ∂t  0  ∂q 0 ∂q 0 ∂q 0  ∂q4,j − c ∂x4,j = √c2 ∂y1,j + √c2 ∂z2,j . ∂t

(C.16)

The first two equations with corresponding zero eigenvalues represent the nonpropagation waves with respect to the x-direction. The third and fourth equations are the one-way wave equations in the -x and x directions respectively with wave speed c. Non-negative values on the right sides of the third and fourth equation represent the effect of oblique incident wave components. With

136


only plane wave propagation in the ±x-direction, these terms vanish and only two homogeneous one-way wave equations remain. To treat an interface between the two different media 1 and 2, suppose that (as in [98]) 1 0 q4,1 (a, t) = √ (ρ1 c1 u1 − p1 ) , 2 1 0 qˆ3,1 (a, t) = √ (ρ1 c1 uˆ1 + pˆ1 ) , 2 1 0 qˆ4,2 (a, t) = √ (ρ2 c2 uˆ2 − pˆ2 ) , 2 1 0 q3,2 (a, t) = √ (ρ2 c2 u2 + p2 ) , 2

(C.17)

0 0 where the characteristic variables qˆ3,2 (a, t) and qˆ4,1 (a, t) are unknown. The char0 0 acteristic variables q3,1 (a, t) and q4,2 (a, t) on the other hand are supposed to be calculated correctly by the Chebyshev PS method, since they represent wave components that have travelled from within the domain toward the boundaries, i.e. they are the outward-propagating characteristic variables. The unknown characteristic variables can be found by the conditions of continuity of pressure and normal velocity across the interface, i.e.: 0 0 0 0 qˆ3,1 (a, t) − q4,1 (a, t) = q3,2 (a, t) − qˆ4,2 (a, t), ρ1 c1 0 0 0 0 qˆ3,1 (a, t) + q4,1 (a, t) = q3,2 (a, t) + qˆ4,2 (a, t) , ρ2 c2

(C.18)

forming two equations for two unknown variables. This yields: 0 qˆ3,1 (a, t)

=

0 qˆ4,2 (a, t) =

2 ρρ21 cc12

0 ρ1 c1 q3,2 (a, t) 1 + ρ2 c2 2 ρρ12 cc21 0 q (a, t) 1 + ρρ21 cc21 4,1

− −

ρ2 c2 ρ1 c1 ρ2 c2 ρ1 c1 ρ1 c1 ρ2 c2 ρ1 c1 ρ2 c2

−1

0 q4,1 (a, t),

−1

0 q3,1 (a, t),

+1 +1

(C.19)

where we recognize plane wave transmission and reflection coefficients. The corrected normal velocity component and pressure fulfilling the boundary conditions can now be found by:

C.3. The k-space method

0 0 qˆ3,1 (a, t) − q4,1 (a, t) √ , 2 0 0 q3,2 (a, t) − qˆ4,2 (a, t) √ p2,corr = , 2 0 0 qˆ3,1 (a, t) + q4,1 (a, t) √ , u1,corr = 2ρ1 c1 0 0 q3,2 (a, t) + qˆ4,2 (a, t) √ u2,corr = . 2ρ2 c2

p1,corr =

137

(C.20)

Note that correcting the pressure and normal velocity values at the boundary as suggested here is the correctional method in the strong form. For an interface between two sub-domains with the same media properties, Eq. (C.19) shows that the unknown characteristic variables are updated by adopting the incoming characteristic variables from the other domain. For the outer boundaries, a first-order approach is to force the ratio between pressure and the velocity component to the boundary to be equal to the plane wave impedance [87]. This could be done as follows for the boundaries at x = −1: p1,corr (−1, y, z, t) = −u1 (−1, y, z, t)ρ1 c1 .

(C.21)

For a more accurate treatment of the outer boundaries, the perfectly matched layer (PML) can be included, as has been done in [177].

C.3 The k-space method Consider solving a three-dimensional acoustic propagation problem for a lossless homogeneous medium using the homogeneous wave equation: 1 d2 − ∆ p(x, y, z, t) = 0, (C.22) c2 dt2 with c the adiabatic speed of sound. Applying Fourier transforms to the x, y and z-direction gives the wave equation in the k-time domain: 2 d 2 2 + c k P (k, t) = 0, (C.23) dt2

138


where P (k, t) =

R∞ R∞ R∞

p(x, y, z, t)e−j(kx x+ky y+kz z) dx dy dz, and with

−∞ −∞ −∞

k = [kx , ky , kz ], k = |k|. The solution to this equation can be written as: P (k, t) = A(k)ejckt ,

(C.24)

with A(k) a complex amplitude. Using this relation, we can write: P (k, t + ∆t) + P (k, t − ∆t) = A(k) ejck(t+∆t) + ejck(t−∆t) = A(k)ejckt ejck∆t + e−jck∆t

(C.25)

= 2P (k, t) (cos (ck(∆t)))

This relation can be rewritten as:

= 2P (k, t) 1 − 2 sin2 (ck∆t/2) .

1 P (k, t + ∆t) − 2P (k, t) + P (k, t − ∆t) = −k 2 P (k, t). 2 2 c2 ∆t sinc (ck∆t/2)

(C.26)

The left-hand side of Eq. (C.26) is similar to a finite-difference representation of the time derivative with the sinc-term as difference. Since Eq. (C.24) is an exact solution of the wave equation in the k-time domain, the time step ∆t in Eq. (C.26) can be chosen arbitrarily long in contrast to a classical finite-difference representation based on a Taylor series expansion. The spatial derivative can be recognized on the right-hand side, where the derivative operator is represented by k 2 . To obtain the solution in the spatial domain, an inverse Fourier transform over the wave number k is applied: 1 p(x, y, z, t + ∆t) − 2p(x, y, z, t) + p(x, y, z, t − ∆t) c2 ∆t2 −1 =F −k 2 sinc2 (ck∆t/2) P (k, t) ,

(C.27)

where F −1 stands for the three-dimensional inverse Fourier transform. Utilizing Eq. (C.27), the solution can be iterated in time by using the solutions at the former two time steps and evaluating the spatial derivatives using the Fourier PS method at the former time step. This method is called the k-space method and is exact for a pressure distribution with compact support in the calculation domain with its spectrum respecting the constraint of two points per shortest wavelength, and for a homogeneous lossless medium. When the coupled firstorder equations are used instead of the single wave equation to solve a problem, first derivatives of the pressure and velocity variables appear, which are calcu-

C.3. The k-space method lated by " # " #! u(x, y, z, t) U(k, t) ∂ = F −1 jkx sinc (ck∆t/2) . ∂x p(x, y, z, t) P (k, t)

139

(C.28)

The advantage of the k-space method over the conventional finite-difference method in time appears for a weakly moving inhomogeneous medium, where the method has been shown to allow for larger time steps [110]. The k-space method has here also been used for the extended Fourier PSTD method where the transform to the full wave number domain is done, as in Paper IV and Paper V.

140


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Paper I

ACTA ACUSTICA UNITED WITH Vol. 94 (2008) 265 – 281

ACUSTICA DOI 10.3813/AAA.918030

A Scale Model Study of Parallel Urban Canyons Maarten Hornikx, Jens Forssén

Applied Acoustics, Chalmers University of Technology, 41296 Göteborg, Sweden. [email protected]

Summary Shielded urban areas are of importance regarding urban citizens’ annoyance and adverse health effects related to road traffic noise. This work extends the existing knowledge of sound propagation to such areas by a scale model study, rather than by model calculations. The scale model study was executed for two parallel urban canyons at a 1 to 40 scale, with a point source located in one canyon. Cases with acoustically hard façades and absorption and diffusion façade treatments were in vestigated. To correct for excess air attenuation of the measurements, a wavelet-based method has been applied. The measurement results in the shielded canyon show that, in contrast to the directly exposed street canyon, the levels and the decay times are quite constant over the length of the canyon. The energy-time curve in the shielded canyon is characterized by a rise time, which can be related to the sound pressure level. The rise times and decays can be explained by separate reflection, diffraction and diffusion processes. A closed courtyard situation enlarges the level difference between acoustically hard façades and applied façade absorption or diffusion treatments at both the directly exposed and shielded side. A comparison between measurements with two different diffusion mechanisms, horizontal and vertical diffusion, reveals that vertical diffusion yields lower levels at the shielded side compared to horizontal diffusion for the investigated situations. PACS no. 43.28.En, 43.38.Ja, 43.58.Bh, 43.58.Gn, 43.60.Hj

1. Introduction Urban acoustics has been an attractive research topic for several decades. The main motive for studying this field has been the high sound pressure levels mainly due to road traffic noise, a large amount of people populating cities are exposed to [1]. Research campaigns aim to reduce these levels, and thereby the related annoyance and adverse health effects. These studies center at investigating, besides the sound excitation part, the sound propagation transfer path between source and receiver: urban sound propagation. Measures that can be taken concerning the transfer path include shielding (e.g. by a barrier), increasing diffusion (e.g. by non-specular reflections at façades) and absorption (e.g. by highly absorptive façade materials). Urban sound propagation models can classically be divided in macroscopic and microscopic models, referring to models covering a part of a city or one city unit, like a street or a square. This second group is useful for cases where short distance traffic is important and to help understanding what happens physically in the less precise macroscopic models. In the 60ies and 70ies, geometrical acoustics models were developed to calculate the sound pressure level in a street canyon with geometrically reflecting boundaries [2, 3, 4]. Lyon noticed the Received 1 February 2007, accepted 17 September 2007.

© S. Hirzel Verlag · EAA

difference between calculations with models based on geometrical reflections and measurement results [5]. Since then, several models have been developed that accounted for non-specular reflections. Bullen and Fricke [6] came with a modal approach accounting for non-specular reflections. Later, the Boundary Element Method was used by Hothersall et al. [7]. Based on the assumption of diffuse reflections, the diffusion equation [8] and a radiosity based model were presented [9]. For narrow street canyons, Iu and Li showed that a coherent image sources model is necessary rather than an energy based model [10]. Other valuable contributions to modelling sound propagation in a street canyon can be found in references [11, 12, 13, 14]. The knowledge of the sound field in a street canyon gained by the models has been supported by measurements. Early measurements include the work of Weiner et al. [4], and Steenackers et al. [15]. Recently, Picaut et al. presented the results of a successful measurement campaign in a single street canyon [16]. Also, scale model measurements have been conducted to show the usefulness of the developed models [10, 16, 17, 18, 19]. In contrast to street canyon situations, studies on squares are more rare. Kang reported on the work done in that field [20]. Among others, Kihlman pointed out that a solution to the urban noise problem lies at the presence of shielded sides like closed inneryards [21]. Access to these shielded areas has been shown to yield a positive effect on health and well-being for inhabitants of dwellings bordering such

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areas [22, 23]. Though, the long term levels or temporal variations in the levels at these shielded urban sides can still yield annoyance and adverse health effects. Therefore, models to predict the levels at shielded urban sides were developed. Research on the sound field at the shielded side is however not as established yet as that for the directly exposed side. Two-dimensional (2-D) wave based models have previously been developed for this purpose: the Finite Difference Time Domain method (FDTD) by van Renterghem and Botteldooren [24] and the Equivalent Sources Method (ESM) by Ögren and Kropp [25]. Also, an approach based on the parabolic equation was suggested [26]. These models predict the sound pressure level at a shielded canyon due to a sound source in a parallel street canyon. Recently, the 2.5-D ESM has been presented [27]. This model calculates the solution of a point source in the geometry of parallel canyons that are invariant along the length of the canyons. The models have previously not been compared with measurement results. Therefore, measurements of sound pressure levels and decay times at the shielded side including varying positions along the canyon will provide a valuable extension to the existing research results. It is in addition interesting to investigate three-dimensional (3-D) effects in order to evaluate the necessity of a 3-D prediction model. A scale model study is very suitable to accomplish such measurements. In contrast to full-scale measurements, most parameters can be controlled there, reducing the level of uncertainty. A scale model study of a point source in the geometry of two parallel canyons was therefore executed and the results are presented here. The used geometry extends the former studied cases of a single street canyon and a single barrier. Situations with acoustically hard façades (further denoted as rigid façades), façades with absorption patches applied and rigid façades with patches causing a non-specular reflection (hereon called diffusion patches) have been examined. The aim of this work is to get new information on the sound field at the shielded canyon by comparing results with directly exposed canyon results, as well as data on the directly exposed and shielded square. The paper is organized as follows. The next section describes the set-up of the scale model, the materials used and the way the obtained data have been analyzed. A wavelet method has thereby been developed to correct for excess air attenuation. In section 3, the accuracy of the scale model results are displayed. In section 4, the sound field at the shielded side is discussed by comparing its properties with those of the sound field at the directly exposed side. The influence of added damping (by façade absorption) and increased diffusion is compared with the case of rigid façades. In section 5, the usability of a 2.5-D calculation model is discussed by comparing canyon results with those from a closed courtyard, and situations with vertically and with horizontally applied diffusion patches.

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Hornikx, Forssén: Urban canyon scale model

2. Scale model set-up and data analysis 2.1. The physical model and used materials The accuracy that can be obtained from scale model measurements increases with the chosen scale. The upper scale is however limited by practical handling of the model and the dimensions of the measurement chamber. The multiple aspects that influence the accuracy of the results – microphone, source, measurement system, excess air attenuation and materials – determined the choice of the scale and frequency range. A scale of 1 to 40 was chosen and a frequency range of the 1/3-octave bands from 4 kHz to 40 kHz (100 Hz to 1000 Hz at full scale) was used. To minimize the effect of background noise and unwanted reflections, the measurements were carried out in the anechoic chamber of Chalmers University of Technology. A rectangular ground floor structure of 3 m×3 m bearing the scale model was placed in this chamber. Figure 1 shows sketches and pictures of the model as well as the used coordinate system. The geometry represents two city canyons with a single point source in one canyon. The source represents a road traffic vehicle and its height of 0 m is a good low frequency approximation for tyre road noise, yet breaks down at higher frequencies and for the engine noise. When considering 1/3-octave band sound pressure levels, differences with real sound source heights are not expected to be large. This is supported by the close results for two source heights in street canyon measurements of Picaut et al. [28]. The second canyon represents a shielded side, without traffic. The height of all three building rows, the street canyon widths and the central building width were chosen to be 0.5 m (20 m at full scale). These dimensions are representative for European city centers [29]. The scale model was as a start designed as a 2.5-D geometry. The source is placed off center in the canyon to prevent exciting one of the first modes at their position of minimum velocity level. Receiver positions were chosen over the height and length and are also shown in Figure 1. By symmetry, results along the y-direction, i.e. over the length of the canyon, have been found by pair-wise averaging over receivers with equal positive and negative y-values. When mentioning dimension, time and frequency in the results, full scale values will be used unless stated differently. Rigid materials were chosen in the study since they yield an extreme case that gives a good understanding of the sound propagation without additional effects as diffusion and absorption. Later, these effects have been added. The ground material consisted of a chipboard plate covered with a thin layer of melamine (a plastic). Plexiglas boxes of 250 mm×250 mm×250 mm (10 m×10 m×10 m in reality) were used for the building blocks. The boxes had an opening at one side, which facilitated the placement of the microphone at the façade level. For the diffusion patches in the canyons, Plexiglas was used as well. Applied absorption patches consisted of 3 mm thick velvety felt. The positions of the absorption patches and diffusion patches are shown in Figures 1b and 1h. Two types

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of diffusion treatments were investigated: cases with horizontally and vertically oriented patches. The impedance of the felt material has been determined using the excess attenuation method as is described in Appendix A1. Figure A1 displays the absorption coefficient for a normal incident sound wave obtained by the method. 2.2. Acquisition system For both frequency and time domain analyses, an impulse response was desirable as the output of the measurements. This was obtained by using the MLSSA system: a measurement program that computes the impulse response from a correlation between the received signal and the known emitted source signal using the MLS technique [30]. The system is useful in the chosen frequency range. An impulse response could also be obtained using a spark type source. The repeatability in the MLS technique as well as its ease of increasing the signal to noise ratio constitute a preferable technique than by using a spark

type source. A sample frequency of 133 kHz was required (3 times the highest frequency of interest) since the MLS power falls by 1.6 dB at 1/3 of the sample frequency. Because its rather flat frequency response and its low directionality in the frequency range of interest, an 1/8 inch condenser microphone, type B&K 4138, was used as the receiver [31]. The microphone was mounted from within a Plexiglas box or from below the ground and positioned with its membrane in the plane of the façade or ground surface to lower the directionality and reduce the disturbing effect of the presence of the microphone and preamplifier on the measurement results. To ensure that the microphone was mounted identically at various positions of each measurement session as well as in the reference measurements, a Plexiglas box (or a ground surface patch) with the microphone mounted in it was moved for each measurement and reference measurement position. A tweeter source with a smooth free field response in the chosen frequency range was chosen for the sound source

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Figure 2. Horizontal (top) and vertical (bottom) directionality of the used sound source. Stars: 10 mm source hole at the 10 kHz 1/3-octave band, Circles: 3 mm source hole at the 40 kHz 1/3octave band. The level increment between gridlines is 5 dB.

[32]. To allow for comparison with 2.5-D ESM calculations, an omnidirectional sound source was desirable. The sound source was mounted in the ground floor below a Plexiglas plate with a circular hole to obtain such a sound field. This hole now acted as the source. A 10 mm hole was used for the 1/3-octave bands 4 up to 12.5 kHz and a 3 mm hole was used for the bands 16 up to 40 kHz. Figure 2 shows the vertical and horizontal directionality of the source with the 10 mm hole at 10 kHz and the 3 mm hole at 40 kHz. Measured results for the full set-up have been corrected for the source directionality if possible. The response of the loudspeaker was another important design parameter. The frequency response should be smooth to get frequency insensitive reference measurements and the impulse response short such that the impulse response of the system is unaffected. The cavity size above the tweeter source was therefore reduced as much as possible and was supplied with damping material to suppress the effect of cavity resonances. The pressure amplitudes were chosen such that non-linear effects could be neglected. 2.3. Data analysis From the MLSSA system, the unfiltered impulse responses have been taken and analyzed in Matlab [33]. All measurement results presented in the next Sections are neither influenced by disturbing room reflections nor by a low signal to noise ratio. Post-processing the unfiltered impulse responses from the scale model requires knowledge on aspects that scale linearly with frequency and, more im-

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Figure 3. The level relative to the free field level of sound propagation over a rigid boundary. The horizontal source-receiver distance is 1 m, the vertical distance varies with angle of incidence. a) The frequency dependent Lre free with an angle of sound wave incidence relative to the normal with the surface of 89◦ , b) The angular dependent Lre free with f = 45 kHz. Solid: A zero admittance boundary, Dashed: A boundary with an admittance due to the boundary layer effect.

portant, aspects that do not scale linearly with frequency. Two effects that do not scale linearly are of relevance here: the finite surface impedance due to the acoustic boundary layer and air absorption. In a thin layer close to a rigid boundary, the acoustic boundary layer, viscosity and heat conduction play a role in the wave field. This wave field can be decoupled in an acoustic, viscous and thermal wave, of which the latter two die out exponentially with the distance from the boundary. The wave fields together satisfy the boundary conditions, and yield a finite admittance, which adds to the admittance of the surface and can be approximated for air at 20◦ C by [34]:

(1) β ≈ 2.01 · 105 (1 − i) f sin2 θ + 0.48 , where θ is the angle of incident sound wave with the normal to the surface. The equation shows the non-linear behaviour in frequency. Equation (1) has with satisfying agreement been verified by scale model measurements [35]. Figure 3 shows the calculated level relative to the free field level with a source and a receiver above a rigid surface with and without including the effect of the acoustic boundary layer. The calculation has been done using a ray model with a spherical wave reflection coefficient. According to equation (1), the effect of the boundary layer



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increases with frequency and angle of incidence. In the audible acoustics range, the boundary layer effect is of negligible importance, but it is not at the high frequencies of the scale model measurements. Since the admittance due to the boundary layer effect is small compared to the admittance of other materials having finite impedance, the boundary layer effect is in principle only of interest in cases of sound propagation over rigid surfaces. It is hard to correct for this finite impedance in post-processing the measured data and the effect of the acoustic boundary layer admittance has not been taken into account here. When using the measurement data for a comparison with model calculations, the effect of the acoustic boundary layer should be included there. The second non-linear effect is the attenuation of sound waves by air. The attenuation is mainly caused by classical absorption (with a power loss proportional to f 2 ), and by relaxation and compression effects of nitrogen and oxygen molecules (with a power loss non-linearly proportional to f ) [36]. A large air absorption may result in a low signal to noise ratio, especially for long impulse responses. Classical air absorption is hard to avoid, whereas changing the constitution of air can influence the second effect. Considering Figure 4 for the relevant frequencies of the scale model (4 kHz–45 kHz), it is worth reducing the relative humidity of the air towards a humidity of 0 %. The facilities to achieve these humidities were not available. The measurements have been corrected for the excess air attenuation. This has been done towards no air attenuation. Several methods to correct for excess air attenuation in scale models were published before (see e.g. [16, 37, 38, 39]). These methods are based on a short-time Fourier transform or by small band filtering the time signal. The drawback of these methods regarding correcting a signal for a factor depending on time and frequency is inherent in the Fourier transform pair: strictly, only a correction in time or fre-

quency is possible. An alternative is to apply the wavelet technique, which transforms a time signal into the timescale plane, where the signal is localized in both scale (which is proportional to the frequency) and time. An ideal localization in frequency and time is however not possible due to limited duration of the frequency-bandwidth product of a signal [40]. A continuous wavelet transform has here been applied to the measured time signal. In the obtained time-scale plane, the signal is corrected for the time and frequency dependent excess air attenuation calculation according to ISO 9613-1:1993. A reconstruction returns the original signal, corrected for the excess attenuation. Appendix A2 describes the details and accuracy of this method. The sound field in the canyons has been evaluated regarding the level and decay time, since both are assumed to be related to the rate of annoyance. Pressure values used to calculate the levels have been obtained by a fast Fourier transform of the impulse responses. Reference receiver positions were located in the street canyon where the first part of the impulse response was used to obtain the reference sound pressure level. There are three types of levels that have been used throughout the paper. The first is the excess attenuation level, i.e. the level relative to the level without any objects, pmeas (f )rmeas , (2) LEA (f ) = 20 log pref (f )rref where pmeas (f ) is the complex pressure at the receiver position, rmeas the line of sight source-receiver distance, pref (f ) the complex pressure at a reference position without obstacles and rref the line of sight source-reference receiver distance. Second, the sound pressure level, relative to the free field level, 2pmeas (f )rmeas . (3) Lrefree (f ) = 20 log pref (f )rref The factor 2 is introduced since the source was located at the ground surface for the reference measurement. Third, the sound pressure level relative to the free field level at one meter from the source, 2pmeas (f ) (4) Lrefree,1m (f ) = 20 log pref (f )rref . This level is introduced to have the same reference level for various receiver positions. To calculate a broad band level, levels have been A-weighted and weighted for a road traffic noise spectrum, 11 Li +Ai +Ci 10 i=1 Bi 10 (5) , LA = 10 log 11 Ai +Ci 10 i=1 Bi 10 where Ai and Ci are the A-weighting and traffic spectrum weighting for the 1/3-octave band i, corresponding to the bands with center frequencies 100 Hz–1000 Hz. Finally, Bi is the 1/3-octave band bandwidth. For the road traffic noise spectrum, a distribution of 90% light and 10% heavy

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vehicles with a speed of 50 km/h has been chosen. The traffic spectra have been taken from Danish measurement data [41]. To investigate the decay properties of the signals, the unfiltered impulse responses have been filtered by a 1/3octave band filter. When the bandwidth-time product of an impulse response is small, the impulse response is sensitive to be influenced by the impulse response of the used filters. Because of its short impulse response, the wavelet transform with the Morlet mother wavelet also used for the correction of excess air attenuation has been used as a 1/3-octave band filter here (with ω0 = 6, see Appendix A2) [42]. The parameter used to quantify the decay of the sound field is the decay time T 10. Section 4.1 describes how the T 10 has been determined in the directly exposed and shielded canyons.

