Time Domain Simulation of Sound Diffusers Using ...

ACTA ACUSTICA UNITED WITH Vol. 93 (2007) 611 – 622

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Time Domain Simulation of Sound Diffusers Using Finite-Difference Schemes J. Redondo, R. Picó, B. Roig

Escuela Politécnica Superior de Gandía, Spain, Universidad Politécnica de Valencia, Carretera Nazaret-Oliva S/N. Grao de Gandía, Espana. fredondo@fis.upv.es

M. R. Avis

Acoustics Research Centre, University of Salford, Salford M5 4WT, UK

Summary Since the invention of sound diffusers three decades ago a substantial effort has been made to predict the acoustic behaviour of these structures, for auralisation and prediction purposes as well as in response to the large costs inherent in anechoic measurements. Volumetric methods such as Finite Element Methods (FEM) or the Finite Difference Time Domain method (FDTD) are not often used, due to their large computational cost. However Near Field to Far Field Transformations (NFFFT) can overcome that problem. The main advantages of the FDTD method are that a single calculation is sufficient to study a wide frequency band, and that the time domain behaviour of the reflected sound can be directly inspected. In this paper we present a comparison between the prediction techniques commented above in the context of sound diffusers, paying special attention to the FDTD method. Having demonstrated that the FDTD method can generate results comparable to more established techniques, early results concerning the modelled performance of diffusers in the time domain (‘time spreading’) are reported, opening a new field of research. PACS no. 43.55.Ka, 43.55.Br, 43.58.Ta, 43.20.Fn

1. Introduction

Received 10 October 2006, accepted 2 April 2007.

Appropriate windowing of the signal allows the elimination of the direct sound. The complete characterization of the diffuser is performed where the incidence angle is also varied from −90 to 90 degrees. As an alternative approach, near-field holography can be used to avoid the need for large anechoic environments needed to ensure far field conditions. Scale models offer a simpler solution to this physical problem - but the technique remains time consuming if several angles of incidence are considered. The parameter measured using this technique is known as the ‘diffusion coefficient’. The second kind of measurement technique was first proposed by Mommertz and Vorlander [4] and has been recently standardized by the ISO [5]. This method allows the direct extraction of a parameter known as a ‘scattering coefficient’ under the assumption that scattered sound is non-coherent. The technique therefore exploits certain time features of the scattered sound. A test sample is introduced into a reverberant chamber, and impulse responses for different sample orientations are obtained. Using synchronous averaging of these impulse responses, the diffuse reflected sound is eliminated and a virtual impulse response is obtained. From that virtual impulse response a pseudo-absorption-coefficient can be obtained in an analogous way to the Sabine method. Finally, a scattering coefficient is obtained from this pseudo-absorption-coefficient. One of the advantages of the AES procedure is that polar patterns of the reflected sound can be obtained.

© S. Hirzel Verlag · EAA

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It is now three decades since Schröeder proposed sound diffusers for the first time [1, 2]. It is well-known that such structures can improve the acoustic performance of rooms in a number of different ways, sometimes described by an increase in sound field diffuseness, subjective impressions of ‘spaciousness’, and the removal of echoes, focalizations and frequency-domain coloration. However, relating diffuser scattering coefficients to these measures is not at all straightforward. Notwithstanding this problem, the quantitative evaluation of sound diffusers is useful because it may facilitate the optimisation of their design, and also provide the scattering coefficients used as input parameters in room acoustics simulation programs based on geometrical techniques. Measurement techniques used to characterise rough surfaces can be divided in two categories. The first, standardized by the AES [3], is based on the measurement of reflected sound over a range of angles. For this purpose an impulse response must be obtained (using MLS, logsweep sine chirp or another impulse measurement technique). A microphone is moved along a semi-circumference (or over a hemisphere for full three dimensional evaluation), centred on the middle point of the test sample.

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Redondo et al.: Time domain simulation of sound diffusers