3. Accuracy of the scale model measurements To get insight in the errors of the scale model measurement results, measurement set-ups with a single thin barrier and single thick barrier were studied, see Figure 5. The source was positioned at (9,0,0) and the receiver positions were at the shielded side at positions y = 0 m and y = 40 m, and for two or three different heights according to the coordinate system of Figure 1. Figure 5 shows Lrefree,measured (f ) compared with Lrefree,calc (f ) using the diffraction model of Pierce [43]. Measurement results have been corrected for source and microphone directionality by making use of the source–barrier angle, barrier–receiver angle and the measured source and receiver directionalities. The single

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Figure 6. Lrefree calculated as a function of the distance from a 20 m wide and high barrier in 1/3-octave bands. Source located at (9,0,0), receivers at (x,40,0). Thick: Infinitely long screen calculation [43], Circles: 100 m long screen calculation [44]. Diffraction around the barrier ends is neglected. Upper results are for 100 Hz, lower results are for 1000 Hz.

screen material was equal to the ground material, and Plexiglas boxes were used for the thick barrier. Most of the measurement results agree with the calculated results within 2 dB. The agreement is worst at the lowest 1/3octave bands, where deviations could be caused by possible leakage through the barrier or radiation by the barrier material. Due to the many different angles of sound wave incidence from multiple reflections in the situation of parallel canyons, source and microphone directionality cannot accurately be corrected for. Some deviations can therefore be expected at the highest frequencies. Previous scale model studies for similar geometries and scales also displayed the limited accuracy that can be obtained from scale model studies in 1/3-octave bands [10, 16]. The scale model was set up as a 2.5-D model, yet the length of the buildings in the scale model is finite. Figure 6 shows a comparison of a calculation with a finite barrier, calculated by a time domain formulation by Svensson [44] and an infinite barrier calculated with the model of Pierce [43]. The receiver is in the shadow zone of the barrier and horizontal diffraction around the barrier has not been included. The results show that the effect of the finite barrier length is small and that the error is smallest at higher fre-



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Figure 7. Measured Lrefree,1m results for: a) Directly exposed street canyon, z = 5 m, b) Directly exposed street canyon, z = 15 m, c) Shielded canyon, z = 0 m, d) Shielded canyon, z = 5 m, e) Shielded canyon, z = 15 m.

quencies, and hence shows that the scale model is good a representation of infinitely long parallel canyons.

4. The sound field in the parallel canyons Several authors reported on the behaviour of the sound field in the directly exposed street canyon before (see the Introduction for references). The directly exposed street canyon results of the current scale model study will be discussed along with these literature results and serve as a reference to place the shielded canyon results in perspective. In this section and section 5, the 1/3-octave band levels and decay times T 10 are the tools of comparison. The level results have to be interpreted with the shown measurement accuracy of section 3 in mind. 4.1. Directly exposed street canyon 4.1.1. Sound pressure levels Figure 7 shows the measured Lrefree,1m for all receiver heights and façade types. The plots are a function of the y-distance and frequency. When regarding Figure 7, we notice that the levels in the directly exposed street canyon

decrease over the y-distance of the street canyon. The decrease is approximately 3 dB per distance doubling for the rigid façades case. This is similar to a line source decay, which would be obtained in the limit of a narrow street canyon flanked by high buildings. Over frequency, the levels for the rigid façades case are quite constant, indicating that the absorption coefficients are constant over the frequency. Figure 8 shows the level difference between receiver positions at z = 5 m and z = 15 m for the various cases with rigid façades, façades with applied absorption patches and façades with diffusion patches (horizontally or vertically oriented). The level at z = 5 m is larger than the one at z = 15 m closer to the source due to the importance of the direct field. At a further y-distance from the source, the levels at z = 5 m and z = 15 m are closer. This behaviour holds for all cases and was also found by Kang [9]. The larger difference in the absorption case for a small y-distance is caused by the high frequency results at receiver positions at z = 5 m that have high values (see the left part of Figure 8). Figure 9 shows the level difference over distance and frequency between rigid façades and the other cases. The

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Figure 8. Level differences between receiver positions at z = 15 m and z = 5 m in the directly exposed street canyon (left) and shielded canyon (right). Solid thick: Rigid façades, Dashed: Horizontally oriented façade diffusion patches, Solid thin: Vertically oriented façade diffusion patches, Dotted: Horizontally oriented façade absorption patches.

Figure 9. Levels relative to the levels with rigid façades. a) Levels averaged over all receiver height positions, b) Levels averaged over all positions. Dashed: Horizontally oriented façade diffusion patches, Solid thin: Vertically oriented façade diffusion patches, Dotted: Horizontally oriented façade absorption patches.

Some extra attention will be paid to the calculation of T 10 here. A decay time in general, and the T 10 in particular here, is determined from the decay of the sound field after a sound source, turned on a long time ago, has been turned off. The T 10 can be calculated from a single impulse response using Schroeder’s backwards integration method in order to determine the reverberation time (Schroeder’s curve, [46]). According to the standard ISO 3382, the relation between the decay of the sound pressure level and the time should be a straight line. The T 10 is then the best fitted straight line between the decay of the sound level from -5 dB to -15 dB, extrapolated to -65 dB. When the sound field is diffuse, the decay of the energy-time curve (the squared impulse response) is theoretically a straight line when plotted as a level and equals the shape of the decay of Schroeder’s curve. However, when the sound field is not diffuse, as in our case, the decay of the energy-time curve is not equal to the decay of the sound field as defined for the T 10. Thus, Schroeder’s curve has to be taken from an impulse response to obtain the true decay. A problem arises when the measured impulse response of a nondiffuse field is not complete: the Schroeder curve of such an impulse responses differs in shape from the complete Schroeder curve. The here measured impulse responses have to be truncated since correction for air absorption starts blowing up the background noise at the tail of the impulse response (especially for the highest 1/3-octave bands). Only the part of the impulse response unaffected by this correction has been taken to calculate the T 10. In order to calculate the T 10, the 1/3-octave band Schroeder curves of the truncated measurements have been fitted by Schroeder’s curves calculated by an Image Sources Model (ISM). The absorption coefficient of the façades has been used as the fitting parameter for all types of façades. The number of reflections included in the ISM corresponds to the time length of the truncated measurement. The ISM and measurement curve were fitted up to a distance were the Schroeder curves start to drop quickly, see Figure 10. The ISM with the fitted absorption coefficient has then been used to compute the T 10s by extending the number of reflections up to a value that guaranteed convergence. 4.1.3. Reverberation time T10 results

results have been averaged over z = 5 m and z = 15 m. The level difference increases with distance for all cases, since higher order reflections are of more importance at a larger distance from the source. This was also shown by Kang ([45], Figure 11). The level difference between rigid façades and applied absorption patches over frequency is obvious, clearly governed by the frequency dependence of the absorption material. The level difference between rigid façades and diffusion patches is more subtle. The results agree with previous ones in that the levels with diffusion patches close to the source are higher than the levels with rigid façades, whereas at a larger y-distance, the levels with diffusion are lower than the level with rigid façades ([45], Figure 6).

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Due to the sensitivity of the T 10 calculations in the case with rigid façades (the small amount of damping leads to large T 10s), the averaged T 10s are presented in Figure 11. The average T 10s for z = 5 m and z = 15 m are in Figure 11a shown for the case with rigid walls. The T 10 at z = 15 m is longer than the T 10 at z = 5 m for a small ydistance and vice versa for a large y-distance. This holds for all cases (not shown in the figure here) and is due to the influence of the direct field. At y = 0 m, the level difference between direct and reflected wave fields is larger due to the path length difference for z = 5 m than for z = 15 m. At y = 40 m, this level difference is larger for z = 15 m, since higher order reflections have a lower amplitude due to the finite height of the street canyon.


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Figure 11. T 10s in the exposed street canyon. a) T 10s averaged over 1/3-octave bands for two receiver heights in the rigid façades case. Solid thick: z = 5 m, Solid thin: z = 15 m, b) T 10s averaged over 1/3-octave bands and receiver height positions, c) T 10s averaged over all receiver positions. Solid thick: Rigid façades, Dashed: Horizontally oriented façade diffusion patches, Solid thin: Vertically oriented façade diffusion patches, Dotted: Horizontally oriented façade absorption patches.

Figure 11b shows the T 10s averaged over 1/3-octave bands and receiver height positions. All T 10s increase with distance (also found before in [16] and [45]). The difference in T 10 between rigid façades and diffusion patches is remarkably large when having their small level difference in mind. This emphasizes that the direct contribution determines the levels for a large part in the directly exposed street canyon. Figure 11c shows the T 10s averaged over all positions. The large T 10 difference at low frequencies between the diffusion patches and the rigid façades cases is even


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more remarkable, since the applied diffusion patches have depths that are much smaller than the wavelengths at the low frequencies (λ/15 at 100 Hz). It may be explained by the effect of multiple reflections: the diffuse part of a multiple reflected sound wave is dominating over the specular reflected part even if the diffuse reflection part for a single reflection is small. The frequency dependence of T 10 for the absorption case shows a correlation with the absorption coefficient of the applied felt. The higher T 10 for absorption compared with horizontally oriented diffusion patches over the y-distance in Figure 11b is caused by the dominance in the low frequencies for the absorption case. The frequency dependence for the rigid case (see Figure 11c) again shows the dominance of the direct contribution to the level, since the levels did not show this frequency dependent trend. 4.2. Shielded canyon 4.2.1. Sound pressure levels In the shielded canyon, the level difference over the 40 m length of the canyon is small (see Figure 7); the amplitude reduction due to spherical spreading is compensated for by the smaller diffraction angle. Over frequency, a clear level decrease is visible for all cases. This decrease is, for the rigid façades and diffusion patches, caused by the higher shielding for high frequencies (diffraction effect). The frequency dependence in the absorption case is governed by both diffraction and absorption effects. In Figure 8, we notice that the level differences between z = 5 m and z = 15 m are quite constant. This level difference is largest for the absorption case, since there, the multiple reflected sound waves have more been damped before they reach the lower receiver position. Figure 9 shows the level differences between the case with rigid façades and the other cases. We notice that the level difference over distance is also rather equal. Over frequency, Figure 9b, the level difference is rather equal regarding diffusion as well. For the investigated geometry and source and receiver positions, the level difference in the shielded canyon is larger than the level difference in the directly exposed street canyon. This is caused by the fact that the higher order reflections are of more importance in the shielded canyon. From Figure 9b, we notice that the levels with absorption patches are lower than the levels with diffusion patches for the higher frequencies. The effect of diffusion is here equal to the effect of the applied absorption patches with an absorption coefficient of around 0.2. From calculations with the FDTD method using a coherent line source, van Renterghem et al. found a level difference between rigid façades and façades with an absorption coefficient of 0.33 (for normal wave incidence) of around 20 dB in the shielded canyon [47]. Their canyons had a 10 m x 10 m wide cross section and were 100 m apart. The use of balconies (causing diffuse reflections) could give a gain of 5–10 dB for all 1/3-octave bands compared to a situation without balconies. The level differences found here are of the same order of magnitude.

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Figure 12. Energy-time curves for receiver positions in the directly exposed and shielded canyon, 250 Hz 1/3-octave band. Thick: Receiver position (60, 0, 5), thin: Receiver position (0, 0, 5). The time is shown at the 1 to 40 scale and an integration time of 0.001 s has been used.

Figure 13. Rise time and T 10 results in the shielded canyon. a) Results averaged over 1/3-octave bands and receiver height positions, b) Results averaged over all receiver positions, c) Results averaged over 1/3-octave bands and for y = 0 m. Line indices as in Figure 11c.

4.2.2. Rise time The energy-time curves in the shielded canyon have a different shape than the ones in the directly exposed street canyon: after the first arrival at time t0 , the energy in-

274


creases until a certain time t1 , after which the energy decreases, see Figure 12. The time ts = t1 − t0 is here called the rise time. The rise times have been calculated per measurement position and 1/3-octave band and averaged results are plotted over y-distance, frequency and for three measurement heights: z = 0 m (x = 49 m), z = 5 m and z = 15 m, see Figures 13a, 13b and 13c. The rise times show clear trends: • The rise time is rather constant over the length of the street canyon; • The rise time decreases with receiver height; • The rise time increases with frequency, except for absorption patches. In the case of absorption patches, the rise time does not increase with frequency, due to the increasing absorption coefficients with frequency. The energy-time curve at the shielded side can be explained using the following ray model approach. Consider the cross section in Figure 14. The impulse response can be approximated by the convolution of the impulse responses from the source to diffraction corner 1 and from diffraction corner 2 to receiver point, weighted by a diffraction factor and phase and amplitude correction. The dashed-dotted line in Figure 14 shows the energy-time curve for the sketched case when the diffraction coefficients would have unity values, i.e. no screening. The absorption coefficient of the façades has been set to 5%, and a time integration of 50 ms has been applied. The thin lines in Figure 14 show the diffraction coefficients for 100 Hz and 1000 Hz. The diffraction coefficient increases with distance since the diffraction angle decreases with increasing order of reflection. The product of the dashed-dotted curve and a diffraction coefficient curve gives the shape that we found from our measurements (Figure 12). Depending on the position of the source and receiver (and thus the shape of the diffraction coefficient curve), the shape of the energy-time curve after its maximum value will be concave, straight of convex. Due to the fact that the diffraction coefficient for higher frequencies increases slower with time in the early part, the rise time will be larger for 1000 Hz than for 100 Hz. This is in accordance with the measurements. When a higher absorption coefficient would be assigned to the façades, the diffraction coefficient is unchanged, yet the decays in the separate canyons are faster. The rise time will then be shorter. The rise time is thus correlated to the level: a longer rise time (smaller absorption coefficient) yields a higher level. This is confirmed by the rise time differences between the rigid façades case and the other cases. 4.2.3. Reverberation time T10 Since the levels in the shielded canyon are low compared to the levels in the directly exposed street canyon, the signal to noise ratio in the shielded canyon is lower and the influence of the correction for excess attenuation is larger. This implicated that the useful length of the impulse response was too short to determine the T 10 in the same way as done for the directly exposed canyon. To have an



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Figure 13 shows the averaged T 10 over the length of the canyon. Again, the decay times are quite constant over the length of the canyon. The T 10 for rigid façades is clearly larger than for the other cases, which do not differ much.

20 10

power level (dB)

0 -10

5. Three-dimensional geometrical effects in canyons

-20 -30 -40 -50 -60

0

0.5

1

1.5 time (s)

2

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3

The 2.5-D ESM prediction model is not able to predict 3-D geometrical effects. For urban sound propagation, 3-D geometrical effects like a closed courtyard situation and vertically oriented diffusion patches are typical. These cases were chosen to investigate the difference between a 2.5-D and 3-D geometry. 5.1. Canyon versus closed courtyard 5.1.1. Directly exposed side

Figure 14. Calculated energy-time curves in the shielded canyon for the sketched cross section. Dash-dotted: Energy-time curve with diffraction factor equal to 1, Dashed thin: Diffraction coefficient for 100 Hz, Solid thin: Diffraction coefficient for 1000 Hz, Dashed thick: Energy-time curve with the diffraction coefficient for 100 Hz, Solid thick: energy-time curve with diffraction coefficient for 1000 Hz.

Figure 15. Differences in level and T 10 between closed courtyard and canyons. Levels are for receivers positions with y = 0 m. The T 10 results are 1/3-octave band averaged. First bar: Rigid façades, Second bar: Absorption patches applied, Third bar: Horizontal oriented diffusion patches applied, Fourth bar: Vertical oriented diffusion patches applied.

impression of the decay of the sound field after the rise time, the T 10 has been calculated using the energy-time curve for the lowest 1/3-octave bands (100 Hz–315 Hz).

The results for a source and receiver in a closed courtyard are compared with the ones of the directly exposed street canyon. The closed courtyard has the dimensions 20 m×20 m×20 m, see Figure 1. Figure 15 shows the level differences LA,courtyard − LA,canyon in dB(A) and the decay time differences T 10courtyard − T 10canyon at y = 0 m for z = 5 m and z = 15 m for the various cases. Schroeder’s curve for a street canyon and closed courtyard are shown in Figure 10. The number of reflections arriving at the receiver within a certain time slot is almost constant over time in the street canyon, whereas the number increases with time in the closed courtyard. The energy decay is thus much slower in the closed courtyard. The concave decay of the sound energy over time in a street canyon was also shown by Steenackers [15] and Kang [9]. The Schroeder curve difference between street canyon and closed courtyard is retrieved in the level difference and the averaged T 10 difference, also shown in Figure 15. The closed courtyard T 10s for rigid façades are similar to the calculated reverberation time by Kang for a larger square 50 m×50 m×20 m [20]. Since the higher order reflections are of more importance in the closed courtyard case than in the street canyon case, the level differences between the rigid façades case and the other cases increase compared to the street canyon case. The level difference is larger for z = 15 m, whereas the averaged T 10 difference is largest for z = 5 m. 5.1.2. Shielded side At the shielded side, a closed courtyard was built as well. The directly exposed side was still kept as a street canyon, see Figure 1h. All receiver positions were at y = 0 m. Results can be found in Figure 15 and show that also at the shielded side, the closed courtyard levels are higher than the canyon levels. It is however much less obvious than at the directly exposed side. Compared to a canyon situation, a closed courtyard situation increases the sound pressure level reduction due to absorption and diffusion relative to the rigid façades case. The negative values in the

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absorption case can be concluded to be a measurement error. Whereas the rigid façades case displays an increase in level and T 10 for the courtyard case, the differences for the other cases are negligible concerning T 10 and within the range of uncertaincy concerning the level. Note that the difference between street canyon and closed courtyard at the shielded side is not studied for a source position different from y = 0 m. The shielding due to the diffraction effect would then increase for the shielded closed courtyard compared to the canyon. On the other hand, the rays are horizontally trapped once the sound energy has reached the closed courtyard. 5.2. Vertical diffusion versus horizontal diffusion 5.2.1. Directly exposed side The second effect not captured by 2.5-D model is when vertically oriented façade diffusion patches are applied, creating horizontal diffusion. For such a case, the same amount of patches was applied as in the case of horizontally oriented façade diffusion patches (40% of the façades were covered), yet now they were mounted vertically oriented. The effect of horizontal diffusion (vertically oriented patches) is investigated by comparing the results with the vertical diffusion case (see Figures 8 and 9). A similar level difference between z = 5 m and z = 15 m as well as a similar level difference with the rigid façades case is found for both types of diffusion. The degree of diffusion could be decisive for this similarity. The T 10 with vertically oriented diffusion patches is higher than with horizontally oriented patches. This can be attributed to the type of diffusion mechanism: when the elements are oriented horizontally, more energy is scattered out of the street canyon than in the case with elements oriented vertically. The T 10 is therefore higher in the latter case. This is even more obvious in the closed courtyard case, see Figure 15, where the difference in T 10 is larger for the vertically oriented patches than for the horizontally oriented ones. 5.2.2. Shielded side At the shielded side, the levels with vertically oriented patches are higher than with horizontally oriented patches, both over distance and frequency (see Figure 9), whereas the level difference between z = 5 m and z = 15 m is rather similar (see Figure 8). The effect of the level difference is also indicated in the rise time plots, Figure 13, where the rise times for horizontal diffusion are longer than the rise times for vertical diffusion. Vertical diffusion is thus shown to be more effective in reducing levels at the shielded side than horizontal diffusion. It can be explained by the fact that vertical diffusion (horizontal patches) leads to a more diffuse sound field in the x-z plane that horizontal diffusion. In the shielded closed courtyard, the level difference between the two types of diffusion is similar to the canyon case.

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6. Conclusions A 1 to 40 scale model of parallel urban canyons was built. It has been shown that this model represents a twodimensional (2-D) situation of infinitely long canyons. This geometry, with a point source, is called 2.5-D if it is invariant along the canyons. The measurements have been made for rigid façades, horizontally oriented façade absorption patches and horizontally oriented façade diffusion patches. A monopole source was located in one street canyon and receiver positions were located over the canyon length of both the directly exposed street canyon and the parallel shielded canyon. Since, according to the authors’ knowledge, this is the first extensive measurement campaign for a shielded canyon, the measurement results also give new insight in the sound field there. 6.1. Sound pressure levels in the shielded courtyard The sound pressure level and the reverberation time T 10 are, in contrast to the values at the directly exposed side, quite constant over the length of the canyon, indicating the importance of taking into account distant sources when predicting noise levels. This holds for all investigated cases. The level differences between rigid façades and applied absorption or diffusion patches are larger in the shielded canyon than in the directly exposed street canyon. It should be noted however, that using rigid façades for the reference case can over predict the effect of noise mitigation measures compared to using real existing building façades. 6.2. Decay times in the shielded courtyard Energy-time curves have been obtained, which show a distinct difference between the directly exposed and shielded canyon. Whereas the energy-time curve in the directly exposed street canyon decreases after the first arrival, the curve in the shielded canyon first increases to a maximum, and thereafter decreases. The time from the first arrival to the maximum level, the rise time, has been shown to be related to the level. The rise time is also constant over the length of the canyon, yet decreases with receiver height and increases with frequency, which has also been explained by separate reflection and diffraction processes. 6.3. Three-dimensional geometrical effects in canyons Besides canyons, closed courtyard situations have been investigated and compared to canyon results. A clear increase of the level and T 10 is visible at the directly exposed side for the closed courtyard. The increase is present, yet less obvious at the shielded side. With the rigid façades case as a reference, absorption and diffusion façade treatments at a shielded closed courtyard reduce the levels more than in a canyon. Only one source position was however used for the closed courtyard situations. Since existing prediction methods are 2.5-D (or 2-D) and only can capture vertical diffusion effects, a comparison between



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horizontal and vertical diffusion has been made from the measurement results. In the directly exposed street canyon, the levels are similar whereas the T 10 with horizontal diffusion is slightly larger than the T 10 with vertical diffusion. At the shielded side however, the levels with vertical diffusion are significantly lower than with horizontal diffusion. The rise times indicate this difference as well. When using a 2.5-D model for urban canyons, one thus has to keep in mind that façade treatments could be more effective in a closed courtyard situation than in the calculated canyon case. Also, the effect of diffusion could be different from reality, since no horizontal diffusion can be modelled with the 2.5-D model. Acknowledgements The work behind this paper has been funded by the Swedish Foundation for Strategic Environmental Research (MISTRA). The authors would like to thank Wolfgang Kropp for his ideas and fruitful discussions.

Appendix A1. Material impedances determined by the excess attenuation method The impedance of the used materials in the scale model study has been determined by the excess attenuation method. The excess attenuation was obtained from the free field complex frequency response of a monopole sound source and the complex frequency response of the same geometrical source-receiver configuration, but now in the presence of the material under investigation. The source as mounted in the ground floor was used, creating approximately an omnidirectional sound field over an half-sphere (see section 2.2). The measured excess attenuation level is pmat,meas (f ) , (A1) LEA,meas = 20 log pref,meas (f ) where pmat,meas is the measured complex pressure in the presence of the material under investigation and pref,meas is the measured complex pressure in the absence of the material under investigation. The excess attenuation levels are at the other hand calculated using pref,calc = e ikr1 /r1 and pmat,calc = e ikr1 /r1 + Qe ikr2 /r2 , where k is the wavenumber, r1 is the direct source-receiver distance, r2 the image source-receiver distance in the presence of the material under investigation and Q(f, θ, Zn ) the spherical wave reflection coefficient implemented according to Chien and Soroka [48]. This is a function of the frequency f , angle of sound wave incidence θ and the normalized surface impedance Zn , Zn =

Zmaterial , cos(θm )Zair

(A2)

where θm is the propagation angle in the material relative to the normal of the surface and Zmaterial and Zair are specific impedances. Since the felt material is mounted on a

hard backed surface (Plexiglas), the impedance model becomes [49]

(A3) Zn,hardback = iZn cot k1 d cos(θm ) , where k1 = kn is the wave number in the felt layer, with n the refraction index, θm = arccos 1 − 1/n2 + cos2 (θ)/n2 , and d the thickness of the felt layer. Felt is a porous material. Several impedance models have been developed for porous materials with a varying number of physical parameters included. An extensive model, the four material parameter model developed by Attenborough [50], has been used here. This model includes material flow resistivity, porosity, tortuosity and the standard deviation of pore size distribution. The normalized surface impedances Zn are derived by a minimization procedure of the following function using the Nelder-Mead Simplex Method, which is a direct search method that does not make use of gradients [51], X=

i=1125

LEA,meas (i) − LEA,calc (i).

(A4)

i=89

The algorithm is an optimization over the whole frequency region of interest where the set of four parameters is fitted to get the minimum value of X. Figure A1a shows the measured excess attenuation and the calculated excess attenuation using the fitted results. In Figure A1b, the found absorption coefficient for a wave incident in normal direction to the felt material is plotted. The third part of the Figure displays the way the microphone was mounted during the measurements. For the Plexiglas and the ground material, the absorption coefficient for normal wave incidence was found to be lower than 0.05 for all frequencies of interest.

A2. Excess air attenuation correction using the continuous wavelet transform method A2.1. The continuous wavelet transform Similar to the Fourier transform, the wavelet transform decomposes a signal onto a set of functions, a basis. In the wavelet transform, these functions all stem from a mother function, the mother wavelet. In contrast to the Fourier transform, the time signal is not transformed to separate frequencies, yet to coefficients centered on a certain time and scale (related to the frequency). The continuous wavelet transform is used, which has no restriction of scales. The continuous wavelet transform (CWT) [52] ∞ 1 t − a dt, (A5) F (a, s) = p(t) √ ψ s s −∞ where t is the time, p(t) the time signal, a the translation time, s the scale factor, F the wavelet coefficients and

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with :

k=0

2πk/(Nδt) for k ≤ N/2 −2πk/(Nδt) for k > N/2 P (k) = Fourier transform of the time signal p(t). ωk =

Figure A2a shows a typical measurement signal in an urban street canyon with source and receiver in the same street canyon. The signal displays the multiple reflections in a street canyon. Figure A2b shows the wavelet transformed signal in the time-frequency plane. Figure A2c shows a part of the energy-time curve of the signal and the wavelet representation, a weighted summation over all scales [53]. Figure A2d shows a part of the Fourier and wavelet power spectrum, the summation of the squared wavelet coefficients over time. Both Figures A2c and A2d show that the wavelet power spectrum and wavelet energytime curve are a smoothed representation of the real power spectrum and energy-time curve. A2.2. Excess air attenuation correction Since the energy content at a certain position in the timescale plane does not have a perfect localization, it is not possible to exactly correct for the excess air attenuation. The correction is calculated as a factor according to the

278

a (-)

(dB) EA

-5

(A6)

where η is the non-dimensional time and ω0 the nondimensional frequency. The coefficient ω0 determines the width of the localization of the transformed signal in the time and the frequency domain; the larger ω0 , the worse the time localization yet better the frequency localization. A value of 20 has been used here. The Morlet wavelet is a complex function, which implies that phase and amplitude information of the signal is kept in the transformed domain. Since the representation in the transformed domain entails that spectral energy is also present for wavelet coefficients outside the relevant frequency range, the smallest and largest scales used in the transform should be outside the frequency range of the signal. It is convenient to use logarithmically spaced scales sj = s0 2jδs , where j are integer numbers and a value of 0.1 will be used here for δs. The continuous wavelet transform of a measured signal is calculated in the frequency domain, where the convolution is replaced by a multiplication. The wavelet coefficients are then obtained by an inverse Fourier transform of the product of the Fourier transformed signal and daughter wavelet [53], N−1 2πs ψ (sωk )e iωk nδt , (A7) P (k) F (a, s) = δt

0

0.6 0.4 0.2

-15

125

250 500 frequency (Hz)

0

1000

125


1000

Figure A1. Excess attenuation of 3 mm velvety felt. Source position (x,y,z)=(9,0,0), receiver position (0,0,15) and felt at (0,0,46). Coordinates according to Figure 1. a) Fitted excess attenuation curve. Solid line: Fitted results, Circles: Measured, b) Obtained absorption coefficients for normal wave incidence, c) Close up of the microphone position during the impedance measurement.

0.06 40

0.04

frequency (kHz)

,

0.8

0.02 0 -0.02 -0.04 -0.06

a)

30 20 10

0

0.01 0.02 time (s)

0.03

2.5

c)

0.01 0.02 time (s)

0.03

10 frequency (kHz)

15

10 0

2 1.5 1 0.5 0 0

0

b)

magnitude (dB)

e

1

10

-10

amplitude (Pa)

ψ (η) = e

iω0 η −η02 /2

15

5

L

the overbar denotes the complex conjugate. The mother wavelet ψ ((t − a)/s) is thus scaled and translated in time during the transform: the daughter wavelets. The coefficient before the wavelet function s−1/2 normalizes the energy content of the daughter wavelets. The mother wavelet used here is the Morlet wavelet, which is one of the most common wavelets,

amplitude (mPa 2 )


-10 -20 -30 -40

1

2 time (ms)

3

-50

d)

5

Figure A2. Example of a signal transformed by the wavelet transform, its energy-time representation and its power spectrum representation. The time is at the 1 to 40 scale.

International Standard ISO 9613-1:1993(E) with a frequency and time corresponding to the center frequency and time of the wavelet coefficient. Since the correction is linear with the energy in time, the time correction is a power of the correction factor per time sample c. The corrected wavelet coefficients, Fcor (nδt, s), become Fcor (nδt, s) = F (nδt, s)cn (s).

(A8)



a)

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20

c)

e)

3 2

0

1

-20 0

0 -40

-1

-60

0

0.1

b)

-80

-2 -3 1

2

3

1.4

4

1

4 75

d)

2

3

4 4

f)

1.2 1 0.8

0

70

0.6 0.4 0.2 0

0.1

0

1

2

3

4

65 4

Figure A3. Synthesized signal with a frequency content corresponding to a scale model measurement. a) Synthesized signal, b) Damped signal, c) Frequency content of undamped signal, d) Air attenuation applied to the signal, e) Phase difference between signal and damped signal, corrected by the wavelet method, f) Levels of the the signal in a) (left pair), levels of the signal in b) (right pair). First and third bar represent the levels of the original signals, the second and fourth bar correspond to the level of the reconstructed signal using the wavelet method.

A2.3. Reconstruction Since the continuous wavelet transform is not based on an orthogonal basis, the transform is redundant and a reconstruction is not perfect. Farge has however shown that the redundancy of the CWT allows for reconstructing the signal using a completely different wavelet function, of which the delta function is the easiest [54]. The reconstruction will be an approximation of the original signal. The used reconstruction equation is [53] J

δt δj Fcor (a, s) , (A9) f (nδt) = Cδ δsj j=0

where J the total number of scales used, δj the weighting of scales, Cδ the reconstruction factor of a delta-function

and δt/δsj the removal of the energy normalization. The accuracy of the reconstruction is displayed for a synthesized signal created from the frequency content of a measured signal. Figure A3a, A3b and and A3c show the created undamped and damped signal and its frequency content which lies within the frequency content of a measurement signal in the scale model. The signal was composed by frequencies with a random phase. Air attenuation has in figure A3b been applied for T = 20o C and RH = 30%, which are typical conditions during scale model measurement. The calculated attenuation loss by air attenuation is

shown in Figure A3d. Figure A3e shows the phase error between original and reconstruction of the damped signal. Finally, Figure A3f shows the levels of the signals. The first and second bar show the level of the undamped signal of Figure A3a and the damped signal corrected by the wavelet method. The third and fourth bar denote the level of the damped signal of Figure A3b and the level of the damped signal when a wavelet forward and inverse wavelet transform have been applied without a correction. The introduced errors are small. Note that the presented way to correct for excess attenuation does not pretend to be superiour over the former presented methods, since a perfect time-frequency localization is not possible. The method does however have the advantage over the former methods that the time-frequency resolution varies with frequency and is generated implicitly with the method (no choices of filter or window widths have to be made). References [1] D. Stanners, P. Bordeau (eds.): Europe’s environment. European Environment Agency, Copenhagen, Denmark, 1995. [2] H. G. Davies: Noise propagation in corridors. J. Acoust. Soc. Am. 53 (1973) 1253–1262.

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[3] K. Lee, H. G. Davies, R. H. Lyon: Prediction of propagation in a network of sound channels, with application to noise transmission in city streets. Report, Acoustics and Vibration Laboratory, Massachusetts Institute of Technology, 1974.


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[36] E. M. Salomons: Computational atmospheric acoustics. Kluwer Academic Publishers, 2001. [37] H. A. Akil, D. J. Oldham: Digital correction for excessive air. Absorption in acoustic scale models. Proc. of the Institute of Acoustics 16 (1994) 525–536. [38] H. A. Akil, D. J. Oldham: Digital compensation for excess air absorption in acoustic scale models. Proc. Euro Noise, Lyon, France, 1995, 73–78. [39] J. D. Polack, A. H. Marshall, G. Dodd: Digital evaluation of the acoustics of small models: the MIDAS package. J. Acoust. Soc. Am. 85 (2001) 185–193. [40] R. N. Bracewell: The Fourier transform and its applications. McGraw-Hill, 2000.

[19] W. R. Schlatter: Sound power measurements in a semiconfined space. M.Sc. Thesis, Department of Mechanical Engineering, MIT, 1971.

[41] H. Jonasson, S. Storeheier: Nord 2000: New Nordic prediction method for road traffic noise. SP Rapport 2001:10, Borås, Sweden, 2001.

[20] J. Kang: Numerical modeling of the sound fields in urban squares. J. Acoust. Soc. Am. 117 (2005) 3695–3706.

[42] S. K. Lee: An acoustic decay measurement based on timefrequency analysis using wavelet transform. J. Sound Vib. 252 (2002) 141–153.

[21] T. Kihlman: National action plan against noise. Summary of "Handlingsplan mot buller" (in Swedish). Allmänna Förslaget, Stockholm, Sweden, 1993. [22] A. Skånberg, E. Öhrström: Adverse health effects in relation to urban residential soundscapes. J. Sound Vib. 250 (2002) 151–155. [23] T. Kihlman, E. Öhrström, A. Skånberg: Adverse health effects of noise and the value of access to quietness in residential areas. Proceedings of Internoise, Dearborn, USA, 2002, paper 484.

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[43] A. D. Pierce: Diffraction of sound around corners and over wide barriers. J. Acoust. Soc. Am. 55 (1974) 941–955. [44] U. P. Svensson: EDBtoolbox. Matlab implementation of the algorithms in [55]. [45] J. Kang: Sound propagation in street canyons: Comparison between diffusely and geometrically reflecting boundaries. J. Acoust. Soc. Am. 107 (2000) 1394–1404. [46] M. R. Schroeder: New method of measuring reverberation time. J. Acoust. Soc. Am. 37 (1965) 409–412.


[47] T. Van Renterghem, E. Salomons, D. Botteldooren: Parameter study of sound propagation between city canyons with a coupled FDTD-PE model. Appl. Acoust. 67 (2006) 487– 510. [48] C. F. Chien, W. W. Soroka: Sound propagation along an impedance plane. J. Sound Vib. 43 (1975) 9–20. [49] J. F. Allard: Propagation of sound in porous media: modeling sound absorbing materials. Chapman & Hall, London, 1993. [50] K. Attenborough: Models for the acoustical properties of the air saturated granular media. Acta Acustica 1 (1993) 213–226. [51] J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright: Convergence properties of the nelder-mead simplex method


Vol. 94 (2008)

in low dimensions. SIAM J. Optimization 9 (1998) 112– 147. [52] A. Bultheel: Wavelets with applications in signal and image processing. Course material University of Leuven, Belgium, 2003. [53] C. Torrence, G. P. Compo: A practical guide to wavelet analysis. Bulletin of the American Meteorological Society (1998) 61–78. [54] M. Farge: Wavelet transforms and their applications to turbulence. Ann. Rev. Fluid Mech. 24 (1992) 395–457. [55] P. U. Svensson, R. I. Fred, J. Vanderkooy: An analytic secondary source model of edge diffraction impulse responses. J. Acoust. Soc. Am. 106 (1999) 2331–2344.

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ACUSTICA

Notices and Announcements

Erratum

DOI 10.2392/AAA.918074

Acta Acustica united with Acustica 94 (2008) 265–281

A Scale Model Study of Parallel Urban Canyons Maarten Hornikx, Jens Forssén

ys = 0 m

0.48 m

ys = 0 m

yr = 0 m / 40 m

5m (1) source

11 m

Due to a misunderstanding a part of Figure 5 was omitted in the article. Please find the complete figure on the right-hand side.

y r = 0 m / 40 m

receivers (2)

20 m

5m (1)

10 m 5m

15 m

receivers (3) (2)

source

11 m

20 m

9m

10 m 5m

11 m

(3)

We kindly ask our readers to take note of this. S. Hirzel Verlag

Figure 5. Lrefree for a thin (left) and thick (right) barrier in 1/3octave bands. Circles: Measurements, Solid lines: Pierce diffraction theory. For positions 2 (and position 3 in the thick barrier case), the direct diffracted and ground reflected waves have been summed.

Doctoral Thesis Abstracts Tobias Lentz: Binaural Technology for Virtual Reality. Supervisor: Michael Vorländer. Publisher: Logos Verlag Berlin, 2008. ISBN 978-3-8325-1935-3 The generation and use of artificial virtual environments is gaining more and more importance. Virtual environments are used in a variety of fields such as product design or evaluation of prototypes. Moreover they turned out to be very effective when it comes to the visualization of complex data sets. In the past, investigations were focused mainly on the visual reproduction technique to present geometrical data in a three-dimensional way (stereoscopic representation). However, the human perception consists not only of visual input but is based on a number of sensations and thus it would be worthwhile to create a multi-modal and interactive virtual environment. In this thesis, first of all the techniques required to include the acoustic component into a virtual environment are described and assessed. Furthermore the implementation of a software system is described, which takes advantage of these techniques to cre-

© S. Hirzel Verlag · EAA

DOI 10.2392/AAA.918075

ate complex acoustical scenes. This system features spatially distributed sound sources which are utilized to create in real time an environment that is as authentic as possible. The system, mentioned above, is based on the binaural technique (binaural: “concerning both ears”) and aims at reproducing a sound for the ears of the user, that is equivalent to the sound in an original surrounding. It is essential to set up a sound field that is as exact as possible, to achieve a simulation with the highest degree of authenticity. This compromises a description of the source, including its relevant angle-, distance- and timedependent radiation, the sound distribution in the virtual scene (room acoustics), the hearing-related consideration of all sound field components, as well as the exact reproduction of the artificial sound at the ear of the user. Therefore, the focus of the thesis is also put on the reproduction technology. In this context, an approach for dynamic crosstalk cancellations is presented, which enables a loudspeaker-based reproduction for binaural acoustical imaging instead of using headphones. Filters are necessary to ensure the required channel sep-

641

Errata The following correction need to be observed regarding Paper I: ◦ Page 269, Eq. (5): 

11 P

Li +Ai +Ci 10

Bi 10   LA = 10 log  i=111 Ai +Ci  P Bi 10 10 i=1

should be replaced by: 

11 P

Li +Ai +Ci 10

10   i=1 LA = 10 log  11  P Ai +Ci 10 10 i=1





  , 

  . 

Paper II

The 2.5-dimensional equivalent sources method for directly exposed and shielded urban canyons Maarten Hornikxa兲 and Jens Forssén Department of Civil and Environmental Engineering, Division of Applied Acoustics, Chalmers University of Technology, SE-41296 Göteborg, Sweden

共Received 1 March 2007; revised 16 August 2007; accepted 20 August 2007兲 When a domain in outdoor acoustics is invariant in one direction, an inverse Fourier transform can be used to transform solutions of the two-dimensional Helmholtz equation to a solution of the three-dimensional Helmholtz equation for arbitrary source and observer positions, thereby reducing the computational costs. This previously published approach 关D. Duhamel, J. Sound Vib. 197, 547–571 共1996兲兴 is called a 2.5-dimensional method and has here been extended to the urban geometry of parallel canyons, thereby using the equivalent sources method to generate the two-dimensional solutions. No atmospheric effects are considered. To keep the error arising from the transform small, two-dimensional solutions with a very fine frequency resolution are necessary due to the multiple reflections in the canyons. Using the transform, the solution for an incoherent line source can be obtained much more efficiently than by using the three-dimensional solution. It is shown that the use of a coherent line source for shielded urban canyon observer positions leads mostly to an overprediction of levels and can yield erroneous results for noise abatement schemes. Moreover, the importance of multiple façade reflections in shielded urban areas is emphasized by vehicle pass-by calculations, where cases with absorptive and diffusive surfaces have been modeled. © 2007 Acoustical Society of America. 关DOI: 10.1121/1.2783197兴 PACS number共s兲: 43.28.Js, 43.20.El, 43.28.En 关VEO兴

I. INTRODUCTION

Sound propagation in urban areas has been studied extensively during the last few decades. A main key in urban sound propagation research at a scale up to hundreds of meters is, according to the number of publications, believed to lie in understanding the sound field in a single urban canyon.1–8 Studies were also addressed to sound propagation in several streets9,10 and from one street to a shielded canyon, where sound propagates over a single building block.11–14 The last category is of special interest here. Sound pressure levels at sides of dwellings that are directly exposed to road traffic noise are difficult to reduce to the desired level regarding annoyance and adverse health effects. Therefore, access to relatively quiet areas became a strategy towards offering inhabitants of city centers a healthy living environment regarding road traffic noise.15 Such areas are often shielded from direct road traffic noise, e.g., courtyards. For this purpose, accurate models have been developed to understand the behavior of sound propagation from a directly exposed urban street canyon to a parallel canyon shielded from direct road traffic noise.12,14 The idealized situation of parallel urban canyons can be considered as typical for large city centers and the geometry extends the widely studied cases of single and double noise screens. Scattering and absorption of sound waves at the façades of the canyons were shown to play an important role in weakening the sound propagation from the directly exposed to the shielded canyon, and meteorological effects, such as refraction and scattering by atmospheric turbulence, were found to influence the sound pressure levels in

a兲

Electronic mail: [email protected]

2532

J. Acoust. Soc. Am. 122 共5兲, November 2007

Pages: 2532–2541

the shielded canyon significantly.12,14 The results of these studies were based on the solution of the two-dimensional 共2D兲 wave equation in the time domain12 and frequency domain.14 These solutions therefore represent the response from a coherent line source. An incoherent line source is, however, a better model for traffic on a straight road. For situations with a single screen, the attenuation of a coherent line source was shown to be larger than that of an incoherence line source.16,17 It is therefore relevant to investigate whether these differences also appear in the parallel canyons case. Possessing a point source solution offers the possibility to calculate the response from an incoherent line source. A pass-by of a single vehicle to compare various geometrical situations can then also be modeled. A solution of the threedimensional 共3D兲 Helmholtz equation 共or wave equation兲 in the situation of canyons is computationally still expensive, however. When assuming the length of the streets as infinite and the cross section invariant over this length, a transform of solutions of the 2D Helmholtz equation to a solution of the 3D Helmholtz equation is possible. The approach, which is called a 2.5D method, was first presented by Duhamel16 and used later by Salomons et al.,18 Duhamel and Sergent,19 Jean et al.,17 Godinho et al.,20 and Defrance and Jean,21 among others. The method has been applied to compute point source solutions for various types of noise screens. Methods like the boundary element method 共BEM兲 and the image sources method 共ISM兲 were used to calculate the 2D solutions. In this paper, the transform is applied to obtain a point source solution in the case of parallel urban canyons. Road traffic noise is modeled by a sound source in one canyon, called the directly exposed canyon, and the observer

0001-4966/2007/122共5兲/2532/10/$23.00

© 2007 Acoustical Society of America

FIG. 1. Geometry of parallel urban street canyons with the used coordinate system. Arbitrary source and observer positions are shown: 共a兲 cross section; and 共b兲 top view.

positions are located in the same canyon or in a parallel shielded canyon. The equivalent sources method 共ESM兲, previously developed to model sound propagation in geometries with urban canyons in the frequency domain,14 is used to generate the 2D solutions. No atmospheric effects are included in the calculations of this study. This paper is organized as follows. In Sec. II, a transform of solutions of a 2D Helmholtz equation to that of a 3D Helmholtz equation is presented. Compared to previous work, including atmospheric absorption in the transform is novel. The relevant sources of error in the transform are investigated for the free field solution. The accuracy of the method is further discussed by studying the case of a point source and an observer between two semi-infinite walls, an extreme case of a narrow street canyon. Then, the transform is applied to the ESM for the parallel canyons case with observer positions both in the directly exposed and in the shielded canyon. The transformed ESM is used for calculations in Sec. III. In Sec. III A, the pass-by of a single vehicle is modeled for a single screen and the parallel canyon situation, where several façade properties in the canyons are studied. Section III B focuses on the difference between the excess attenuation of a coherent and an incoherent line source for the various cases of urban street canyons. II. THE 2.5D ESM

Three-dimensional sound propagation in an unbounded medium due to a mono frequency monopole with unity source strength is described by the Helmholtz equation. The time convention ei␻t has been adopted with ␻ the angular frequency and t the time. The effect of air absorption is included by an imaginary part to the wave number. In Cartesian coordinates 共x , y , z兲, we write

冋

册

⳵2 ⳵2 ⳵2 + + + 共k − i␣共f兲兲2 p ⳵x2 ⳵ y 2 ⳵z2 = − ␦共x − xs兲␦共y − y s兲␦共z − zs兲,

共1兲

where p共x , y , z , k兲 is the complex pressure due to the monopole source located at 共xs , y s , zs兲, and ␣共f兲 the frequency dependent air absorption coefficient. Applying a Fourier transform in the y direction, we obtain

冋

册

⳵2 ⳵2 + + 共共k − i␣共f兲兲2 − k2y 兲 P ⳵x2 ⳵z2 = − eikyys␦共x − xs兲␦共z − zs兲,

共2兲

with P共x,ky,z兲 =

冕

⬁

p共x,y,z,k兲eikyy dy,

−⬁

A. Transform from a 2D to a 3D solution of the Helmholtz equation

The geometry under study is shown in Fig. 1. A point source and an observer are located in a situation of parallel urban canyons. The observer may be placed in either the directly exposed or the shielded canyon. Except for the point source, the geometry is assumed to be invariant in the y direction: a 2.5D geometry. The purpose is to solve the 3D wave equation for this geometry in the frequency domain in a fast way. Solving a 3D equation for a 2.5D geometry is here further called a 2.5D method. It is done by applying a Fourier transform to the 3D Helmholtz equation with respect to the y coordinate. The resulting equation is set equivalent to a 2D Helmholtz equation, which is then solved efficiently. To obtain the desired 2.5D solution, an inverse Fourier transform over wave number ky is finally applied. J. Acoust. Soc. Am., Vol. 122, No. 5, November 2007

where k = 冑k2x + k2y + kz2. This equation is equivalent to the 2D Helmholtz equation

冋

册

⳵2 ⳵2 + 共K − i␣2D共f,ky兲兲2 q = − ␦共x − xs兲␦共z − zs兲, 2 + ⳵x ⳵z2 共3兲

where q共x , K , z兲 = Pe−ikyys, K = 冑k2 − k2y , and ␣2D共f , ky兲 = i共冑共k − i␣共f兲兲2 − k2y − K兲. The solution to Eq. 共1兲 is obtained by solving Eq. 共3兲 and applying an inverse Fourier transform with respect to ky. For future use, we write the transform pair p共x,y,z,k兲 =

1 2␲

冕

⬁

q共x,K,z兲eiky共ys−y兲 dky ,

共4兲

−⬁

Hornikx and Forssén: 2.5D calculation method for urban canyons

2533

FIG. 2. Integration paths in the complex plane for the wave numbers ky and K in Eq. 共4兲 for unbounded wave propagation. 共--兲 Integration along ky with singularities at ky = −k and ky = k; 共··兲 integration along K with a singularity at K = 0. Integration is not done around the singularities.

q共x,K,z兲 =

冕

⬁

p共x,y,z,k兲e−iky共ys−y兲 dy.