This similarity with techniques used to evaluate sound source directivity has resulted in widespread acceptance of this technique by the scientific community. More usefully, results can be easily predicted by several simulation techniques. Cox, D’Antonio and other authors have performed a wide range of comparisons between predictions and measurements [6, 7, 8], and have demonstrated that amongst those prediction techniques tested, Boundary Element Methods (BEM) are the most accurate. These accurate predictions of the behaviour of rough surfaces have further allowed the optimisation of diffuser performance [9]. In contrast to the AES method, the ISO measurement technique cannot be simulated with conventional frequency-domain BEM and FEM methods. However, recently, transient BEM and FEM models have been proposed - in particular, a transient BEM for the simulation of sound diffusers [10]. Currently, this technique represents the only published method for a study of the time features of sound reflected from a surface, with the possible exception of on the one hand analytical models recently proposed by Svensson and other authors [11, 12, 13, 14, 15], and on the other hand the use of Transmission Line Matrix (TLM) in combination with Finite Elements (see for instance [16]), to obtain the time domain features of edge diffraction. An alternative to the transient BEM technique is the Finite Difference Time Domain Method (FDTD). FDTD is a well known “Volumetric method” that was first introduced by Yee in 1966 [17] in order to study the scattering of electromagnetic waves. More recently, a counterpart method for acoustic applications has been developed. A non-exhaustive list would include applications in room acoustics [18, 19], environmental acoustics [20], ultrasounds [21], phononic crystals [22, 23] and recently the simulation of the time domain behaviour of porous media [24, 25]. An additional feature of a FDTD technique is that active diffusers [26, 27, 28] can be simulated in a relatively straightforward way, including a simulation of the signal processing which controls the devices. These devices have been shown to be useful in situations where space reduction and l.f. bandwidth extension are important, and the FDTD simulation technique is here applied to produce successful simulations. Even taking into account the progressive increase in calculation speed of computers, today it remains unviable to simulate a full 3D room up to high frequencies with FDTD. As Svennson reports [29] in a comparison of different simulation techniques, a FDTD simulation of a large auditorium could result in a total computation time of about 44 years! Nevertheless, it seems reasonable to suppose that speed increases and cost reductions in computing equipment will continue, and that as a result many kinds of numerical solution methods will become increasingly viable at audio frequencies. Most of the prediction techniques discussed up to this point have been “surface methods” [30]. “Volumetric methods” such as Finite Element Methods (FEM) have not

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been used, due to the inherent computational cost in a far field computation. However, Near Field to Far Field Transformation (NFFFT) methods can minimize the difference in computational cost between “Surface” and “Volumetric” techniques because only a small part of the space in the vicinity of the surface under test must be meshed. The main advantage of “Volumetric” methods is that fluid motion and structural vibration can be coupled in a straightforward manner. In this paper we validate FDTD simulations of the scattering provided by sound diffusers, comparing it with other well-known frequency domain techniques. Specifically, we compare results derived using a) The Finite Difference Technique in the Time Domain transformed from Near Field to Far field. b) The Finite Element Method (FEM) in the Frequency Domain transformed from Near Field to Far Field. c) The Boundary Element Method (BEM) in the Frequency Domain. The object is to validate the FDTD method combined with NFFFT, given its attractiveness in terms of solution time. Since this solution offers an intrinsic time domain analysis of the method, an investigation of the ‘time spreading’ effect produced by sound diffusers is also discussed. This effect has been reported by several authors (e.g. [31, 32]) but there is still an open question as to its subjective importance and method of evaluation. The main contribution of this work is to present and validate a method which allows the simulation of sound diffusers in the time domain, and to generate time-spreading results from which the beginnings of a subjective evaluation method can be postulated. The structure of the paper is as follows; in section 2 a two-dimensional FDTD scheme is briefly outlined; in section 3 the numerical setup is presented, in section 4 we present a comparison of the results obtained with BEM, FEM, and FDTD for different surfaces; in section 5 the time spreading provided by the evaluated surfaces is investigated; finally section 6 describes the main conclusions of this paper.

2. FDTD scheme 2.1. Fundamentals The starting point of an acoustic FDTD model with no sound sources is given by the equations for conservation of momentum and continuity. In a homogeneous medium with no losses, these can be written as: ∂p + k · u = 0, ∂t ∂u p + ρ0 = 0, ∂t

(1a) (1b)

where p is the pressure field, u = (ux , uy ) is the vector particle velocity field, ρ0 is the mass density of the medium and k = ρ0 c2 is the compressibility of the medium. The spatial and time derivatives of pressure and particle velocity can be approximated by central finite difference. For


instance, the derivative of the sound pressure respect to x can be approximated as ∂p ∂x

x=x0

≈

p x0 +

Δx 2

− p x0 − Δx

Δx 2

,

(2)

where Δx is the spatial interval between two closelyspaced points in the x direction. In two dimensions, three grids must be defined; one grid for pressure and two for the different particle velocity components. To minimize the significance of higher order terms in equations (3a)–(3c) below, those grids are ‘staggered’. For instance, the mesh for the x component of the particle velocity is shifted a distance of Δx/2 with respect to the pressure mesh. The same applies for time meshing; the particle velocity meshes are shifted Δt/2 in time with respect to the pressure mesh. In doing so one can obtain a set of update equations to obtain the values of pressure and particle velocity after repetition for a given number of steps: n+ 1 pi,j 2

=

n− 1 pi,j 2

uxni+ 1 ,j − uxni− 1 ,j 2

− k Δt +

uxn+1 i+ 1 ,j 2

n+1 uyi,j+ 1 2

2

Δx n n uyi,j+ 1 − uy i,j− 1 2

2

Δy

 n+ 1  n+ 1 2 − pi,j 2 pi+1,j Δt  , = uxni+ 1 ,j − ρ Δx 2  n+ 1  n+ 12 2 − p p Δt  i,j+1 i,j n , = uyi,j+ 1 − ρ Δy 2