共5兲

−⬁

According to Eq. 共4兲, the 2D solutions q共x , K , z兲 over all wave numbers ky are needed to calculate p共x , y , z , k兲. Integration over all wave numbers ky implies an integration path over a part of the real and imaginary axis of K for a certain wave number k 关see Fig. 2兴. The solution of Eq. 共3兲 in an unbounded domain, is

q共x,K,z兲 =

冦

i − H共2兲共Kr2D兲 K2 艌 0 4 0 1 共1兲 K 共兩K兩r2D兲 K2 ⬍ 0 2␲ 0

冧

p共x,y,z,k兲 =

1 2␲ +

冕冕

k/␤

,

共6兲

J. Acoust. Soc. Am., Vol. 122, No. 5, November 2007

+

q共x,K,z兲·共eiky共ys−y兲 − e−iky共ys−y兲兲dky

0

1 2␲

− e−i

where r2D = 冑共x − xs兲2 + 共z − zs兲2, H共2兲 0 the Hankel function of order zero and second kind and K共1兲 0 the modified Bessel function of order zero and first kind. For a real positive K value, the 2D solution is a traveling wave, whereas for imaginary K 共i.e., for k2y ⬎ k2兲, the Helmholtz equation gets the form of a diffusion equation approximately obeying an exponentially decaying Green’s function. Salomons18 showed that the major contribution of the integral in Eq. 共4兲 lies around the point of stationary phase, which is situated at ky = k sin ␪, with ␪ = arctan关共y − y s兲 / r2D兴关see Fig. 1共b兲 for angle ␪兴. Since the point of stationary phase is bounded for ky by −k and k, integration limits with respect to ky may be restricted to −␥k and ␥k, with ␥ a number slightly larger than 1 关see Fig. 2兴. Integration from −␥k to ␥k implies integration over a singular value of q 共both for the Hankel and modified Bessel function兲 in the origin of the complex plane 关see Fig. 2兴. This happens either when r2D = 0 or when K = 0 关see Eq. 共6兲兴. For the integration variable ky, the singularity appears when ky = −k or ky = k. The integral in Eq. 共4兲 could be calculated applying residue calculus, yet since the 2D equation is solved numerically later 共using the ESM兲, a numerical approach similar to the one presented by Salomons is used.18 This approach embraces substitution of the integration variable for values around the singularity. The integration variable ky is substituted by u = 冑k2 − k2y for 兩k / ␤ 兩 ⬍ 兩ky 兩 ⬍ 兩k兩 and by v = 冑k2y − k2 for 兩k 兩 ⬍ 兩ky 兩 ⬍ 兩k␤兩, where ␤ ⬎ 1 and a real number. The integral can now be divided into four parts along the real ky axis 共since q is independent of the sign of ky兲. Solutions of q are calculated in each part with a 共differ2534

ent兲 frequency resolution ⌬f and are spline interpolated to an equidistant wave number resolution ⌬k

0+

冑k2−u2共ys−y兲

1 2␲

− e−i

k冑1−␤−2

冕

k冑␤2−1

0+

q共x,u,z兲 · 共ei 兲

u

冑k2 − u2 du

q共x,iv,z兲 · 共ei

冑k2+v2共ys−y兲

兲

冑k2−u2共ys−y兲

v

冑k2 + v

冑k2+v2共ys−y兲

dv + 2

1 2␲

冕

k␥

q共x,K,z兲

k␤

· 共eiky共ys−y兲 − e−iky共ys−y兲兲dky .

共7兲

Then, the integrals are approximated by sums L−1/2

pn共x,y,z,k兲 =

1 兺 q共x, 冑k2 − 共n⌬ky兲2,z兲 2␲ n=1/2 · 共ein⌬ky共ys−y兲 + e−in⌬ky共ys−y兲兲⌬ky M−1/2

1 n⌬u + q共x,n⌬u,z兲 2 兺冑k − 共n⌬u兲2 2␲ n=1/2 · 共ei

冑k2−共n⌬u兲2共ys−y兲

+ e−i

冑k2−共n⌬u兲2共ys−y兲

兲⌬u

N−1/2

+

1 n⌬v 兺 q共x,in⌬v,z兲冑k2 + 共n⌬v兲2 2␲ n=1/2

· 共ei

冑k2+共n⌬v兲2共ys−y兲

+ e−i

冑k2+共n⌬v兲2共ys−y兲

兲⌬v

O−1/2

+

1 兺 q共x, 冑k2 − 共k␤ + n⌬ky兲2,z兲 2␲ n=1/2

· 共ei共k␤+n⌬ky兲共ys−y兲 + e−i共k␤+n⌬ky兲共ys−y兲兲⌬ky ,

共8兲

with L = k / ␤⌬ky, M = k冑1 − ␤−2 / ⌬u, N = k冑␤2 − 1 / ⌬v, O = 共k␥ − k␤兲 / ⌬ky, and pn共x , y , z , k兲 the numerical 3D solution. The values for k, ␤, ␥, ⌬ky, ⌬u, and ⌬v should be chosen such that L, M, N, and O get integer values. Hornikx and Forssén: 2.5D calculation method for urban canyons

FIG. 3. Principle of the ESM. Equivalent sources are positioned at interfaces between cavities, impedance boundaries, and the free field.

The transform of a coherent line source solution to a point source solution can be applied to any method fulfilling Eq. 共3兲 and boundary conditions 关see Sec. II C兴. For the geometry of Fig. 1, the ESM is used.14 This method has been validated before by the BEM14 and a scale model study.22 In the ESM, the domain is decoupled into subdomains: closed cavities and a free space above a flat ground surface 关see Fig. 3兴. The domains are coupled by placing equivalent sources at the interfaces of the domains. First, the Green’s functions from the primary source to the equivalent sources and among the equivalent sources in the subdomains are calculated. The Green’s functions used here are a modal solution in the cavities and a semifree field Green’s function above the canyons. To obtain cavity modes with finite amplitude, a small amount of damping has been used in the cavity Green’s functions.14 Then, the strengths of the equivalent sources at the subdomain interfaces are calculated using conditions for pressure and normal velocity over the interfaces. Absorption patches at the boundaries and niches 共e.g., balconies兲 can be modeled by additional subdomains and equivalent sources. For the imaginary frequencies, the Helmholtz equation 共now a diffusion equation兲 is solved by a second-order accurate finite difference scheme. At the rigid boundaries, the pressure gradient is set to zero and at the far away reflection free boundaries of the computational domain, the pressure is set to zero. The size of the domain above the canyons was chosen to be 240 m ⫻ 140 m such that the results at the observer points converged. The transform applied to the 2D ESM is here called the 2.5D ESM. All 2.5D ESM calculations in this paper have been made without including air absorption.

two frequencies and two different y distances, both with and without the imaginary frequencies. The results are plotted as a function of r2D 共which is equal to the x distance, since the z coordinate of the source and observer are equal to zero兲. The error is defined as 20 log共兩pn兩 / 兩pexact兩兲, with pexact 2 = e−ikr / 4␲r, and r = 冑r2D + 共y − y s兲2. The numerical results have been calculated using Eqs. 共6兲 and 共8兲. The results show that omitting the imaginary frequencies in the integral evaluation introduces fluctuations in the result for small r2D values. The error is larger for smaller r2D values and for lower frequencies. These effects are due to the slower decay of the function K共1兲 0 共Kr2D兲 in amplitude for smaller K and smaller r2D. It is also visible that very close to r2D = 0 m, there is even an error when imaginary frequencies are included. This error can be attributed to the location of the point of stationary phase. When ␪ goes towards 90° 关see Fig. 1共b兲 for ␪兴, the point of stationary phase shifts to the singularity point where

B. Accuracy in the free field situation

We will now discuss two aspects of the transform in Eq. 共4兲 that are important regarding accuracy in the current application. Sound propagation in an unbounded domain will be considered first. The first aspect concerns the importance of the integration path for 兩ky兩 ⬎ 兩k兩共i.e., along the imaginary axis of K兲. As mentioned, the solution of the Helmholtz equation for imaginary frequencies obeys a different type of equation and is not straightforward in some calculation models. Figure 4 shows the calculation error of the transform for J. Acoust. Soc. Am., Vol. 122, No. 5, November 2007

FIG. 4. Error in the transform in Eq. 共4兲 for sound propagation in an unbounded domain, both with and without imaginary frequencies. The error is 20 log共兩pn兩4␲r兲, pn calculated according to Eq. 共8兲 with a frequency discretization of ⌬f = 0.1 Hz. Source at 共0,0,0兲 and observer at 共x , y , 0兲: 共a兲 f = 100 Hz, y = 0 m; 共b兲 f = 1000 Hz, y = 0 m; 共c兲 f = 100 Hz, y = 100 m; and 共d兲 f = 1000 Hz, y = 100 m. Hornikx and Forssén: 2.5D calculation method for urban canyons

2535

FIG. 6. Source at 共9,0,0兲 and observer at 共0 , y , 5兲 above a rigid ground surface between two semi-infinite rigid façades.

the ground have an infinitely high acoustic impedance, further denoted as a rigid surface. In general, the boundary condition that belongs to Eq. 共2兲 reads

⳵P ik = − P, ⳵n Zs FIG. 5. Error in the transform in Eq. 共4兲. Source at 共0,0,0兲 and observer at 共x , y , 0兲. Two frequency discretizations have been used for the evaluation of the integral. The calculated r2D,max are 344 m for ⌬f = 0.2 Hz and 688 m for ⌬f = 0.1 Hz 共c0 = 344 m / s兲: 共a兲 f = 100 Hz, y = 0 m; 共b兲 f = 1000 Hz, y = 0 m; 共c兲 f = 100 Hz, y = 100 m; and 共d兲 f = 1000 Hz, y = 100 m.

the numerical integration is still more sensitive. The fluctuation of the error is slower for y = 100 m than for y = 0 m. This is most obvious at 100 Hz. The reason is again the location of the point of stationary phase; for y = 100 m, the angle ␪ is larger at the same x position than for y = 0 m, which implies that the point of stationary phase gets closer to the imaginary frequencies. The second aspect concerns the frequency resolution ⌬f. The free field solution of the 2D Helmholtz equation that appears in the argument of the integral in Eq. 共4兲 is oscillating over ky. For a larger r2D, this function oscillates more rapidly over ky. To preserve accuracy of the numerical integration, a minimum frequency resolution corresponding to five points per period was therefore proposed,1 ⌬f = c0 / 5r2D,max, where r2D,max is the maximum distance of interest. Figure 5 shows the error as function of the r2D distance for two frequency resolutions and two frequencies. The error is defined as has been done for Fig. 4. It is obvious that the frequency dependence concerning r2D is not strong. Further, we notice that a certain frequency discretization holds for a longer r2D distance for y = 100 m than for y = 0 m. This is due to the fact that, at y = 0 m, the point of stationary phase is located exactly at ky = 0. A constant frequency discretization, and consequently a constant discretization for K, yields a discretization for ky that is biggest for ky = 0. Thus, with the point of stationary phase located at ky = 0, the introduced discretization error is largest here.

C. Accuracy in the directly exposed street canyon

We now turn to a situation with source and observer positions situated between two semi-infinite walls with the source placed at a ground surface 关see Fig. 6兴. The walls and 2536


共9兲

at the boundaries, with the frequency dependent normalized surface impedance Zs共f兲 =

Zboundary共f兲 , Zair

where Zboundary and Zair are specific impedances of the boundary material and air. The 2D Helmholtz equation, Eq. 共3兲, is completed by

⳵q iK =− q ⳵n Zs,2D

共10兲

at the boundaries. For equivalence of Eqs. 共9兲 and 共10兲, we write Zs,2D共f,ky兲 =

Zs共f兲K . k

共11兲

The effect of air absorption has been neglected in this equivalence condition, and a source position of y s = 0 m is assumed. When the surface impedances are infinitely large, as is our main interest, the equivalence condition of Eq. 共11兲 is automatically fulfilled. The situation can be considered as the limit case of a narrow street canyon. Therefore, and since analytical solutions in two and three dimensions are straightforward for this situation, it is worth studying the influence of the errors of the transform in this case. The source and observer positions correspond to those used in a scale model measurement study.22 The 2D solution pn is computed by a summation of 50 image sources using Eq. 共6兲, whereas the 3D solution consists of a summation of the same number of image sources using the free field point source solution e−ikr / 4␲r. Note that the error in the image sources method vanishes when the number of image sources is increased. Figure 7 关panels 共ai兲 and 共aii兲兴 shows the sound pressure level relative to the free field level as function of frequency for one y position, and as function of y for f = 100 Hz. The source and observer positions are given in Cartesian coordinates 共x , y , z兲 according to Fig. 1. Panels 共b兲–共e兲 of Fig. 7 show the errors in the transform defined as 20 log共兩pn兩 / 兩p3D,ISM兩兲, where ISM refers to image sources Hornikx and Forssén: 2.5D calculation method for urban canyons

FIG. 7. Calculations for the source and observer positions as in Fig. 6. 共ai兲 Lrefree for source at 共9,0,0兲 and observer at 共0,0,5兲; 共aii兲 Lrefree for source at 共9,0,0兲 and observer at 共0 , y , 5兲 at 100 Hz. Panels 共b兲–共e兲 show the errors in the transform 共see text兲 for various choices in the numerical scheme to calculate pn: 共b兲 ⌬f = 0.1 Hz; 共c兲 ⌬f = 0.1 Hz, no imaginary frequencies; 共d兲 ⌬f = 0.2 Hz; 共e兲 ⌬f = 0.1 Hz, Zfacades = 17.8.

FIG. 8. Lrefree with a point source at 共9,0,0兲 and the observer at 共0,0,5兲 in the geometry of Fig. 1. Imaginary frequencies have been included in the transform of the 2.5D ESM and 3D ISM calculations have been made without diffraction.

M

n p共x,y,z,k兲 = 2 兺 Qsph 共Zs共f兲,rn兲

method. With a frequency resolution of ⌬f = 0.1 Hz to calculate pn, the errors over frequency are small and occur mostly at resonances 关see Fig. 7共bi兲兴. This error increases with y distance 关see Fig. 7共bii兲兴, which is due to the fact that the contribution of higher order reflections to the sound pressure level increases with increasing y position. These higher order reflections have a larger path length, which results in a larger error 关see Fig. 5兴. Still, the errors at 1000 m are small. Omitting the imaginary frequencies leads to larger errors 关see Figs. 7共ci兲 and 7共cii兲兴, which are more pronounced at the lower frequencies, as observed before. Excluding the imaginary frequencies leads to substantial errors over the y distance at 100 Hz. These errors do not seem to increase substantially over the distance. This can be understood since all observer positions are located at the same r2D position from the source. For a coarser frequency resolution of ⌬f = 0.2 Hz, the errors increase for all frequencies compared to the ⌬f = 0.1 Hz case 关see Fig. 7共di兲兴. Also, the errors increase for y ⬎ 200 m due to the influence of the higher order reflections at larger y position, 关see Fig. 7共dii兲兴. When the impedance of the boundaries is finite, a distinct set of 2D solutions has to be calculated for each frequency in 3D 共since Zs,2D is a function of f and ky兲, which increases the computation time. To show the influence of an impedance boundary on the error, we use a real valued surface impedance of the boundaries of Zs = 17.8, corresponding to an absorption coefficient of 0.2 for normal wave incidence. The results are shown in Figs. 7共ei兲 and 7共eii兲. The 2D and 3D solutions have now been calculated by the following sums: M

i n 共Zs,2D共f,ky兲,r2D,n兲H共2兲 q共x,K,z兲 = − 2 兺 Qcyl 0 共Kr2D,n兲, 4 n=0 共12兲 J. Acoust. Soc. Am., Vol. 122, No. 5, November 2007

n=0

e−ikrn , 4␲rn

共13兲

where Qcyl is the cylindrical wave reflection coefficient implemented according to Chandler-Wilde and Hothersall,23 Qsph the spherical wave reflection coefficient implemented 2 according to Chien and Soroka,24 and r2D,n = 冑共z − zs兲2 + 共x − xs,n兲2, with xs,n the x distance of the nth image source to the receiver. The above expressions are generally applicable for any 共complex valued兲 impedance. For high frequencies and large distances, Qsph and Qcyl have both been substituted by the plane wave reflection coefficient, Qplane. A frequency resolution of ⌬f = 0.1 Hz has been used. Figure 7共ei兲 shows that the results over frequency are better than the corresponding results without absorption. An explanation is that the largest error in the latter case was due to higher order reflections 共with distances larger than r2D,max from the source兲, which play a smaller role when increasing the absorption. Over the y distance, the error increases from 200 m on. Inaccuracies from the first-order reflections were found to cause this error. Minor errors in the numerically calculated reflection coefficients could cause the calculated error, i.e., the error shown may overpredict the actual error due to the transform. For the directly exposed canyon of Fig. 1, the 2.5D ESM is used. Figure 8 shows the sound pressure level relative to the free field level, Lrefree. Results of the 2.5D ESM including imaginary frequencies are compared with calculations with the 3D ISM for the situation with two semi-infinite walls 关see Fig. 6兴. The agreement is fine and the small error was shown to disappear at higher frequencies. The small error at low frequencies can be attributed to the lack of the diffraction contributions in the ISM model. This is supported by a comparison between 2D results from the ESM, the ISM, and the BEM; see Fig. 9. The ISM results display deviations from the ESM and BEM results. Since the 2D solutions are shown, no transform errors are included. At a larger y disHornikx and Forssén: 2.5D calculation method for urban canyons

2537

TABLE I. Required frequency discretization 共Hz兲 in the 2.5D ESM for a tolerance of 0.5 dB. Situation of Fig. 1 for rigid façades and Fig. 10共a兲 for the case with diffusion patches. Source coordinates are 共9,0,0兲. For the incoherent line source calculations, ⌬f = 0.2 Hz up to 50 Hz.

Observer coordinates Rigid façades Diffusion patches

Point source

Incoherent line source

f = 100 Hz, y = 0 m

f = 1000 Hz

共0,0,15兲 0.4 0.4

共60,0,15兲 0.1 0.2

共49,0,0兲 0.4 1.6

D. Accuracy in the shielded canyon

FIG. 9. Lrefree with a coherent line source at 共9,0,0兲 and the observer at 共0,0,5兲 in the geometry of Fig. 1. The 2D ISM result has been calculated without diffraction.

tance, the deviations between results of the 3D ISM model and the transformed ESM are expected to increase, since the diffraction contribution is larger there. For engineering use of the 2.5D ESM, errors are calculated in the 100 Hz 1 / 3-octave band for an observer point in the directly exposed street canyon and two different façade types; rigid façades and façades with rigid diffusion patches applied. In Fig. 10共a兲, the street canyons geometry with diffusion patches is shown. The patches have a dimension of 0.2 m by 2.0 m and are positioned according to a scale model study.22 The case with diffusion patches is somewhat closer to a realistic case. Table I shows the required frequency descritization for the two cases, where a maximum error of 0.5 dB relative to a converged calculation with a frequency discretization of ⌬f = 0.05 Hz has been used as a tolerance. The same frequency discretization of ⌬f = 0.4 Hz can be used for an acceptable error for both situations. A larger ⌬f will be allowable for larger y distances. The reason has been mentioned before in the discussion of Fig. 5: the discretization in ky is coarsest around ky = 0 and introduces the largest error if the observer is located at y = y s.

For the shielded canyon, no analytical solution exists. To investigate the sources of error in the transform in the shielded canyon of Fig. 1, the frequency resolution error is studied by convergence of the 2.5D ESM results with ⌬f, as for the directly exposed canyon. The source coordinates are 共9,0,0兲 and the observer is located at coordinates 共60,0,15兲. Since the shortest travel path between the source and nearest observer is approximately 67 m, and thereby much larger than the wavelength, the imaginary frequencies are omitted in the calculation. The required frequency discretizations for a tolerance of 0.5 dB are printed in Table I. In contrast to the directly exposed street canyon results, a finer frequency discretization is necessary for an acceptable accuracy. The influence of higher order reflections causes this difference. For the diffusion patches case, the required frequency discretization is coarser than for the case of rigid façades. The reason for this difference is that the higher order reflections are of more importance in the case of rigid façades. Calculation time with the 2D ESM method increases with frequency. For the geometry with diffusion as in Fig. 10共a兲, a single observer position, ⌬f = 0.2 Hz and for 1000 Hz, a 2.5D calculation at the shielded side takes approximately 42 h using MATLAB on a computer with a dual 2.66 GHz processor and a 4 GB memory. Six equivalent sources per wavelength are then used in the method and no parallel calculations are done. Since we have calculated 2D frequencies up to 1000 Hz, this computation time stands for the calculation of all 2.5D results up to 1000 Hz. When

FIG. 10. 共a兲 Cross section of two parallel street canyons with extra patches 共absorption or diffusion兲. 共b兲 The absorption coefficient for normal wave incidence used for the absorption patches. Diffusion patches are rigid. 2538



ban street canyons.22. The situation of a single building block is modeled using Pierce’s solution for a thick barrier.26 The multiple reflections cause a level that is more than 20 dB 共!兲 higher than a situation with a single wide barrier only. Diffusion and absorption have a clear effect on the level. The slope of the excess attenuation curves over distance—it increases for a large 兩y兩 distance—is similar for all cases and governed by the changing diffraction angles. This implies that the level difference over the 兩y兩 distance is smaller for the cases with barrier than without. It indicates why distant sources are important for sound pressure levels at a shielded side. The LEA is larger for the shielded canyon case with rigid façades than in the free field at a large 兩y兩 distance, an effect also described by Forssén and Hornikx.27.

FIG. 11. Vehicle pass-by excess attenuation levels for the geometry of Fig. 1. Source at 共9 , y , 0兲 and observer at 共49, 0, 0兲. Diffusion and absorption patches as in Fig. 10.

façades would consist of material with a finite impedance, 42 h would be the approximate calculation time for 1000 Hz only. Other frequencies require a new calculation over the 2D frequencies. In the absence of air absorption, however, cavity Green’s functions have to be calculated only once per frequency. III. POINT AND LINE SOURCES IN STREET CANYONS

The 2.5D ESM will now be used to investigate some aspects related to two types of sound sources of road traffic. The extreme cases of a single pass-by and a continuous traffic flow will be illuminated. Here we will consider the sound pressure only at a single point in the shielded canyon. In the transform integral in Eq. 共5兲, no imaginary frequencies have been taken into account, and frequency resolutions in the transform of 0.1 Hz for rigid façades and 0.2 Hz for façades with absorption or diffusion patches have been used. A. Vehicle pass-by

Figure 11 shows four curves of the excess attenuation level LEA 关the level minus the level without obstacle共s兲兴 at observer point 共49,0,0兲, due to a pass-by of a vehicle at 共9 , y s , 0兲关see Fig. 1 for the coordinates兴. The results have been A weighted, and a road traffic noise spectrum has been used. A distribution of 90% light and 10% heavy vehicles has been assumed with a speed of 50 km/ h and the road noise spectra have been taken from Danish measurement data.25 The 1 / 3-octave bands 100– 1000 Hz have been used. The curves represent a vehicle on a rigid ground surface in the following situations: a single building block as barrier 共i.e., the central building block of Fig. 1兲, two parallel canyons with rigid façades 共i.e., the “full” situation of Fig. 1兲, two parallel canyons with façade absorption, and two parallel canyons with partially diffusely reflecting façades 关i.e., as in Fig. 10共a兲兴. The diffusion patches are rigid. Figure 10共b兲 shows the absorption coefficient for normal wave incidence used for the absorption patches. It originates from the absorption material used in a scale model study of parallel urJ. Acoust. Soc. Am., Vol. 122, No. 5, November 2007

B. Coherent and incoherent line source

A traffic flow consisting of a continuous flow of vehicles with random distances may be modeled by an uncorrelated row of point sources, being an incoherent line source for predicting the equivalent level. Using a calculation model that generates the excess attenuation at the shielded side for a coherent line source might thus give incorrect results when aiming at predicting the excess attenuation due to road traffic noise from a traffic flow. Former studies of Duhamel16 and Jean et al.17 showed that calculating the excess attenuation of a single screen gives higher losses when modeling for a coherent line source instead of an incoherent line source. They showed that the excess attenuation of a screen due to an incoherent line source is smooth over frequency, whereas it showed a clear interference pattern for the coherent line source solution. Van Renterghem et al.14 show an error in the same direction in the case of two parallel urban canyons. They did not, however, make use of point source solutions to assemble the incoherent line source solution. The excess attenuation for an incoherent line source can be found by integrating the squared absolute value of the calculated point source solutions pn共x , y , z , k兲 over y

冢

LEA,inco,1共x,z,k兲 = 20 log

冕冕

冢

⬇ 20 log

⬁

兩pn共x,y,z,k兲兩2 dy

−⬁ ⬁

兩pfree共x,y,z,k兲兩2 dy

−⬁

冣

N

兩pn共x,n⌬y,z,k兲兩2⌬y 兺 n=−N N

兺

兩pfree共x,n⌬y,z,k兲兩2⌬y

n=−N

冣

.

共14兲

By Parseval’s theorem, the energy of the integrals in Eqs. 共4兲 and 共5兲 are equal Hornikx and Forssén: 2.5D calculation method for urban canyons

2539

FIG. 12. LEA results for the parallel street canyons case with the source at 共9,0,0兲 and the observer at 共49,0,0兲. The coherent line source solution is calculated with the 2D ESM and the incoherent line source solution according to Eq. 共16兲: 共a兲 rigid boundaries as in Fig. 1; 共b兲 diffusion patches as in Fig. 10共a兲; 共c兲 absorption patches as in Fig. 10; and 共d兲 LEA,incoherent − LEA,coherent for 1 / 3-octave bands.

冕

⬁

兩pn共x,y,z,k兲兩2 dy =

−⬁

1 2␲

冕

⬁

兩q共x,K,z兲兩2 dky .

共15兲

−⬁

Therefore, another way to calculate the incoherent line source solution is by using 2D solutions LEA,inco,2共x,z,k兲

冢

= 20 log

N

1 兺兩q共x, 冑k2 − 共n⌬ky兲2,z兲兩2⌬ky 2␲ n=−N N

兺

n=−N

兩pfree共x,n⌬y,z,k兲兩 ⌬y 2

冣

.

共16兲

Both expressions 共14兲 and 共16兲 converge when N goes to infinity and ⌬ky and ⌬y go to zero. LEA,inco,2共x , z , k兲 can be calculated more efficiently by a coarser frequency discretization than the integral in Eq. 共4兲, since the oscillating part is eliminated by taking the squared absolute values of q. When 兩q兩2 is a smooth function, e.g., as from Eq. 共6兲, integration can be done very efficiently. For a resonant sound field however, the discretization has to be treated with more care. From Sec. III A, we know that at the shielded side sources at a larger y distance contribute to the level substantially. The integration limits have therefore to be rather large in Eq. 共14兲. Equation 共16兲 enables a largely reduced computational cost, and is therefore used for further calculations. At the shielded side, the major contribution in the summation in Eq. 共16兲 comes from the lowest 2D frequencies, since they embody the first modes and are less reduced in amplitude by diffraction. The finest frequency discretization in the summand is thus required for the lowest frequencies. Therefore, the necessary frequency discretization is investigated where a fixed frequency discretization of ⌬f = 0.2 Hz for the lower frequency region up to 50 Hz has been chosen. The 1000 Hz 2540


1 / 3-octave band results are shown in Table I for the cases with rigid façades, as in Fig. 1, and partially diffusely reflecting façades, as in Fig. 10共a兲. The errors have been calculated relative to a case with ⌬f = 0.2 Hz for the whole frequency range which gave a convergent result. A smaller ⌬f is necessary for the rigid case. For calculations of an incoherent line source in a realistic environment 共which has some absorption and diffusion兲, a proposal is to use ⌬f = 0.2 Hz up to f = 88 Hz and then 30 logarithmically spaced frequencies per 1 / 3-octave band. For the diffusion patches case, this approach yields a maximum error of 0.4 dB in the 1 / 3-octave band range 100– 1000 Hz, whereas a maximum error of 0.9 dB is found for the rigid façade case. For the geometry of Fig. 10共a兲 with diffusion patches and the observer in the shielded canyon, the calculation time using this frequency discretization and frequency range is 23 h using the implemented 2.5D ESM method on a computer with a dual 2.66 GHz processor and a 4 GB memory. The excess attenuation for a coherent line source and an incoherent line source for the parallel canyons of Fig. 1 are now compared for the cases of rigid boundaries, applied absorption patches, and applied diffusion patches. The cases with absorption and diffusion have again been modeled according to Fig. 10. The source position is at 共9 , y , 0兲 and the observer position at 共49,0,0兲. As literature results for a single screen also show, the incoherent line source results are smoother over frequency than the coherent results; the interference effects are cancelled due to incoherence 共see Fig. 12兲. In 1 / 3-octave bands, the coherent excess attenuation levels are lower than the incoherent attenuation levels for most cases and we see that the differences of the various cases differ by several dB. Even though these are only numerical results and evaluation together with measured data Hornikx and Forssén: 2.5D calculation method for urban canyons

has not been made here, it is clear that the influence of absorption and diffusion can be judged inaccurately using the coherent excess attenuation. IV. CONCLUSIONS

The possession of point source solutions and incoherent line source solutions is of importance in urban sound propagation research when aiming to predict levels at shielded areas. By applying an inverse Fourier transform, coherent line source solutions of the two-dimensional equivalent sources method 共2D ESM兲 were transformed to a point source solution for the situation of parallel urban canyons. This method is called the 2.5D ESM. The theoretical basis of the method should make it possible to model complex valued surface impedances. The integration paths of the transform integral over the imaginary frequencies were shown to be of importance for calculations in the directly exposed street canyon at the lower frequencies. In the shielded canyon, these integration paths may be omitted. The long travel distances by multiple reflections in the canyons impose a requirement of a high-frequency resolution for the evaluation of the transform integral. For a point source, a frequency discretization of ⌬f = 0.4 Hz gives satisfying results in the directly exposed canyon. In the shielded canyon for façades with some diffusion ⌬f = 0.2 Hz can be used, and ⌬f = 0.1 Hz should be used for rigid specular reflecting façades. Vehicle pass-by calculations show the importance of higher order reflections at the shielded canyon. The 2D solution can also be used for the calculation of an incoherent line source. This method is much more efficient regarding the computation time compared to a method making use of point source solutions. The excess attenuation for realistic traffic flows, modeled by incoherent line sources, shows that the effect of absorption and diffusion in street canyons may be predicted incorrectly when a coherent line source solution is used; i.e., the effect of mitigation may be overpredicted when modeling traffic along a straight road by a coherent line source. ACKNOWLEDGMENTS

The work behind this paper has been supported by the Swedish Foundation for Strategic Environmental Research 共MISTRA兲. The authors would like to thank Wolfgang Kropp for his ideas and fruitful discussions. 1

K. V. Horoshenkov, D. C. Hothersall, and S. E. Mercy, “Scale modelling of sound propagation in a city street canyon,” J. Sound Vib. 223, 795–819 共1999兲. 2 M. R. Ismail and D. J. Oldham, “A scale model investigation of sound reflection from building façades,” Appl. Acoust. 66, 123–147 共2005兲. 3 J. Kang, “Sound propagation in street canyons: Comparison between dif-


fusely and geometrically reflecting boundaries,” J. Acoust. Soc. Am. 107, 1394–1404 共2000兲. 4 J. Kang, “Numerical modelling of the sound fields in urban streets with diffusely reflecting boundaries,” J. Sound Vib. 258, 793–813 共2002兲. 5 K. K. Lu and K. M. Li, “The propagation of sound in narrow street canyons,” J. Acoust. Soc. Am. 112, 537–550 共2002兲. 6 J. Picaut, L. Simon, and J. Hardy, “Sound field modelling in a street with a diffusion equation,” J. Acoust. Soc. Am. 106, 2638–2645 共1999兲. 7 J. Picaut and L. Simon, “A scale model experiment for the study of sound propagation in urban areas,” Appl. Acoust. 62, 327–340 共2001兲. 8 J. Picaut, T. Le Polles, P. L’Hermite, and V. Gary, “Experimental study of sound propagation in a street,” Appl. Acoust. 66, 149–173 共2005兲. 9 J. Kang, “Sound propagation in interconnected urban streets: A parametric study,” Environ. Plan. B: Plan. Des. 28, 281–294 共2001兲. 10 J. Picaut, L. Simon., and J. Hardy, “Sound propagation in urban areas: A periodic disposition of buildings,” Phys. Rev. E 60, 4851–4859 共1999兲. 11 M. Hornikx, “Towards a parabolic equation for modeling urban sound propagation,” Proceedings 11th Long Range Sound Propagation Symposia, Fairly, Vermont, 2004. 12 T. van Renterghem and D. Botteldooren, “Numerical simulation of sound propagation over rows of houses in the presence of wind,” Proceedings of the 10th International Conference on Sound and Vibration, Stockholm, Sweden, June 2–3, 2003, 1381–1388. 13 T. van Renterghem, E. Salomons, and D. Botteldooren, “Parameter study of sound propagation between city canyons with a coupled FDTD-PE model,” Appl. Acoust. 67, 487–510 共2006兲. 14 M. Ögren, “Prediction of traffic noise shielding by city canyons,” Ph.D. thesis, Chalmers University of Technology, Gothenburg, Sweden, 2004. 15 T. Kihlman, Handlingsplan Mot Buller (National Action Plan Against Noise) 共Allmänna Förlaget, Stockholm, Sweden, 1993兲. 16 D. Duhamel, “Efficient calculation of the three-dimensional sound pressure field around a noise barrier,” J. Sound Vib. 197, 547–571 共1996兲. 17 P. Jean, J. Defrance, and Y. Gabillet, “The importance of source type on the assessment of noise barriers,” J. Sound Vib. 226, 201–216 共1999兲. 18 E. M. Salomons, A. C. Geerlings, and D. Duhamel, “Comparison of a ray model and a Fourier—boundary element method for traffic noise situations with multiple diffractions and reflections,” Acust. Acta Acust. 226, 35–47 共1997兲. 19 D. Duhamel and P. Sergent, “Sound propagation over noise barriers with absorbing ground,” J. Sound Vib. 218, 799–823 共1998兲. 20 L. Godinho, J. António, and A. Tadeu, “Sound propagation around rigid barriers laterally confined by tall buildings,” Appl. Acoust. 63, 595–609 共2002兲. 21 J. Defrance and P. Jean, “Integration of the efficiency of noise barrier caps in a 3-D ray tracing method. Case of a T-shaped diffracting device,” Appl. Acoust. 64, 765–780 共2003兲. 22 M. Hornikx, “Sound propagation to two-dimensional shielded urban areas,” Licenciate of Eng., thesis, Chalmers University of Technology, Gothenburg, Sweden, 2006. 23 S. N. Chandler-Wilde and D. C. Hothersall, “Efficient calculation of the Green’s function for acoustic propagation above a homogeneous impedance plane,” J. Sound Vib. 180, 705–724 共1995兲. 24 C. F. Chien and W. W. Soroka, “Sound propagation along an impedance plane,” J. Sound Vib. 43, 9–20 共1975兲. 25 H. Jonasson and S. Storeheier, “Nord 2000. New Nordic prediction method for road traffic noise,” SP Rapport No. 2001:10, Borås, Sweden 2001. 26 A. D. Pierce, “Diffraction of sound around corners and over wide barriers,” J. Acoust. Soc. Am. 55, 941–955 共1974兲. 27 J. Forssén and M. Hornikx, “Statistics of A-weighted road traffic noise levels in shielded urban areas,” Acta. Acust. Acust. 92, 998–1008 共2006兲.


2541

Errata The following corrections need to be observed regarding Paper II:

◦ Page 2539, Eq. (14): 

R∞

(x, y, z, k)|2



|pn dy    −∞  LEA,inco,1 (x, z, k) = 20 log  ∞  R   2 |pf ree (x, y, z, k)| dy 

−∞

N P

(x, n∆y, z, k)|2 ∆y



|pn     n=−N ≈ 20 log  N .   P 2 |pf ree (x, n∆y, z, k)| ∆y n=−N

should be replaced by: 

R∞

(x, y, z, k)|2



|pn dy    −∞  LEA,inco,1 (x, z, k) = 10 log  ∞  R   |pf ree (x, y, z, k)|2 dy 

−∞

N P

(x, n∆y, z, k)|2 ∆y



|pn     n=−N ≈ 10 log  N .   P 2 |pf ree (x, n∆y, z, k)| ∆y n=−N

◦ Page 2540, Eq. (16): LEA,inco,2 (x, z, k)   N p P 1 2 2 2 |q(x, k − (n∆ky ) , z)| ∆ky   2π  n=−N  = 20 log  . N   P 2 |pf ree (x, n∆y, z, k)| ∆y n=−N

should be replaced by: LEA,inco,2(x, z, k)   N p P 1 2 − (n∆k )2 , z)|2 ∆k k |q(x, y y  2π   n=−N ≈ 10 log  . N P   |pf ree (x, n∆y, z, k)|2 ∆y n=−N

Paper III

Applied Acoustics 70 (2009) 267–283

Contents lists available at ScienceDirect

Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust

Noise abatement schemes for shielded canyons M. Hornikx *, J. Forssén Chalmers University of Technology, Department of Civil and Environmental Engineering, Division of Applied Acoustics, SE-41296 Göteborg, Sweden

a r t i c l e

i n f o

Article history: Received 12 October 2007 Received in revised form 20 March 2008 Accepted 2 April 2008 Available online 27 May 2008 PACS: 43.28.Js 43.50.Gf Keywords: Noise abatement schemes Shielded urban areas Equivalent sources method Incoherent line source

a b s t r a c t Access to quiet areas in cities is important to avoid adverse health effects due to road traffic noise. Most urban areas which are or can become quiet (LA,eq < 45 dB) are shielded from direct road traffic noise. By transfer paths over roof level, many road traffic noise sources contribute to the level in these shielded areas and noise abatement schemes may be necessary to make these areas quiet. Two real life shielded courtyards in Göteborg have been selected as reference cases for a numerical investigation of noise abatement schemes. The selected areas are modelled as canyons with a road traffic noise source modelled outside the canyon by a finite incoherent line source, which is more realistic than both a coherent and an incoherent line source of infinite length. The equivalent sources method has been used for the calculations. For all studied noise abatement schemes in the shielded canyon, the reductions are largest for the lower canyon observer positions. Façade absorption is the most effective when placed in the upper part of the canyon and can typically yield a reduction of 4 dB(A). Constructing 1 m wide walkways with ceiling absorption reduces the level typically by 3 dB(A). These effects are most effective for narrower canyons. For treatments at the canyon roof, reductions are independent of the canyon observer position and amount to 4 dB(A) for a 1 m tall screen and 2 dB(A) for a grass covering of a saddle roof. Downward refracting conditions increase the levels for the lower canyon observer positions and higher frequencies. For sources located in canyons, abatement schemes therein are more effective for noise reduction in the shielded canyon than similar abatement schemes in the shielded canyon itself, given that all contributing source canyons are treated. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Noise from road traffic can lead to annoyance and adverse health effects such as sleep disturbance, disturbed relaxation and stress related effects [1]. For a ‘good and healthy’ environment for citizens, the sound level in LA,eq outdoors should be below 45 dB1 [2]. Equivalent sound level often exceed 45 dB, however, especially in urban areas directly exposed to road traffic noise. Öhrström et al. [2] give evidence on health benefits of the presence of a relatively quiet outdoor section bordering a dwelling; when the sound level in LA,eq due to road traffic noise does not exceed 60 dB at the most-exposed side of the dwelling, 80% of the occupants are protected from annoyance and adverse health effects if they also have access to a side where the LA,eq does not exceed 45 dB. Within the Swedish research program Soundscape Support to Health, an area

* Corresponding author. Tel.: +46 31 772 8605; fax: +46 31 772 2212. E-mail address: [email protected] (M. Hornikx). 1 We remark that 45 dB is here used as a guideline level. Since other effects, as the frequency content of noise, its time variations and non-acoustical aspects as, e.g. individual perception, also influence the adverse effects, 45 dB does not guarantee a ‘good and healthy’ environment for all citizens. Here, LA,eq means the 24 h equivalent sound level, given as a so-called free field value. 0003-682X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2008.04.002

bordering a dwelling where the LA,eq does not exceed 45 dB is defined as a quiet side [3]. Access to quiet areas can thus be concluded to be a contribution to solve the urban road traffic noise problem. Quiet urban areas are often shielded from direct road traffic noise, e.g. closed courtyards. When inspecting maps of urban city centres in Europe, such enclosed areas are often present, or could be created by filling in openings in building blocks. The LA,eq at shielded sides does however often exceed 45 dB. As an example, the environmental administration of the city of Göteborg executed more than 600 indicative measurements at courtyards in the city [4]. The percentage of cases where LA,eq exceeded 45 dB was 60%. If noise from fans is disregarded and no openings to streets are assumed, the background noise level at shielded sides is determined by noise from road traffic in a wide area that reaches the shielded sides by propagation over the building roof level [5]. To be able to create shielded urban areas where the background noise level due to road traffic noise does not exceed 45 dB, a study of possible noise abatement schemes is an obvious issue. The accurate prediction of the effect of such noise abatement schemes requires a calculation model which includes material impedance, boundary roughness, edge diffraction and multiple reflections. Also, the influence of meteorological effects such as wind, temperature and turbulence on noise abatement schemes is of relevance

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and should be included. An accurate sound level prediction at shielded urban areas is also of importance to assess shielded sides psycho-acoustically, since such studies are still ongoing. The study of sound propagation to shielded urban areas has got its main attention in recent years. Accurate two-dimensional (2D) prediction models have successfully been developed: the finite difference time domain model by, e.g. Van Renterghem and Botteldooren [6], and the frequency domain equivalent sources method (2D ESM) by Ögren and Kropp [7]. Both methods model a coherent line source of infinite length in the geometry of parallel urban canyons, a source canyon and a shielded canyon. Since it has been shown in e.g. [8] that the use of a coherent line source instead of the more realistic incoherent line source to represent a traffic flow could lead to different results of a noise abatement treatment, a 2.5D ESM was recently presented by Hornikx and Forssén [9]. This method allows for calculating the response of a point source or an incoherent line source of arbitrary length in the geometry of parallel urban canyons, with an arbitrary position of the observer. Concerning noise abatement schemes, Hornikx and Forssén showed in a scale model study of parallel urban canyons that façade treatments are more effective for observer positions in the shielded canyon compared to observer positions in the directly exposed canyon due to the larger importance of higher order façade reflections at the shielded side [10]. Façade treatments as absorption and diffusion were shown to lead to a reduction of the level at the shielded canyon [11,12]. Besides the effect of absorption and diffusion, Van Renterghem et al. showed that a wind velocity gradient above the urban roof level influences the level at the shielded canyon substantially [11]. In addition, the level at the shielded canyon was found to increase due to scattering by atmospheric turbulence [11,13]. The above results were mainly based on studies with reference situations of acoustically hard and smooth façade surfaces and, concerning the numerical studies, a coherent line source. Real life façades already have some degree of absorption and diffusion, leading to a different effect of adding treatments. To make a step forward from the existing valuable studies of the benefits of absorption and diffusion for noise abatement in shielded areas, it is essential to use reference geometries and materials taken from real life cases and use a more realistic line source outside the shielded canyon. This is what this paper aims at. As an additional extension to the former studies, treatments at roof level are studied, the favourable location of façade absorption in the shielded canyon is investigated and the effects of treatments are evaluated at multiple observer positions in the shielded canyon. The effect of the canyon height to width ratio on treatments is also regarded, and the influence of a downward refracting wind condition is investigated for some situations. Finally, mean results per treatment are shown and the effect of treatments in a source canyon is discussed separately from treatments in the shielded canyon. It is important to mention that the main limitation of this study is the approximated geometry of the shielded areas. They have been treated as long canyons, instead of closed yards in a three-dimensional sense. Results from scale model measurements indicated that façade treatments for canyons are less efficient than corresponding ones for closed yards. The results of the current study can thus be seen as an underestimation of the real treatment effects [10]. The paper is organised as follows. In Section 2.1, the two selected city environments for study are presented. The source and observer positions in the environments are discussed in Section 2.2 and the chosen noise abatement schemes are shown in Section 2.3. Section 3 gives an explanation of the used calculation model, the 2.5D ESM. The chosen source type, a finite incoherent line source, is motivated by comparing its results with those from a coherent and an incoherent line source of infinite length. The results of the investigated noise abatement schemes are presented in Section 4 and are summarized in a table.

2. Situations of study 2.1. Selected courtyard geometries From the city of Göteborg, two areas have been selected as reference geometries for investigating the effect of noise abatement schemes. Göteborg is a city located at the west-coast of Sweden, as of 2006 populated by 489,000 inhabitants. Several parts of the city are characterised by building blocks containing a closed or a partially closed yard. A typical part is the area Linnéstaden. Fig. 1 shows a map of a part of this area, and a picture of the façades of the buildings at Linnégatan, the busiest road in the neighbourhood with an average traffic flow of 12,800 vehicles per 24 h (as of 2006) [14]. For this study, the courtyard marked in the map has been selected. The buildings consist of 8 floors with a total height of around 26 m. Their façades mainly consist of brickwork, windows and doors (see Fig. 2). Other typical areas in Göteborg characterised by their closed courtyards are the ones containing houses called Landshövdingehus. There are more than 20 areas in Göteborg with such buildings. These buildings were erected in the first half of the 20th century and are characterised by a ground floor façade of plastered brickwork and two floors with a wooden façade. The second chosen courtyard is located in the street Bomgatan and is indicated in Fig. 3, its cross-section is shown in Fig. 4. The busiest road close to Bomgatan is Mölndalsvägen, with an average traffic flow of 12,400 vehicles per 24 h (as of 2004) [14]. The height of the buildings is around 14 m. Both at Bomgatan and Linnégatan, the courtyard width is approximated by 25 m. The presence of narrower courtyards leads to the choice of a second geometry with a courtyard width of 14 m. The total of four courtyard geometries are modelled as canyons and summarised in Table 1 including the height to width ratio (H/W) of the canyons. For all reference canyon geometries, asphalt has been chosen as the ground surface and (saddle) roofs have been treated as being non-absorptive. Table 2 shows the used absorption coefficients of the materials, mostly taken from literature. 2.2. Source and observer positions Noise abatement schemes are studied in two parts: (I) By shielded canyon treatments (see Fig. 5a). (II) By source canyon treatments (see Fig. 5b). Source and shielded canyon situations have been chosen such that, by reciprocity, parts of the calculations are equal, which cuts computational costs. There is a notable difference between the two parts. Whereas shielded canyon treatments will affect levels with a weak dependence on the source position(s), source canyon treatments only affect the levels due to that particular source canyon; contributions from other source canyons are not reduced in level at the immission point. By separating source and observer by a large distance (here >500 m), the effect of treatments in both canyons can be added (taking into account the double roof reflection [5]), which can be seen as a far field approximation. The calculated results can from that point of view be generalized and they do not suffer from near-field effects as present when the distance to the closest busy road as of Figs. 1 and 3 would have been used in the calculations. The effect of treatments in the shielded canyon has been evaluated at a number of relevant observer positions in the shielded canyon (see Figs. 2 and 4); along the canyon ground surface at a height of 1.5 m and at window surfaces. The sources are modelled as incoherent line sources of finite length. For the source spectrum, a road traffic distribution of 90%


269

Fig. 1. (a) View in Linnégatan and (b) map with the chosen Linnégatan courtyard, indicated by a dot.

Fig. 2. Cross-section of the investigated courtyard geometries at Linnégatan with source and observer positions. (a) Linnégatan narrow (LN) and (b) Linnégatan wide (LW). The source positions at ground level are only used when studying abatement schemes within a source canyon.

270


Fig. 3. (a) Façade of a Landshövdingehus in Göteborg [15] and (b) map with the chosen Bomgatan courtyard, indicated by a dot.

Fig. 4. Cross-section of the investigated courtyard at Bomgatan with source and observer positions. (a) Bomgatan narrow (BN) and (b) Bomgatan wide (BW).

light and 10% heavy vehicles with a speed of 50 km/h has been chosen. The traffic spectra have been taken from Danish measurement data [19]. Concerning the source canyon treatments, road traffic may be represented by sources at several heights (see e.g. [19]). However, since 1/3-octave band results are of interest here, the dif-

ference between modelling one or multiple source heights is not expected to be significant. One source height of 0.15 m will therefore be used here and two road traffic lanes are modelled. For the shielded canyon treatments, a single line source is placed at roof level.

M. Hornikx, J. Forssén / Applied Acoustics 70 (2009) 267–283 Table 1 The four chosen canyon geometries Location

Initials

Height (m)

Width (m)

H/W (–)

Linnégatan Linnégatan Bomgatan Bomgatan

LN LW BN BW

25 25 14 14

14 25 14 25

1.8 1.0 1.0 0.6

H/W stands for the height to width ratio of a canyon.

Table 2 Absorption coefficients for a normal incident sound wave used in the calculations Material

1/1-Octave band (Hz) 125

250

500

1000

Cement plaster [17] Windows Brickwork, unglazed [17] 16 mm Wood at 4 cm framework [18] Asphalt (r = 2 107 Pa s/m2) [19] Grass ground (r = 5 105 Pa s/m2) [19] Grass roof, 15 cm layer (r = 1 105 Pa s/m2) [19]

0.02 0.10 0.03 0.18 0.00 0.06 0.18

0.02 0.10 0.03 0.12 0.01 0.10 0.28

0.03 0.10 0.03 0.10 0.01 0.15 0.42

0.04 0.10 0.04 0.09 0.02 0.24 0.58

The impedances of asphalt and grass have been calculated using the Delany and Bazley impedance model [16]. The grass materials have been used in abatement schemes.

2.3. Noise abatement schemes Treatments to reduce the sound pressure level in shielded canyons are here divided into absorption and screen treatments. For some screen treatments, additional absorption has been applied as well. The investigated noise abatement schemes are additions to the reference canyon geometries of Figs. 2 and 4 and are shown in a pictorial description in Fig. 6. The scheme numbers (e.g. A2) are used in Table 3, where results have been averaged per scheme. The effect of increasing façade absorption is investigated by successively increasing the absorption coefficient of the façade materials to the frequency independent values 0.4, 0.6 and 0.8, for a normal incident sound wave (schemes A1, A2 and A3). Notice that already 0.4 is larger than the largest absorption coefficient of the existing materials (see Table 1). The new absorption coefficient has only been assigned to the parts of the façades not being windows. In the geometries of Linnégatan, 32% of the façades are covered by façade absorption treatments and in the Bomgatan geometries, this portion is 48%. The effect of the ground surface has been investigated by changing its impedance from that of asphalt to that of grass (scheme A4). The combined change of the ground surface and an increase of the façade absorption coefficient to 0.8 has also been studied (scheme A5).

271

When applying façade absorption material, it is of interest to know its most favourable position. With schemes A1, A6, A7, A8 and A9, this preferable position has been investigated. A constant absorption area has been used and the absorption material has been located on both façades, the upper part of both façades, the lower part of both façades, the left façade or the right façade. Since the total absorption area of these schemes are equal, the absorption coefficient of the applied absorption varies over the schemes. The effect of a change of the rigid roof to that of a grass covered roof, e.g. a vegetation roof, has been studied in scheme A10. The impedance of a layer of 0.15 m soft grass on a rigid surface has then been used, see Table 2. The first type of screen treatment is the addition of horizontal screens above the windows (schemes S1, S2, S3 and S4). A thin horizontal screen may represent a (stiff) sunscreen or a walkway (note that we do not model geometrical changes along the canyon, i.e. in the y-direction). Three screens have been used per façade for the Bomgatan geometries and seven per façade for the Linnégatan geometries. Absorption material covers the lower face of the screens, with an absorption coefficient of either 0.1 or 0.8, for a normal incident sound wave. The screens are 1 m wide in schemes S1 and S2, and 2 m wide in schemes S3 and S4. With schemes S5 and S6, the influence of a thin vertical screen at the roof top is studied, either 1 or 2 m tall, respectively. The canyon shielding could be reduced under downward refracting conditions. To model this, a linear horizontal wind velocity profile with a gradient of 1 s1 is assumed which causes the sound waves to bend over the roof top (see Fig. 7). For the reference canyon geometries and scheme A3, this downward refracting atmosphere has been studied. These situations are referred to as situations D1 and D2, respectively. 3. The 2.5D ESM calculation model 3.1. Description of the calculation model The method used to assess the proposed noise abatement schemes is the 2.5D ESM for urban canyons [9]. With this method, the response of a point source or an incoherent line source in a geometry that is invariant in one direction (here the y-direction) can be calculated in the frequency domain. The 2.5D ESM solution is obtained by an integration over 2D ESM solutions. In the ESM, the computational domain is split into sub-domains in which the Green’s functions are known. The interfaces of the sub-domains are populated by equivalent sources. Fig. 8 shows the position of the equivalent sources at the interfaces for scheme S4 and exemplifies the various sub-domains; a rectangular cavity (the canyon), a free space above the cavity, minor rectangular cavities and impedance patches.

Fig. 5. The effect of noise abatement schemes is evaluated by separately considering treatments in (a) the shielded canyon and (b) the source canyon. The source is modelled as a finite incoherent line source, and the used coordinate system is shown in (a).

272


Fig. 6. Pictorial description of the investigated noise abatement schemes. The figures exemplify calculation part I (see Fig. 5a).

Table 3 Effect of the investigated shielded canyon noise abatement schemes DLref in dB(A), averaged over observer positions and all four canyon geometries Scheme Absorption treatments A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 Screen treatments S1 S2 S3 S4 S5 S6

Treatment description

FILS

Stdev

FILSsc

ILS

CLS

afac = 0.4 afac = 0.6 afac = 0.8 Façade absorption in upper canyon half Façade absorption in lower canyon half Absorption at left canyon façade Absorption at right canyon façade Grass roof

1.8 2.7 3.6 0.8 4.1 2.1 1.4 1.9 1.6 2.2

0.8 1.3 1.8 0.5 2.0 0.9 0.8 1.0 0.8 0.4

2.4 4.0 5.5 2.0 7.1 2.5 2.8 2.5 2.7 2.2

2.3 2.9 3.2 1.1 4.8 2.5 1.2 1.8 1.6 4.6

1.6 2.6 3.5 0.7 3.9 2.0 1.4 1.8 1.6 2.0

horizontal screen, ascreen = 0.1 horizontal screen, ascreen = 0.8 horizontal screen, ascreen = 0.1 horizontal screen, ascreen = 0.8 vertical screen at roof vertical screen at roof

0.9 2.5 3.0 5.5 4.3 5.6

1.0 1.8 2.2 3.7 0.4 0.4

1.7 4.4 4.5 9.4 4.4 5.7

1.4 2.7 3.2 4.8 4.3 5.4

0.8 2.3 2.8 5.4 4.3 5.6

0.6 1.3

2.6 2.6

1.2 2.8

0.5 0.2

0.9 1.5

Grass ground

afac = 0.8 and grass ground

1m 1m 2m 2m 1m 2m

Reference scheme Downward refraction D1 D2

Reference canyon geometry Scheme A3

Results are given for the finite incoherent line source (FILS), the corresponding standard deviation (Stdev), the incoherent line source (ILS) and the coherent line source (CLS). Results of source canyon treatments for the finite incoherent line source (FILSsc) are averaged over the two source positions and four canyon geometries.

Fig. 7. For the reference canyon geometries and scheme A3, a downward refracting wind velocity is modelled such that canyon shielding is absent. The figures exemplify calculation part I (see Fig. 5a).

273


Fig. 8. Equivalent sources in the ESM, placed in the LN geometry with 2 m wide screens (scheme S4). The equivalent sources are placed at sub-domain interfaces. Note that the screens only have absorption at their lower face.

The amplitudes of the equivalent sources are calculated by fulfilling the Helmholtz equation in the sub-domains, the locally reacting boundary conditions at the impedance patches and the continuity of pressure and normal velocity across the sub-domain interfaces. The pressure for an arbitrary observer position in the computational domain can then be calculated by the contributions from the equivalent sources bordering the respective sub-domain (and the primary source if located in the same sub-domain). The façade absorption has here been modelled as a real valued impedance and the ground surface material as a complex impedance.

Another different Green’s function concerns the one above the roof level. In the current application, we model a saddle roof and, in some schemes, a thin screen. Numerically, a wedge diffraction solution is used to obtain the Green’s function from a point outside the canyon to the equivalent sources at canyon roof level. Calculating a 2D wedge diffraction solution is however much slower than calculating a 3D wedge diffraction solution. Since the insertion loss of a 3D wedge equals the insertion loss of a 2D wedge in the far field (kr > 1) and for a spherical wave divergence correction [21], a 3D wedge diffraction solution has here been used according to [22]. The Green’s function becomes


G2D;wedge Compared with a previous ESM [7], some different Green’s functions have been used here. The major part of the ESM calculation time is caused by calculating the Green’s functions between the equivalent sources in the cavities. It is therefore computationally attractive to use a faster Green’s function than the formerly used modal summation in two-dimensions; a combination of a modal and a wave approach. A single term expression based on a wave approach is used in one direction, the x-direction, and a modal summation is used in the z-direction. Since the expression for the wave approach is a single term, only one summation remains in this method. Formulas for such a Green’s function have been published among others by Morse and Feshbach [20]. For the situation studied here, the Green’s function can be formulated as

Gðxjxs ; zjzs Þ 8 N P > cosðnpzs =lz Þ cosðnpz=lz Þ 2 > cosðkn ðlx xs ÞÞ cosðkknn xÞÞ > < lz sinðkn lx Þ n¼0 ¼ N > P > s ÞÞ cosðnpzs =lz Þ cosðnpz=lz Þ > cosðkn ðlx xÞÞ cosðkknnxsinðk : l2z n lx Þ

for 0 6 x 6 xs ; for xs 6 x 6 lx ;

n¼0

ð1Þ where xs and zs are the source coordinates, lx and lz the cavity dimensions and kn the modal wave number, which may include air absorption. Since, for a certain maximum mode number N, this expression is more accurate than the modal solution in both directions, the Green’s function calculation time has been sped up more than quadratically with the number of equivalent sources.

G3D;wedge G2D;free G3D;free

rffiffiffiffiffiffiffiffiffiffiffiffi r wedge ; rfree

ð2Þ

where G2D,wedge and G3D,wedge are the Green’s functions for a wedge in 2D and 3D, G2D,free and G3D,free are the free field Green’s functions in 2D and 3D, rfree is the free field distance between two field points and rwedge the shortest distance between two field points via the wedge. For the Green’s functions among the equivalent sources at the canyon roof level, diffraction components at the wedge top and wedge base have been neglected, i.e. here a flat roof has been assumed. For refraction schemes D1 and D2, the Green’s functions above the canyon roof level have been calculated according to a ray approach as described by Salomons [23, Chapter L]. Air absorption has not been included in this study. Since air absorption increases with propagation distance and frequency, it has to be kept in mind that abatement schemes are of larger importance for lower frequencies here. 3.3. Convergence and accuracy The 2D ESM has successfully been compared with the boundary element method (BEM) for the case of rigid walls with the source and observer in the same canyon [12]. Accuracy and convergence of the 2.5D ESM has been discussed in [9]. In addition, Appendix A shows a comparison between 2.5D ESM results and those from a previously performed scale model study for a point source and two parallel canyons [10]. Cases with rigid boundaries, façade absorption and façade diffusion have been compared, showing good agreement.

274


The number of 2D solutions required to calculate the incoherent line source solutions is taken as proposed by Hornikx and Forssén [9]. Six equivalent sources per wavelength have been used, which is a trade-off between accuracy and computation time. Thirty logarithmically spaced frequencies per 1/3-octave band have been used. For abatement schemes with horizontal screens (i.e. S1, S2, S3 and S4), the sub-domain complexity is higher than for previously implemented ESM codes. To verify these implementations, results in the limit of small horizontal screen widths were found to led to the results without screens. 3.4. Relative sound pressure level DLref Calculation results are presented by the relative sound pressure level DLref

DLref ¼ Lp Lp;ref

ð3Þ

with Lp the level for a certain noise abatement scheme and with Lp,ref the level in the reference canyon geometry. The relative sound pressure levels DLfree and DLrigid are also used and represent levels relative to, respectively, the free field situation and the situation of a canyon with smooth and acoustically hard boundaries. When results averaged over observer positions are presented, the averaging has been done over DL in dB(A), or in dB when frequency dependent results are presented. 3.5. Finite incoherent line source (FILS) The use of a proper type of line source is highly relevant in this study. An incoherent line source (ILS) is more suitable to represent a traffic flow than a coherent line source (CLS), but is not appropriate for the current application. A finite incoherent line source (FILS) has instead been used, which will be motivated below by comparing results from using the three different source types. The FILS solution can be obtained by using the 2.5D ESM, see Appendix B. Fig. 9 illustrates the behaviour of the results from using the three different source types by a frequency excerpt of the relative level for the BN canyon geometry for calculation part I with smooth nonabsorptive façades and the observer at position O7 (see Fig. 4). The dominance of the modes for the CLS is clear whereas modes are hardly visible in the ILS results due to the incoherent addition of point sources. The FILS with a maximum horizontal angle between source and observer of hmax = 45°, see Fig. B.1, does not exhibit the extreme results of either the CLS or the ILS. Fig. 10 shows the effect of absorption (scheme A3 relative to the reference canyon geometry) and diffusion (reference canyon geometry with non-absorbing boundaries relative to reference 20 10

Δ Lfree (dB)

0 –10

4. Results Calculation results are presented in 1/3-octave bands ranging from 100 to 1000 Hz, where the computation time has limited the upper frequency. Even though road traffic noise emission is important above 1000 Hz, the most difficult noise reduction at the immission points is for the lower frequencies. Furthermore, for all cases studied here, the A-weighted noise level is dominated by the 1/3-octave bands up to 1000 Hz. The results of all noise abatement schemes are presented and discussed in Sections 4.1– 4.4 for treatments in the shielded canyon, i.e. calculation part I as of Fig. 5a. The results have been averaged over all observer positions and four canyon geometries (LN, LW, BN, BW) and are summarised in Table 3. The results of the noise abatement schemes in the source canyon, i.e. calculation part II as of Fig. 5b, are also included in Table 3 and are discussed in Section 4.5. 4.1. Reference canyon results for the shielded canyon The average results for the reference canyon geometries relative to free field, DLfree, are shown in Fig. 11a. The thick line is DLfree and the thin line its standard deviation (with value axis on the right hand side). The DLfree decreases with frequency owing to the canyon edge diffraction. Fig. 11b shows the DLfree averaged per canyon geometry in dB(A) with the standard deviation illustrated by error bars. The DLfree is similar for all canyon geometries. To illustrate the variation of the level over the observer positions, Fig. 11c shows the average DLfree for observer positions along the ground floor for the wide canyons, i.e. BW and LW, and Fig. 11d displays the average DLfree for observer positions over the height of the canyon. For the latter, an average for façade positions in Linnégatan has been taken over both canyon widths, i.e. LN and LW, and both façades. The results in Fig. 11c exhibit a rather weak dependence over the width of the canyon, whereas the levels decrease with decreasing observer height in Fig. 11d. The high values for the upper observer positions in Fig. 11d are caused by the large depression of the windows there, see Fig. 2. One has to keep in mind that, owing to the presence of the boundary, levels for façades observer positions are higher than for canyon observer positions (up to 3 dB higher levels for a diffuse sound field). 4.2. Absorption schemes for the shielded canyon

CLS FILS, θmax = 45 °

–20

Results of noise abatement schemes in this Section and Sections 4.3 and 4.4 are presented in a similar way as in Fig. 11. Fig. 12a shows DLref as a function of frequency for the three absorption treatments A1, A2 and A3. The effect of absorption increases with

ILS –30 –40 65

canyon geometry with smooth and non-absorbing boundaries) for several maximum angles in the FILS. For a zero maximum angle, the FILS technically represents a point source solution, which has a relative level similar to the relative level of the CLS for screened observer positions [21]. The results indeed show that the FILS tends to the CLS results for a decreasing hmax. On the other hand, the FILS results approach the ILS results when hmax approaches 90°. Studying the FILS and the CLS results, a substantial deviation can be seen for hmax > 70°. However, for most real life canyons, with finite dimensions, the use of sources at these large angles is probably not realistic, but could be modelled for specific situations. A moderate choice of hmax = 45° has therefore been made here,2 with the aim of receiving more generalized results.

70

75 frequency (Hz)

80

85

Fig. 9. Relative levels, DLfree, for three different source types. Canyon geometry BN for calculation part I (Fig. 5a) with observer position O7. Façades are smooth and non-absorbing.

2 Note that for the modelled FILS, 45° stands for the maximum angle between source and observer as well as all image observers in the canyon, which are implicitly included in the ESM solution.

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2

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ref

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ILS CLS FILS, θ = 45° max FILS, θ = 70° max FILS, θ = 85° max FILS, θ = 89° max

–10

1000

125


1000

Fig. 10. Relative levels, DLref and DLrigid, for various source types, for (a) an absorption coefficient of 0.8, i.e. scheme A3, and (b) the reference canyon geometries with nonabsorbing boundaries. Results are for calculation part I (Fig. 5a) and averaged over all observer positions and all canyon geometries.

–6

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free

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Fig. 11. Average relative levels, DLfree, for the reference canyon geometries for calculation part I (see Fig. 5a). (a) Observer positions for all canyon geometries, (b) observer positions per canyon geometry with standard deviations, (c) observer positions at a height of 1.5 m in the wide canyons geometries, and (d) observer positions in the Linnégatan canyon geometries for façade observer positions. Thick line: DLfree. Thin line: standard deviation of DLfree.

the absorption coefficient and is rather constant over frequency. Fig. 12b shows the average of scheme A3 per geometry. Although differences are smaller than 1 dB(A), it can be concluded that the effect of façade absorption is more effective for the narrower canyons. Fig. 12c shows that the DLref is rather equal over the width of the canyon. From Fig. 12d, it can be concluded that the reduction weakens with increasing observer height. The reason is that the levels at the lower observer positions are more dependent on the higher order reflections (as explained by the rise time in [10]). The standard deviations show to be larger for larger reductions. Fig. 13 shows the results of the treatments A4 and A5. The upper curve in Fig. 13a is the effect of changing the ground surface from asphalt to grass. Although the absorption of grass is larger for the higher frequencies (for normal incidence), this is not very obvi-

ous from the results. Compared to the façade absorption scheme A3, the change to a grass ground surface (scheme A5) does not lead to a large further reduction. Fig. 13b displays the results for the scheme with a grass ground surface, scheme A4. Although differences are smaller than 1 dB(A), it shows that the ground surface has a larger effect in the wider canyons. This is an expected result, since the grass absorption area is larger for the wider canyons. Ground absorption does not lead to a large variation of DLref over the width of the canyon, see Fig. 13c, and affects the lowest observer positions most, see Fig. 13d. Fig. 14 shows the effect of five different absorption material placements, according to the sketches in Fig. 6. The differences among the five schemes are in general not large. Note that, for a result averaged over all observer positions, the effect of absorption at

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Fig. 12. Average relative levels, DLref, for façade absorption schemes in the shielded canyon. (a), (c) and (d) as in Fig. 11. The results in (b) are for scheme A3. Thick lines: DLref. Thin lines: standard deviation of DLref.

A4 A5

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Fig. 13. Average relative levels, DLref, for ground absorption schemes in the shielded canyon. (a), (c) and (d) as in Fig. 11. The results in (b) are for scheme A4. Thick lines: DLref. Thin lines: standard deviation of DLref.

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A1 A6 A7 A8 A9

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ref

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6

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Δ Lref (dB)

Fig. 14. Average relative levels, DLref, for schemes with various absorption material locations in the shielded canyon. (a), (c) and (d) as in Fig. 11. The results in (b) are per scheme. Thick lines: DLref. Thin lines: standard deviation of DLref.

0 25

Fig. 15. Average relative levels, DLref, for the shielded canyon grass roof scheme. (a), (c) and (d) as in Fig. 11. The results in (b) are for scheme A10. Thick line: DLref. Thin line: standard deviation of DLref.


the left and at the right façade have to be averaged when multiple roads with arbitrary location contribute to the canyon level. According to Fig. 14b, absorption material placed in the upper part of the canyon façades gives the best results and absorption material placed in the lower half of the canyon gives the smallest effect. One reason is that for the latter scheme, only the lower observer positions are largely affected by the façade absorption material. The effect of a grass covered saddle roof is presented in Fig. 15. The frequency dependence in Fig. 15a is caused by the properties of the used material (see Table 2). The grass roof causes larger losses for narrower canyons, see Fig. 15b, and does not exhibit a clear dependence on the observer position. This is in contrast with the height dependent results of the former schemes. The losses due to the grass roof are likely additive to in-canyon treatments. 4.3. Screen schemes for the shielded canyon

Fig. 19 shows the influence of a downward refracting atmosphere on the canyon observer levels, schemes D1 and D2. For the D1 scheme, the results are presented relative to the reference canyon geometry, whereas for the D2 scheme, results are presented with scheme A3 as reference. There are basically two effects occurring in the downward refracting case that explain the results (see Fig. 18). (1) The role of the absence of canyon shielding. Whereas all wave contributions to canyon observer positions in the non-refracting case are diffracted by the saddle roof, canyon observer positions have direct or reflected wave contribu-

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4.4. Downward refraction for the shielded canyon

0

stdev (dB)

Δ Lref (dB)

The horizontal screens make the sound field more diffuse by breaking up the modes in the canyon. In addition, the screens prevent the sound waves from being scattered down into the canyon by reflecting waves upwards and absorbing them at the lower screen faces. Finally, façade observer positions will become more shielded for noise immission. Fig. 16 shows the results for the treatments S1–S4. For the 1 and 2 m wide screens with an absorption coefficient of 0.1 (schemes S1 and S3, respectively), there is no clear frequency dependent effect (see Fig. 16a). The increase of the absorption coefficient at the lower faces of the screens has a large effect as can be seen in the results for a 1 and 2 m wide screen with an absorption coefficient of 0.8 (schemes S2 and S4, respectively). This effect is larger for the lower frequencies. Fig. 16b shows the results of scheme S4 and illustrates that the effect of horizontal screens at façades is more effective for the narrow canyons. The stronger effect for Linnégatan may be attributed to the fact that a larger number of screens has been applied than for Bomgatan. Over the width of the canyon, the effect of the treatments is rather even,

see Fig. 16c. From Fig. 16d, we see that the effect of the screens is strongly height dependent. The highest observer positions undergo a negative effect due to reflections from the screens. The effect of a thin vertical screen on top of the canyon roof, schemes S5 and S6, is shown in Fig. 17. Fig. 17a shows that the effect of a 1 m tall screen (scheme S5) is somewhat more that 4 dB and quite even over frequency. The 2 m tall screen reduces the levels only a little more compared to the 1 m tall screen. The excess screen reduction can be attributed to the rooftop being elevated and the shape of the roof being changed. The small difference between the effect of the 1 and 2 m tall screens might indicate the larger influence of the roof shape change. Fig. 17b displays the effect of the 2 m tall screen per canyon geometry. The narrow canyons benefit to a slightly larger extent from the screens than the wide canyons. Fig. 17c and d show that the effect of the screens is rather independent on the observer position, owing to the fact that the screen is located outside the canyon. The low standard deviation of the results also gives evidence to the weak position and frequency dependence. The effect of the roof top screen is likely additive to in-canyon treatments.

6

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2

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20

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Fig. 16. Average relative levels, DLref, for horizontal screens schemes in the shielded canyon. (a), (c) and (d) as in Fig. 11. The results in (b) are for scheme S4. Thick lines: DLref. Thin lines: standard deviation of DLref.

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S5 S6

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ref

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20

0 25

Fig. 17. Average relative levels, DLref, for the vertical screens schemes outside the shielded canyon. (a), (c) and (d) as in Fig. 11. The results in (b) are for scheme S6. Thick lines: DLref. Thin lines: standard deviation of DLref.

Fig. 18. Sketches of the two dominant effects under a downwind refracting atmosphere: (1) the absence of canyon shielding for canyon observer positions and (2) the different number of reflections for the downwind refracting atmosphere and the non-refracting atmosphere.

tions in the calculated downward refracting case. Since canyon shielding is stronger for the higher frequencies and lower observer positions, downward refraction is impairing for these regions, see Fig. 19a and d. (2) The role of multiple reflections. In the downward refracting case, the angle of the incoming sound waves is such that waves are being reflected out of the canyon after a small number of reflections, if reflections are being regarded as specular. In the non-refracting case, waves diffracted by the wedge top under low angles will reflect many times in the canyon and excite the canyon modes very well. The result of this phenomenon is that the downward refracting

case can be favourable for low frequencies (see Fig. 19a) and high observer positions (see Fig. 19d), where canyon shielding is low in the non-refracting case. The second effect also explains the fact that the influence of downward refracted sound waves on the façade absorption scheme A3 is more impairing than on the reference canyon geometry, see Fig. 19; in a downward refracting case, the sound pressure level at the canyon observer positions is less influenced by the higher order reflections than for the non-refracting case, making façade treatments less effective. Finally, the first effect explains the difference between wide and narrow canyon results, see

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4 Δ Lref (dB(A))

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20

0 25

Fig. 19. Average relative levels, DLref, for downward refraction in the shielded canyon calculation part. For D2, DLref = Lp,D2 Lp,A3. (a), (c) and (d) as in Fig. 11. The results in (b) are for scheme D1. Thick lines: DLref. Thin lines: standard deviation of DLref.

Fig. 19b, since the absence of canyon shielding is more relevant for narrow canyons. 4.5. Average results and source canyon schemes The results of all studied shielded canyon noise abatement schemes using the finite incoherent line source (FILS) are summarised in Table 3. The results for the source canyon schemes (FILSsc) as well as the results for the incoherent and coherent line sources (ILS and CLS) for shielded canyon schemes are also listed in the table. The standard deviations are given for the shielded canyon FILS results and display what we have seen before: for in-canyon treatments, the standard deviation is larger for treatments with a larger reduction. The standard deviations are smaller for treatments outside the canyon (i.e. grass roof and canyon roof screens). For all in-canyon treatments, the source canyon treatments (FILSsc) give a larger effect than the shielded canyon treatments (FILS). The reason is that the sources for the source canyon treatments are located closer to the ground surface than the chosen observer positions for the shielded canyon treatments. When a single source canyon is dominating the sound pressure level at a shielded canyon, treatments in that particular source canyon are thus more effective than similar treatments in the shielded canyon itself for most observer positions. For most shielded canyon treatments, the FILS results are in-between the ILS and CLS results. Although the CLS was not expected to give realistic results, calculation results show that the use of a more realistic FILS yields similar results. In addition, the effect of absorption treatments is larger using the ILS than using the FILS, and the vertical screen effects are rather similar for all source types. Furthermore, downward refraction is less unfavourable for noise abatement for the ILS case since canyon shielding, as partially absent by downward refraction, was smaller than for the FILS.

5. Conclusions Noise abatement schemes for two real life courtyards in Göteborg, shielded from direct road traffic noise, have been investigated numerically using the 2.5D equivalent sources method (2.5D ESM). The abatement schemes involve façade absorption, horizontal screens at façades, a vertical screen on the roof and grass surfaces on ground and roof, all compared with the real life courtyards as reference. The grass surfaces have been modelled as complex-valued impedances in the ESM. In addition, effects of a downward refracting atmosphere have been studied. The courtyards have been modelled as long canyons. The results therefore represent the effect of noise abatement schemes for road traffic noise sources outside an elongated shielded area. A line source with a road traffic noise spectrum outside the shielded area has been used as the sound source. Calculations with the chosen canyon geometries show that the source modelled as a coherent or incoherent line source represent extreme situations. The more realistic finite incoherent line source (FILS) has therefore been used here with a chosen length corresponding to an angular coverage of ±45°. The results of the noise abatement schemes however show that the coherent line source results are within 0.3 dB (A) of the FILS results. The effects of the noise abatement schemes have been evaluated depending on frequency, observer position and height to width ratio of the canyon geometries. Noise abatement schemes in the source canyon have been separately studied from schemes in the shielded canyon, with the following conclusions from the latter study. The sound pressure level in the reference canyon geometries is the lowest for the lower observer positions in the canyon and for the highest frequencies. Applying additional façade absorption with a frequency independent absorption coefficient can be concluded to yield a:


larger level reduction for the lower canyon observer positions; larger level reduction for the narrower canyons (although differences are smaller than 1 dB(A)); rather similar reduction over frequency. For the situations studied here, a change of the façade absorption coefficient, a, to 0.8 leads to a reduction of around 4 dB(A) for most canyon observer positions. From a study of the preferable façade absorber location, it can be concluded that absorption applied in the upper part of the façades is slightly favourable for noise abatement. A grass ground surface instead of an asphalt ground surface amounts to an average reduction of 1 dB(A). A grass covered saddle roof reduces the levels rather independently on the canyon observer position by 2 dB(A). A combination of horizontal screens, like walkways, and absorption at the lower face of the screens has an effect that strongly decreases with canyon observer height. In addition, the effect of horizontal screens is largest for the narrower canyons and more effective for the lower frequencies when absorption has been applied. An average level reduction of 2.5 dB(A) is obtained for 1 m wide horizontal screens with absorption material underneath them (a = 0.8). Applying a vertical screen at the top of the saddle roof is more cost effective than applying the horizontal screens; a 1 m tall screen reduces the level for all canyon observer positions by more than 4 dB(A). The effect of a vertical screen at the rooftop is slightly more effective for narrower canyon geometries. For downward refracting conditions, the canyon shielding effect is partly cancelled for the highest frequencies and lowest observer positions. For lower frequencies and high observer positions however, downward refraction could result in lower levels. The narrow canyons are, on average, the most negatively affected by downward refracting conditions. Also, a façade absorption treatment is less effective under downward refracting conditions. The effect of the noise abatement schemes in the source canyon, for an observer outside the canyon, is larger than for shielded canyon schemes owing to the low source positions. The relevance of a source canyon abatement scheme however depends on the contribution of this source canyon to the total noise level in the shielded canyon. This study is limited to the chosen courtyard geometries modelled as 2D canyons. For a more accurate prediction of the effect of noise abatement schemes for a closed courtyard in a 3D sense, a 3D calculation model is necessary. Acknowledgements The work behind this paper has been supported by the Swedish Foundation for Strategic Environmental Research (MISTRA). The authors would like to thank Wolfgang Kropp for his ideas and fruitful discussions. Appendix A. Validation of the 2.5D equivalent sources method (2.5D ESM) by scale model measurements A 1–40 scale model study of parallel street canyons was executed to investigate the sound field in both streets [10], with a

281

point source located in one of the canyons. The results of this study are now used to validate the 2.5D ESM. Fig. A.1 shows two crosssections of the scale model geometry with full scale dimensions. In Fig. A.1a, all boundaries are smooth and rigid. The locations of source and observers are also shown. Fig. A.1b shows the cross-section with applied façade patches and represent two more situations. In one situation, the patches are rigid and façade reflections are partly diffuse. In the second situation, the patches consist of a porous material. The impedance of the used material has been obtained in [10]. A comparison is carried out in 1/3-octave band levels. Fig. A.2a and b show the measured and calculated level relative to the free field level at 1 m from the source, Lfree,1m, for the case of rigid façades. Both the calculated and the measured results are presented in surface plots as a function of y-distance and frequency. The results are further compared over y-distance and over frequency (frequency and position averaged results are presented, respectively). For the case with rigid façades, an absorption coefficient amodel of 5% over the whole frequency range for the façades and ground surface was used in the calculations. Using the measured impedances of the façades and ground surface (and including the boundary layer effect) would lead to an over-prediction of the sound pressure levels. The used amodel is a combination of absorption coefficients of the materials and diffusive and absorptive effects by roughness elements in the source canyon (i.e. used tape, gaps between building blocks, transitions in the ground surface). A best fit of amodel is not the aim here, since that would also cover errors in the calculation model and measurements. The case with rigid façades shows a fine agreement both in the source canyon and in the shielded canyon. Fig. A.2c and d are for the case with absorption patches. In this case, we find fairly well matching curves over frequency in both directly exposed and shielded canyon, characterised by the absorption coefficient of the felt material. The deviations at the highest frequencies can be caused by the directionality of the microphone (see [10]). This difference was also visible between the measured and modelled absorption coefficient of the felt material [10]. The high frequency deviations cause the level deviation over the distance. Fig. A.2e and f show the results in the case with diffusion patches at the façades. The results in the source canyon are quite well matching. Over the y-distance, the model however over-predicts the measurements from a certain distance. Extra damping caused by the small gaps behind the patches could be the reason for the deviation. To account for this extra damping, the ESM calculations as presented in Fig. A.2e and f were executed with an absorption coefficient of 7% for all façades and the ground surface. This is obviously not high enough. Also, back diffusion effects from the façades at a y-distance larger than 50 m are not included in the measurements. The agreement is less good in the shielded canyon, where the measured levels are below the calculated ones. The deviation at the highest frequencies could be caused by the microphone directionality. Appendix B. The finite incoherent line source solution (FILS) The finite incoherent line source solution can be obtained using the 2.5D ESM. This will here be shown for the case of an observer placed at an infinitely large wall with a normalised surface imped-

Fig. A.1. Cross-section of parallel street canyons used in a 1–40 scale model. Streets are 100 m long, ranging from 50 to 50 m with the point source located at 0 m. (a) Crosssection with rigid façades and (b) cross-section with extra patches (rigid or absorptive).

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Fig. A.2. Comparison between results from a scale model study and 2.5D ESM calculations. (a) Rigid façades, observer position O1, (b) rigid façades, observer position O2, (c) façades with absorption patches, observer position O1, (d) façades with absorption patches, observer position O2, (e) façades with diffusion patches, observer position O1, and (f) façades with diffusion patches, observer position O2. Circles: measurements and solid lines: 2.5D ESM results.

reflection coefficient, (Z cos(h) 1)/(Z cos(h) + 1), which is the locally reacting far field approximation. The integration variable will now be changed to h. Using the relation y = x tan(h), we find

LFILS ðx; 0; 0Þ ¼ 10 log

Z

hmax

hmax

¼ 10 log

Z

hmax

hmax

Fig. B.1. Finite incoherent line source calculation setup. Observer placed at an infinitely large wall with normalized surface impedance Z.

ance Z. An incoherent line source is located at x, the shortest distance to the observer, and the maximum transverse distance to the observer is ymax (see Fig. B.1). Both source and observer are placed at a hard ground surface. The finite incoherent line source level (re. 1 Pa) now reads

LFILS ðx; 0; 0Þ ¼ 10 log

Z

!

ymax

¼ 10 log

Z

jð1 þ RÞG3D ðx; ys ; 0Þj dys ymax

ymax

j1 þ Rj2 dys x2 þ y2s

x secðhÞ2 ! j1 þ Rj2 dh : x

! dh

ðB:2Þ

We will now show that this solution for a finite incoherent line source can be calculated using a 2.5D calculation method. In a 2.5D calculation method (the 2.5D ESM has been used in the paper), we have a Fourier transform pair (see [10]). Using Parseval’s theorem, we can calculate the incoherent line source level, denoted by LILS,ESM

LILS;ESM ðx; 0; 0Þ ¼ 10 log

1 2p

Z

1

jð1 þ RESM ÞG2D ðx; K; 0Þj2 dky

1

ðB:3Þ

2

ymax

j1 þ Rj2 secðhÞ2

! ðB:1Þ

with half-space G3D the Green’s function of the inhomogeneous 3D Helmholtz equation with source strength 2p and R the plane wave

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 with K ¼ k ky , the wave number ky is the projection of k onto the y-axis such that ky = k sin(h) and the reflection coefficient RESM is the plane wave reflection coefficient using the impedance Z2D as in the 2.5D ESM [9]

RESM ¼

Z 2D 1 Kk Z 1 ¼ : Z 2D þ 1 Kk Z þ 1

ðB:4Þ


We approximate pG 2D(x, K,ffi 0) by its far field approximation ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2D ðx; K; 0Þ ¼ p=j ð2=KxpÞejKx ejp=4 and use integration limits ky,max and ky,max to get the finite incoherent line source level

! Z ky;max 1 2 LFILS;ESM ðx; 0; 0Þ ¼ 10 log jð1 þ RESM ÞG2D ðx; K; 0Þj dky 2p ky;max ! Z ky;max j1 þ RESM j2 ¼ 10 log ðB:5Þ dky : Kx ky;max

As for the calculation of the first finite incoherent line source level, we will here substitute the integration variable to h

LFILS;ESM ðx; 0; 0Þ ¼ 10 log

Z 0

B ¼ 10 log B @

¼ 10 log

hmax

hmax

Z

hmax

hmax

Z

hmax

hmax

! j1 þ RESM j2 cosðhÞk dh Kx 1 j1 þ RESM j2 cosðhÞ C rffiffiffiffiffiffiffiffiffiffiffiffi dhC A k2

1 ky2 x

! j1 þ RESM j2 dh : x

ðB:6Þ

Since RESM = R for ky = k sin(h), both methods are identical. Since the latter method is based on solutions of the 2D Helmholtz equation, whereas the former is based on solutions of the 3D Helmholtz equation, the latter is much less time consuming than the former in the canyon geometries of this paper. Note that, since ky,max has been kept constant during 2.5D ESM calculations, the line source length virtually increases for image source contributions. References [1] Berglund B, Lindvall T, Schwela D, editors. Guidelines for community noise. Geneva: World Health Organization (WHO); 2000. [2] Öhrström E, Skånberg A, Svensson H, Gidlof-Gunnarsson A. Effects of road traffic noise and the benefit of access to quietness. J Sound Vib 2006;295(1):40–59. [3] The Swedish Research Programme Soundscape Support to Health. .

283

[4] Data obtained from the environmental administration of Göteborg. . [5] Thorsson PJ, Ögren M, Kropp W. Noise levels on the shielded side in cities using a flat city model. Appl Acoust 2004;65(4):313–23. [6] Van Renterghem T, Botteldooren D. Numerical simulation of sound propagation over rows of houses in the presence of wind. In: Proceedings of the 10th international conference on sound and vibration, Stockholm, Sweden; 2003. p. 1381–8. [7] Ögren M, Kropp W. Road traffic noise propagation between two dimensional city canyons using an equivalent sources approach. Acust Acta Acust 2004;90(2):293–300. [8] Jean P, Defrance J, Gabillet Y. The importance of source type on the assessment of noise barriers. J Sound Vib 1999;226(2):201–16. [9] Hornikx M, Forssén J. The 2.5-dimensional equivalent sources method for directly exposed and shielded urban canyons. J Acoust Soc Am 2007;122(5):2532–41. [10] Hornikx M, Forssén J. A scale model study of parallel urban canyons. Acust Acta Acust 2008;94(2):265–81. [11] Van Renterghem T, Salomons E, Botteldooren D. Parameter study of sound propagation between city canyons with a coupled FDTD-PE model. Appl Acoust 2006;67(6):487–510. [12] Ögren M. Prediction of traffic noise shielding by city canyons. Göteborg, Ph.D. thesis. Chalmers University of Technology; 2004. [13] Ögren M, Forssén J. Modelling of a city canyon problem in a turbulent atmosphere using an equivalent sources approach. Appl Acoust 2004;65(6):629–42. [14] Traffic flow data obtained from the traffic agency of the Göteborg Municipality. . [15] Photo material obtained from the Building Department of the Göteborg Municipality. [16] Delany ME, Bazley EN. Acoustical properties of fibrous absorbent materials. Appl Acoust 1970;3(2):105–16. [17] Cox TJ, D’Antonio P. Acoustic absorbers and diffusers: theory, design and application. Spon Press; 2004. [18] Brandt O. Akustisk Planering, Handbok nr 1 Statens Nämd för Byggnadsforskning, Stockholm; 1958 [in Swedish]. [19] Jonasson H, Storeheier S. Nord 2000. New Nordic Prediction Method for Road Traffic Noise. SP Rapport 2001:10, Borås, Sweden; 2001. [20] Morse P, Feshbach H. Method of theoretical physics. McGraw-Hill; 1953. p. 1369. [21] Van Renterghem T, Salomons E, Botteldooren D. Efficient FDTD-PE model for sound propagation in situations with complex obstacles and wind profiles. Acust Acta Acust 2005;91(4):671–9. [22] Hadden JW, Pierce AD. Sound diffraction around screens and wedges for arbitrary point source locations. J Acoust Soc Am 1981;69(5):1266–76 [Erratum, J Acoust Soc Am 1982;71(5):1290]. [23] Salomons EM. Computational atmospheric acoustics. Dordrecht: Kluwer Academic Publishers; 2001.

Errata The following corrections need to be observed regarding Paper III: ◦ Page 271, Fig. 5(b):

should be replaced by:

◦ Pages 282 and 283. The Expressions "LFILS (x, 0, 0)" from Eqs. (B.1) and (B.2), "LILS,ESM (x, 0, 0)" from Eq. (B.3) and "LFILS,ESM (x, 0, 0)" from Eqs. (B.4) and (B.5) should respectively be replaced by: "LFILS (0, 0, 0)", "LILS,ESM (0, 0, 0)" and "LFILS,ESM(0, 0, 0)".

Paper IV

3739

An eigenfunction expansion method to efficiently evaluate spatial derivatives for media with discontinuous properties M. Hornikxa and R. Waxlerb a

Applied Acoustics, Chalmers University of Technology, Sven Hultins Gata 8a, SE-41296 Gothenburg, Sweden b University of Mississippi, NCPA, 1 Coliseum Drive, University, MS 38677, USA [email protected]

3740 Pseudo-Spectral methods are often used as an alternative to the Finite Difference Time Domain (FDTD) method to model wave propagation in heterogeneous moving media. The FDTD method is robust and accurate but is numerically expensive. Pseudo-Spectral methods make use of the wavelike nature of the solution to obtain more efficient time-domain algorithms. The most straightforward of the Pseudo-Spectral methods is the Fourier method in which a spatial Fourier transform is used to evaluate the spatial derivatives in the wave equation. Whereas this method is accurate for a weakly heterogeneous moving medium, it degenerates for media with discontinuous properties. The eigenfunction expansion method presented here is a way to accurately and efficiently evaluate spatial derivatives in media with interfaces. As in the Fourier method, transforms may be calculated using FFT’s and spatial sampling is limited only by the Nyquist condition. The performance of the method is shown in a time-domain implementation for media with discontinuous density and sound speed.

1

Introduction

Wave propagation problems that do not have analytical solutions may be solved by numerical methods. For problems where Green’s functions are known, the governing acoustical equation in its integral form can be solved by discretizing interfaces separating sub-domains (e.g. the boundary element method). For problems where Green’s functions are numerically too expensive to evaluate, domain discretization methods can be used (e.g. finite element (FE) or finite difference (FD) methods). Generalized FEM and FD methods can be numerically expensive. A way to more efficiently apply domain discretization methods is to make use of the wavelike nature of the solution. One such method is the PseudoSpectral (PS) method [1]. For a homogeneous medium, spatial derivatives of the solution at a certain time can accurately be calculated by the simplest PS method, the Fourier PS method. Since spatial Fourier transforms are used, the spatial resolution is bounded by the Nyquist criterion (i.e. 2 points per wavelength). The signal is required to have compact support. This method is more efficient and requires less storage than the finite differences time domain (FDTD) method. The Fourier PS method is still accurate for a weakly non-homogeneous medium [2], but fails due to Gibbs’ phenomenon if the medium properties are discontinuous. The Gibbs’ phenomenon can be controlled in the Fourier PS method by low-pass filtering (while sacrificing accuracy at the higher frequencies). A post-processing method can be applied, but is computationally inefficient [3]. For a solution which does not have spatial local support, spatial derivatives can derived using Chebyshev polynomials. This will however require a higher spatial resolution than in the Fourier method (π points per wavelength), a more stringent stability criterion and a multiple subdivision of the spatial domain [1]. A way to solve wave propagation in discontinuous media accurately and efficiently is by using a generalized eigenfunction expansion. This method is an extension of the Fourier method to discontinuous media. The Fourier method appears as the special case of no discontinuity. The generalized (continuum) eigenfunctions are solutions to the wave equation with the discontinuous media. In this paper, the 1D continuum eigenfunction expansion (CEE) for two discontinuous media will be derived for the wave equation. Some numerical keyissues are addressed and results of calculations for dis-

continuous media are shown. Results of calculations for a medium with a slowly varying sound speed are presented. The numerical efficiency of the CEE time domain method is compared with that of the FDTD method.

2

Theory

We consider 1D wave propagation in semi-infinite fluid media, see Fig 1.

Figure 1: The 1D domain is subdivided in two semi-infinite media, 1 and 2. Wave propagation is governed by the wave equation:

d dx

1 d ρ(x) dx

1 d2 − p(x, t) = 0, ρ(x)c(x)2 dt2

(1)

where c(x) and ρ(x) are the piecewise constant wave speed and density (given by cj and ρj for j = 1, 2) and p(x, t) is the pressure. To solve this equation in the time domain, the spatial derivative operator on p(x, t) will be calculated using the continuum eigenfuction expansion method. The continuum eigenfunctions satisfy the eigenvalue equation [L − R] ψ(, x) = 0, (2) d 1 d with L = dx ρ dx , ψ(, x) the eigenfunctions, R = 1 ρc2

and the eigenvalues. Two orthogonal continuum eigenfunctions that are solutions to Eq (2) are: ψ+ (, x) = N+ () ψ− (, x) = N− ()

√

√

−c x c x 1 α√ 1 e 1 + β1 e

x0

c2 x

√

e− c1 x√ α2 e

− c x 2

√

+ β2 e

c2 x

x0

(3)

,

3741 where ∈ (−∞, 0). The coefficients N+ () and N− () in Eq (3) are normalization constants chosen so that the orthogonality condition

∞

−∞

ψ± (, x)ψ± ( , x)

dx = δ( − ) ρc2

where N is the total number of discrete spatial points, NX the number of points for x < 0, NU the number of points for x > 0 and N = NX + NU − 1. At x = 0, half the value of p is taken, corresponding to a triangular integration. By the discontinuity of p(x, t) at x = 0 in the transform of Eq (9), wave number components are erroneously obtained. The errors will be canceled by wave number components of the other two parts in the integrals of Eq. (8). The values of P1 are obtained by taking the complex conjugate of P1 , multiplied by ejk1 (U −X) . For P2 , integration is done over p(x cc21 ), implying that medium 2 has a different spatial sampling than medium 1. It is a consequence of harmonizing the wave number discretization in both media to k1 , which physically means that the implemented spatial discretization of both media captures the same maximum frequency. For P2 , we use zero values for p(x cc21 ) for x < 0 and p(0, t)/2 at x = 0. Values of P2 are obtained similar to P1 . Two Fourier transforms are thus needed to transform p(x, t) to the wave number domain and after having multiplied by coefficients in the wave number domain, two inverse Fourier transforms are left to obtain Lp(x, t). The spatial discretization should obey Nyquist criterion, i.e. 2 points per wavelength.

(4)

is satisfied. Here the overbar denotes the complex conjugate. The coefficients α1 , β1 , α2 and β2 can be calculated by the continuity of pressure and normal velocity at x = 0. We can now decompose a function p(x, t), the solution of Eq (1) at a certain time, onto the orthogonal eigenfunctions by: ∞ dx P± (, t) = p(x, t)ψ± (, x) 2 , (5) ρc −∞ where

p(x, t)

=

0

−∞

±

P± (, t)ψ± (, x) d.

(6)

The operator Lp(x, t) than follows from Eq (2) as:

Lp(x, t) =

±

0

−∞

P± (, t)ψ± (, x) d. ρc2

4 (7)

Thus, the operator including the spatial derivative is calculated by transforming p(x, t) through the orthogonal eigenfunctions, multiplying the transformed function P± (, t) by ρc2 and performing the inverse transform to get Lp(x, t). Inserting the eigenfunctions, we can calculate Lp(x, t) by: Lp(x, t) = 8 2 R < − ∞ k1 0 ρ1 π : − R ∞ k22 0 ρ π 2

“ P1 + “ P2 +

β1 P πα1 1 β2 P πα2 2

+

1 P πα2 2

+

1 P πα1 1

” ”

(8) ejk1 x

dk1

ejk1 x

dk1

x 0,

where = −kj2 c2j and we integrate over k1 , R0 R P1 = −∞ p(x, t)e−jk1 x dx, and P2 = 0∞ p(x cc2 )e−jk1 x dx . 1 From Eq (8), it is clear that the spatial derivative operator is evaluated in the wave number domain k1 .

3

Numerical implementation

To obtain Lp(x, t) numerically from Eq (8), fast Fourier transforms (FFTs) are used because of their computational efficiency. We assume that our p(x, t) has compact support for x between −X and U . When calculating P1 , we therefore need to integrate from −X to U . This is done by using zero values for p(x, t) for x > 0. P1

= ≈

0

−∞

p(x, t)e−jk1 x dx

(9)

N

X −1 2πlm p(0, t) + Δx1 p(lΔx1 − X, t)e−j N , 2 l=0

,

Calculation results

The 1D CEE method to calculate the second spatial derivative will be validated here for several cases. For a homogeneous medium, i.e. α1 = 1, β1 = 0, α2 = 1 and β2 = 0, Eq (8) returns to: Lp(x, t) = R∞ R ∞ k2 P ejkx dk for all x with P = −∞ p(x, t)e−jkx dx. − −∞ 2πρ This is called the Fourier PS method, and only requires one Fourier transform and one inverse Fourier transform to obtain Lp(x, t). The results of the CEE method will be shown along with the results from the Fourier PS method.

4.1

Spatial derivative

We consider a wave with a Gaussian shape propagating from medium 1 to medium 2, which we describe analytically as: p(x, t) = 2 2 α1 e−a(x−(x0 +c1 t)) + β1 e−a(x+(x0 +c1 t)) c1 2 e−a( c2 x−(x0 +c1 t))

(10) for x < 0 , for x > 0.

with α1 and β1 as in Eq (3) and x0 = −64Δx1 . The frequency content of the signal is determined by a, and 2 c1 set to a = 1000Δx . With the CEE method, we can 1 calculate Lp(x, t) at every instant t. Figure 2a shows pn (x, t1 ) with t1 = (68Δx1 /c1 ) s using Eq (10), for ρ2 = 2ρ1 with constant c and c2 = 2c1 with constant ρ. The pressure is normalized as pn (x, t1 ) = p(x, t1 )/|ˆ p(x, t1 )|, where pˆ(x, t1 ) denotes the maximum value. The operator Lpn (x, t1 ) is now calculated analytically, Lpn (x, t1 )an , using the CEE method, Lpn (x, t1 )CEE ,

3742 and using the Fourier PS method, Lpn (x, t1 )F P S . The operators have been normalized as ˆ Lpn (x, t1 ) = Lp(x, t1 )/|Lp(x, t1 )an |. The chosen spatial discretization Δx1 results in a sample frequency as c1 fs = Δx . Figure 2b shows Lpn (x, t1 ) for the two differ1 ent cases. Figure 2c shows absolute error in Lpn (x, t1 ), expressed by 20log10 |Lpn (x, t1 )an − Lpn (x, t1 )CEE | for the CEE method, and Fig 2d shows this absolute error as a function of the frequency up to fs /2. The error using the CEE method is, for the two cases considered, very low and rather flat over the frequency band. The error can be shown to be related to the level of p(x, 0) at fs /2. The grid resolution in the Fourier method is equal to the grid resolution in the CEE method apart from the different sound speed case, where an equidistant grid in the Fourier method is used. In the Fourier PS method for the different density case, the operator L has been evaluated in two steps; i.e. a subsequent calculation of two first spatial derivatives. The error using the Fourier method is substantial compared to the CEE method, and is very large around the media interface. c2 = c1, ρ2 = 2 ρ1

c2 = 2 c1, ρ2 = ρ1 1

n

0 −20

−10

0

10

n

Lp (−) n

100 0 −100 −200 −300 0

20 Error in Lp (dB)

Error in Lpn (dB)

0

−2 −20 −10 0 10 100 0 (c) −100 −200 −300 −20 −10 0 10 Samples from interface (−)

20

(d)

0.1 0.2 0.3 0.4 Normalized frequency (−) Analytical CEE Fourier PS

0.5

0.5 0 −20

20

(b)

2

Error in Lpn (dB)

n

Lp (−)

p (−)

(a) 0.5

Error in Lpn (dB)

pn (−)

1

−10

0

10

20

−10

0

10

20

2 0

−2 −20 100 0 −100 −200 −300 −20 100 0 −100 −200 −300 0

−10 0 10 Samples from interface (−)

20

0.1 0.2 0.3 0.4 Normalized frequency (−)

0.5

Figure 2: Calculation of Lpn (x, t1 ): (a) analytical normalized pressure pn ; (b) normalized operator Lpn from analytical, CEE and Fourier PS methods; (c) 20 log |Lpn (x)an − Lpn (x)CEE | and 20 log |Lpn (x)an − Lpn (x)F P S |; (d) 20 log |Lpn (f )an − Lpn (f )CEE | and 20 log |Lpn (f )an − Lpn (f )F P S |.

4.2

Time-domain implementation

So far, the evaluation of the first operator of Eq (1) has been discussed. For an efficient evaluation of the second operator of Eq (1), use has been made of the kspace method. This method uses the analytical solution of the wave equation in the wave number-time (k − t) domain and has been used in the Fourier PS method before (e.g. [2]). We apply it here to the CEE method. After multiplying Eq (1) by ψ± (, x) and integrating over x we get:

1 d2 k2 + 2 P± (, t) = 0. ρ c ρ dt2

(11)

The solution to this equation can be written as: P± (, t) = A± ()ejc1 k1 t .

(12)

After some algebra, we can write: 1 p(x, t + Δt) − 2p(x, t) + p(x, t − Δt) (13) c2 ρ Δt2

0 k2 − sinc2 (c1 k1 Δt/2) P± (, t) ψ± (, x) d. = ρ −∞ ± The left hand side of Eq (13) can be seen as a finite difference representation of the time derivative operator. The difference between Eq (13) and a traditional second order finite difference representation in time (known as the leapfrog iteration) is the sinc2 (c1 k1 Δt/2) term at the right hand side of Eq (13). In contrast to leapfrog iteration, no error in the time stepping is introduced by the k-space method, since Eq (12) is exact. Given that c is constant in both media, the k-space method is unconditionally stable [2]. The numerical time step Δt is bounded by the Nyquist criterion as well, and has here been chosen to be Δt = Δx1 /(2c1 ). The time domain calculation is started with Eq (10) and t runs from 0Δt to 2500Δt. Figure 3a shows the analytical solution at t = 1000Δt for the two cases considered above, i.e. ρ2 = 2ρ1 with constant c and c2 = 2c1 with constant ρ, showing a reflected and transmitted wave. The accuracy of the CEE k-space method, Eq (13), is studied by comparing the analytical transmission coefficient with the calculated transmission coefficient.

Tan

=

TCEE

=

2ρ2 c2 ρ 2 c2 + ρ1 c1 Ft (p(II, t)) , Ft (p(I, t))

(14)

where Ft is the Fourier transform with respect to time, p(I, t) is the pressure of the incident wave recorded at a position I with x < 0, and p(II, t) the pressure of the transmitted wave recorded at a position II with x > 0. Figure 3b shows Tan , TCEE and TF P S (calculated similarly as TCEE ) for the two cases studied. The accuracy for the CEE k-space method is fine, as expected. The accuracy breaks down close fs /2, where fs is the sample frequency related to the spatial discretization. The Fourier PS k-space method shows to be a low frequency approximation of the correct solution (as also shown in [3]). The leapfrog iteration method is known to be dispersive. To display the accuracy of the CEE k-space method regarding dispersion, the relative phase error φ using both methods is calculated by:

3743

φ (f ) = =

Δφ − Δφan (15) Δφan φ [Ft (p(II, t))] − φ [Ft (p(I, t))] − ωT 100 ωT

100

where φ [y] the phase of y and T the travel time between points II and I. Along with the relative phase error for the k-space method, the relative phase error using the leapfrog iteration method (by omitting the sinc2 (c1 k1 Δt/2) term in Eq (13)) is shown in Fig 3c, using Δt = Δx1 /(10c1 ). The results show that whereas the phase error using the leapfrog method is around 0.4 % at fs /2, the error using the k-space method is negligible. The Fourier PS φ results for the c2 = 2c1 case can be attributed to the fact that the medium interface in this method is situated between two spatial grid points. c =c ,ρ =2ρ 2

1

2

c =2c ,ρ =ρ

1

2

2

1

2

1

2

1

1

n

p (−)

pn (−)

(a) 0 −1 −50

0 50 Samples from interface (−)

0 −1 −50

100

2

0 50 Samples from interface (−)

100


0.5


0.5

2

T (−)

T (−)

(b) 1.5

1 0


1.5

1 0

0.5

1

1 ε (%)

0

−1 0

φ

εφ (%)

(c)


0.5

0

−1 0

CEE, k−space method CEE, leapfrog Fourier PS, k−space method Fourier PS, leapfrog Analytical

Figure 3: Results of time domain calculations: (a) analytical normalized pressure pn (x, 1000Δt); (b) transmission coefficient T from analytical, CEE and Fourier PS methods; (c) relative phase error φ , from CEEk−space , CEEleapf rog , Fourier PSk−space , Fourier PSleapf rog methods.

4.3

Slowly varying medium sound speed

The advantage of the CEE method over boundary discretization methods arises when the medium properties smoothly vary spatially. For smoothly varying medium properties, the Fourier PS method will return a good approximation of the second spatial derivative operator, see e.g. [4]. Since the CEE method is based on Fourier transforms, discontinuous media problems with smoothly varying media properties are expected to be well resolved using the CEE method for media with piecewise constant properties. In this section, it will

be studied whether this is the case by varying the medium sound speed c(x) smoothly in domain 2. Results of calculations with the CEE and Fourier PS methods will be compared to a second order finite difference time domain (FDTD) method, which reads: p(x, t + Δt) − 2p(x, t) + p(x, t − Δt) 2

(16)

2

Δt c(x) (p(x + Δx, t) − 2p(x, t) + p(x − Δx, t)) Δx2 The FDTD has been implemented with fsF DT D = 16fsCEE Hz and ΔtF DT D = ΔxF DT D /(10c1 ), with fs related to the spatial discretization. The FDTD results up to fsF DT D /32 have a small amplitude and phase error and will therefore serve as a reference. The sound speed profile has first been chosen to be a shifted and low pass filtered version of c2 = 2c1 (a first order Butterworth filter with a cut-off frequency of fs /20 has been used). Figure 4a displays the normalized sound speed profile for this case, with cn (x) = c(x)/c1 . The time domain calculations were executed with initial values p(x, 0) as from Eq (10) with a = 2(fs /1000)2. Figure 4b shows the transmission coefficients from the FDTD, CEE and Fourier PS calculations. For the latter two, which are equal for this case, both the k-space method and the leapfrog iteration in time have been calculated for. Since c(x) increases in medium 2, the k-space method has a more stringent stability criterion (see [2]), and Δt = Δx1 /(10c1 ) has been chosen for the k-space and the leapfrog iteration method. The results show that the error in the CEE and Fourier PS methods are small, apart from frequencies close to the sample frequency, as we have seen in the former section. In Fig 4c, the relative phase error is calculated, where the phase change ΔφF DT D has been used as a reference. The relative phase error in the k-space method is smaller than in the leapfrog method, but not as small as in section 4.2. The reason is that the Green’s function in k − t space, as used in the k-space method, is not exact any longer. As a second sound speed profile, we take the former sound speed profile and add c1 to the values in medium 2, see second subplot of Fig 4a. From the T and relative phase error results, we notice that the Fourier PS results show similar deviations as for the single discontinuity as in section 4.2. The error in T and relative phase error in the CEE method are comparable with the errors in the former sound speed profile case. Numerical tests indicate that a more abrupt transition in the sound speed will increase the required number of points per wavelength. =

4.4

Numerical efficiency

The CEE method is developed to obtain an efficient wave propagation computation through discontinuous inhomogeneous fluid media. To indicate the efficiency of the calculations in section 4.3, the computation time from CEE and second order accurate FDTD methods are compared. For a similar accuracy, the number of points per wavelength in the FDTD was found to be 12 times larger than in the CEE method, while keeping the same ratio Δt/Δx in both methods. For this

3744 (a) n

2 1

2 1

−50

0

50

−50

Samples from interface (−)

1.5

1


1.5

1 0

0.5


0.5

References

1

(c)

0.5 ε (%)

0.5

50

2

(b)

1 0

0

Samples from interface (−)

T (−)

T (−)

2

0

φ

εφ (%)

wave propagation in a two fluid problem, where density and wave speed are discontinuous across the media interface. Also, for an additional smoothly varying medium property (here c), the CEE method was shown to be accurate. The CEE, where slightly more than 2 points per wavelength are required, is computationally faster and requires less storage than the FDTD method. The CEE method applied to higher dimensions is currently under development.

3 c (−)

cn (−)

3

−0.5

[1] B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, (1996).

0 −0.5

−1 0


0.5

−1 0


0.5

FDTD CEE, k−space method CEE, leapfrog Fourier PS, k−space method Fourier PS, leapfrog

Figure 4: Results of time domain calculations: (a) normalized sound speed profiles cn (x); (b) transmission coefficient T from analytical, CEE and Fourier PS methods; (c) relative phase error φ , from CEE and Fourier PS methods. 1D problem, the ratio 12 also holds for the required storage capacity of both methods. The number of operations in the CEE method NCEE can be estimated by Nt CCEE Nx log2 (Nx ), with Nt the number of time steps, Nx the number of spatial points, Nx log2 (Nx ) the approximate number of multiplications for a FFT and CCEE a constant. For the number of operations in the FDTD method, NF DT D , we can then write 12Nt CF DT D 12Nx . The ratio NF DT D /NCEE was found to be 16 from the MATLAB implementation, with Nx = 512. This ratio depends on both Nx and the chosen numbers of points per wavelength in both methods. As shown by Liu, PseudoSpectral methods are even more rewarding for higher dimensions [5].

5

Conclusions

To model wave propagation through discontinuous inhomogeneous fluid media in the time domain, the use of a continuum eigenfunctions expansion (CEE) can be useful to calculate the spatial derivative operator of the wave equation of the solution at every time step. The solution is decomposed through the orthogonal set of continuum eigenfunctions, and the spatial derivative operator is taken in the transformed (wave number) domain. As for the Fourier Pseudo-Spectral (PS) method, integral transforms can be evaluated by making use of fast Fourier transforms, but in contrast to the Fourier PS method, the used eigenfunctions account for the discontinuity of the media properties. Calculations for one dimensional problems show that the continuum eigenfunction expansion method yields accurate results for

[2] D. Mast, L. Souriau, D.-L. Liu, M. Tabei, A. Nachman and R. Waag, "A k-space method for large-scale models of wave propagation in tissue," IEEE transactions on ultrasonics, ferroelectrics, and frequency control 48, 341-354 (2001). [3] J. Lu, J. Pan and B. Xu, "Time-domain calculation of acoustical wave propagation in discontinuous media using acoustical wave propagator with mapped pseudospectral method," J. Acoust. Soc. Am., 118, 3408-3419 (2005). [4] B. Cox, S. Kara, S. Arridge and P. Beard, "k-space propagation models for acoustically heterogeneous media: Application to biomedical photoacoustics," J. Acoust. Soc. Am. 21, 3453-3464 (2007). [5] Q. Liu, "The PSTD algorithm: A time-domain method requiring only two cells per wavelength," Microwave Opt. Technol. Lett., 15, 158-165 (1997).

Errata The following corrections need to be observed regarding Paper IV: ◦ Page 3741, Eq. (8):  R  − ∞ 0 Lp(x, t) = R∞  − 0

k12 ρ1 π k22 ρ2 π

P1 + P2 +

 R  − ∞ 0 Lp(x, t) = R  − ∞ 0

k12 ρ1 π k22 ρ2 π

P1 + P2 +

should be replaced by:

β1 πα1 P1 β2 πα2 P2

β1 α1 P1 β2 α2 P2

+ +

+ +

1 jk1 x dk 1 πα2 P2 e 0 1 jk x 1 dk1 πα1 P1 e

1 jk1 x dk 1 α2 P2 e 0 1 jk x 1 dk1 α1 P1 e

x 0,

,

x≤0

x0 ≥ 0.

◦ Page 3741, Eq. (9): P1 = ≈

Z

0

p(x, t)e−jk1 x dx

−∞

NX X −1 2πlm p(0, t) + ∆x1 p(l∆x1 − X, t)e−j N , 2 l=0

should be replaced by: P1

= ≈

cn

Z

0 −∞

∆x1

p(x, t)e−jk1 x dx 0 X

l=−(NX −1)

p(l∆x1 , t) −j 2πlm N , e cn

with 2 l=0 = 1 l = [−(NX − 1), ..., −1].

◦ Page 3741, right column, line 2: "... number of points for x < 0, NU the number of points for x > 0 and ..." should be replaced by "... number of points for x ≤ 0, NU the number of points for x ≥ 0 and ...". ◦ Page 3742, first column, last line: "After multiplying Eq (1) by ψ± (, x) and integrating over x we get:" should be replaced by: "After multiplying Eq (1) by ψ± (, x)R and integrating over x we get:".

Paper V

AN EXTENDED FOURIER PSEUDOSPECTRAL TIME-DOMAIN (PSTD) METHOD FOR FLUID MEDIA WITH DISCONTINUOUS PROPERTIES MAARTEN HORNIKX Department of Civil and Environmental Engineering, Division of Applied Acoustics, Chalmers University of Technology, G¨ oteborg, 41296, Sweden [email protected] http://www.ta.chalmers.se ROGER WAXLER National Center for Physical Acoustics, University of Mississippi, Oxford, MS 38677, USA [email protected]

The Fourier pseudospectral time-domain (PSTD) method has become attractive to efficiently model wave propagation through a weakly inhomogeneous fluid medium. The method however degenerates for fluid media with discontinuous properties. The extended Fourier PSTD method is here proposed to accurately and efficiently model two-dimensional wave propagation through weakly inhomogeneous fluid media with discontinuities in the media properties. The method utilizes generalized eigenfunctions for calculating the spatial derivative operator of the wave equation. Two different ways to iterate in the time domain are explored, where time iteration in the wave number-time domain is shown to give accurate results. A calculation of wave transmission through the water-air interface followed by sound propagation in a downward refracting atmosphere is demonstrated. Keywords: pseudospectral;time-domain method;generalized eigenfunction expansion;water-air interface;atmospheric sound propagation.

1. Introduction Wave propagation problems in fluid media that do not have analytical solutions may be solved by numerical methods. For problems with piecewise homogeneous media for which piecewise analytical Green’s functions are known, the governing wave equation in its integral form can be solved by assigning a source distribution to the interfaces separating the media. For problems with piecewise inhomogeneous media, Green’s functions are numerically expensive to evaluate and domain discretization methods as the finite element method (FEM) and finite-difference method (FDM) become favourable from a computational point of view. Although much effort has been put into developing cost-efficient high-order FEM and FD methods, see e.g.1,2 for an overview, they can numerically still be expensive for short-wave three-dimensional problems. A way to more efficiently apply domain discretization methods is by incorporating the wavelike nature of the solution in the numerical method. One such method is the pseudospectral (PS) method, see e.g.3 . This technique can be utilized in a time-domain method. For large computational domains, time-domain methods could be computationally favourable over frequency-domain methods4 . Also, timedomain methods provide results that can lead to a good understanding of the governing physical phenomena. The pseudospectral time-domain (PSTD) method is a well established method in fields as seismic wave propagation, fluid dynamics, weather prediction and electrodynamics, and application of the PSTD method to acoustic problems has become more popular5−18 . The PSTD method calculates spatial derivatives by a pseudospectral technique 1

2

Maarten Hornikx and Roger Waxler

and uses a different technique, e.g. finite-differences, to iterate the solution in time. For a single homogeneous medium, spatial derivatives of an acoustic variable at a certain time can accurately be calculated by the Fourier PS method. Since spatial discrete Fourier transforms are used in this method, the acoustic variable is required to be periodic in the calculation domain and the spatial resolution is bounded by the Nyquist criterion (i.e. 2 points per wavelength). For a similar accuracy, this method is often computationally more efficient and requires a lower storage capacity than the conventional second-order accurate finitedifferences method, an often used method in atmospheric acoustic propagation problems, see e.g.19 . The Fourier PS method is still accurate for a weakly inhomogeneous medium20 . Although it has been applied for media with discontinuous properties, see e.g.21,22 , it then becomes a low-frequency approximation due to Gibbs phenomenon23 . To suppress the Gibbs phenomenon in the Fourier PS method, several approaches have been taken: low-pass filtering the medium properties (while sacrificing accuracy at the higher frequencies), increasing the spatial grid density globally, increasing the spatial grid density locally and using a nonuniform fast Fourier transform, applying a post-processing method and redistributing the grid points by a mapping curve14 . For non-periodic problems, spatial derivatives can be calculated using Chebyshev polynomials, which is another PS method. This will however require a higher spatial resolution than in the Fourier method (π points per wavelength), a more stringent stability criterion and a multiple subdivision of the spatial domain with the necessary interface update routines, see e.g.3 . A way to accurately and efficiently solve wave propagation in piecewise homogeneous and inhomogeneous fluid media is by making use of a generalized eigenfunction expansion to calculate spatial derivatives. This method can be regarded as an extension of the Fourier PSTD method to discontinuous media as the Fourier PSTD method appears in the special case of no discontinuity. The method is further referred to as the extended Fourier PSTD method. As for the Fourier PSTD method, the extended Fourier PSTD method only requires 2 spatial points per wavelength. To calculate the spatial derivative operator of the wave equation on an acoustic variable at a certain time, the variable is decomposed onto the generalized eigenfunctions, which are solutions to the time-harmonic wave equation for the discontinuous media problem. The spatial derivative operator is then taken on the eigenfunctions in the transformed domain, and an inverse transform returns the spatial derivative of the variable. For one-dimensional cases, the extended Fourier PSTD method was shown to give accurate results for fluid media with discontinuous properties, as well as for additional medium inhomogeneities23 . The extended Fourier PSTD method for the wave equation is here derived for the twodimensional case of a two-fluid problem. Figure 1 shows the geometry studied in the paper. The method is derived for piecewise homogeneous fluids and will also be shown to be a good approximation when the sound speed in one of the fluids is weakly inhomogeneous. An interesting application of the method is wave propagation through the water-air interface, followed by atmospheric sound propagation, where sound waves are propagating through air and are being refracted by a spatially inhomogeneous sound speed. The paper is organized as follows. In Sec. 2, the extended Fourier PS method to evaluate the spatial derivative operator of the wave equation for piecewise homogeneous fluid media is presented. Two numerical approaches to the time iteration in the extended Fourier PSTD method are explained in Sec. 3. In the first method, the solution is obtained by time iteration in the wave number-time (k-time) domain whereas in the second method, time iteration is done in the space-time domain. The methods are studied for piecewise homo-

Extended Fourier PSTD method for fluid media with discontinuous properties

3

geneous fluids and piecewise homogeneous fluids with an additional inhomogeneous sound speed in one fluid. Whereas the piecewise homogeneous case can be solved by more efficient methods, the advantage of the presented method appears for the less trivial situation of the additional inhomogeneities. The method with time iteration in the k-time domain is shown to be accurate. In Sec. 4, atmospheric sound propagation from an underwater source close to the surface is calculated for both a homogeneous atmosphere as for an atmosphere with a typical wind speed profile. 2. Extended Fourier pseudospectral (PS) method We consider two-dimensional wave propagation in semi-infinite homogeneous fluid media, see Fig. 1. In the derivation of the method, we assume that c2 ≥ c1 . Wave propagation in the geometry of Fig. 1 is governed by the wave equation and interface conditions at z = 0: " # 1 ∂2 ∂ 1 ∂ 1 ∂2 p(x, z, t) = 0, (1) + − 2 ρj ∂x2 ∂z ρj ∂z cj ρj ∂t2 p(x, 0, t)|1 = p(x, 0, t)|2 , 1 ∂p(x, 0, t) 1 ∂p(x, 0, t) = ρ2 , ρ1 ∂z ∂z 1

2

where j denotes medium 1 or 2, the density ρj and sound speed cj are piecewise constant (and independent on x) and p(x, z, t) is the acoustic pressure. solve this equation in To i h 1 ∂ ∂ 1 ∂2 on p(x, z, t) will be the time domain, the spatial derivative operator ρj ∂x2 + ∂z ρj ∂z calculated by the extended Fourier PS method, which makes useof a generalized eigenfuction expansion method. The temporal derivative operator

1 ∂2 c2j ρj ∂t2

on p(x, z, t) is examined in

two different ways, as will be discussed in Sec. 3. For all pseudospectral methods, the calculation of a spatial derivative operator on a variable, here the pressure p(x, z, t), relies on the expansion of the variable through a set of basis functions ψn,m (x, z): X X p(x, z, t) = Pn,m (t)ψn,m (x, z), (2) |n|

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