,

(3a)

(3b)

(3c)

where the superscripts represent the time index, and the subscripts the spatial indices, namely: 1 n+ 1 (4a) pi,j 2 = p i Δx, j Δy, (n + ) Δt , 2 1 uxni+ 1 ,j = ux (i + ) Δx, j Δy, n Δt , (4b) 2 2 1 n uyi,j+ ) Δy, n Δt . (4c) 1 = uy i Δx, (j + 2 2 To ensure numerical convergence, the time step should be small enough to describe the wave propagation. The limit relationship between the spatial steps and the time step is given by the so-called Courant number, s, which in 2 dimensions can be defined as follows s = cΔt

1 Δx

2

+

1 Δy

2

≤ 1.

(5)

Another limitation of FDTD, common to other numerical techniques, results from the fact that the maximum element size used in discretization is determined by the frequency, with the sample criterion requiring four elements per wavelength. It has been shown in literature [33] that at least 10 elements per wavelength are required for adequate accuracy. Thus at high frequencies, a large numerical problem has to be solved. In this paper we will use a space step small enough to have accurate results up to 5 kHz.

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2.2. Boundary conditions Assuming a locally reactive boundary, the particle velocity at the boundary can be expressed as follows: u·n=

p , Z

(6)

where n is the unitary vector normal to the boundary and Z is the acoustic impedance of the boundary. Botteldooren [18] has demonstrated that complex impedances can be modelled in a FDTD scheme. For simplicity we will consider here only surfaces in which the acoustic impedance is independent of frequency. A simple implementation of boundary conditions in FDTD schemes leads to a ‘staircase-like’ situation in cases where boundary surfaces are not aligned with the mesh. A better approach for such kinds of surfaces is the use of conformal methods [34]; however, given that in this paper we will be looking at Schröeder diffusers, there will be no need to use those techniques. Due to cumulative errors in using equation (6) for characteristic boundary conditions, allowable absorption values at the boundaries are limited. For practical reasons it is impossible to achieve level differences between reflected and incident waves greater than 60 dB. Therefore a very large grid may be considered, in order to separate, in time, unwanted boundary reflections from the significant reflections inside the integration area. This has clear consequences for calculation times - the larger the integration area is, the slower the calculations which result – and so a technique known as “Perfectly Matched Layers” (PML) [35, 36] may be adopted. This technique defines a lossy medium in locations proximate to the boundaries, which in turn implies the modification of equations (1a) and (1b) to include attenuation factors for each dimension considered (γx and γy in two dimensions), i.e. ∂px + γx px + k ∂t ∂py + γy py + k ∂t

∂ux ∂x ∂uy ∂y

= 0,

(7a)

= 0,

(7b)

∂p +ρ ∂x

∂ux + γx ux = 0, ∂t ∂uy ∂p +ρ + γy uy = 0, ∂y ∂t

(7c) (7d)

where the sound pressure p has been split into two additive components px and py . These components have no physical significance and are defined only to facilitate the numerical solution. The attenuation factors are null inside the integration area, and are increased in areas near the boundaries. 2.3. Total field and scattered field formulation The main point of interest in this work is the reflected sound field, and therefore it is desirable to make use of time windowing to separate the incident sound from the reflected sound. This can cause problems, especially at

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low frequencies and for measurement positions near to the sound diffuser. As we will make use of the Near to Far Field transformation, the receiver locations in the FDTD simulation will be placed extremely close the reflection surfaces, which means time windowing of the signal is rather difficult. However, one can use two different techniques developed for FDTD: the total field/scattered field formulation [37] and the pure scattered field approach [38]. The details of both techniques can be found in reference [33]. Both methods produce the same results, although their implementations differ, and both rely on the linearity of equations (1a-1b) or (7a-7d). The first technique comprises the separation of the integration into two domains. One of these, the total field domain, includes the surface under test, and both incident and scattered field are considered. In the outer domain only the scattered sound is calculated. This implies that the incident field is injected as a pseudo boundary condition in the limit area between both domains. In these pseudo boundary conditions, the incident field is either added to or subtracted from the total field. In the second technique known as the pure scattered field approach, the incident field is injected at the boundaries of the tested surface, and only the scattered field is considered within the integration area. In both techniques the incident field must be known. To obtain the incident field one can use analytical expressions (if available), or a reference simulation in which the scattering object is not included. Conceptually, from the standpoint of a particular receiver location, both techniques are equivalent to performing a ‘normal’ total-field simulation and subtracting from that result, the result obtained in a simulation with no scattering object. It is interesting to note that these “total field and scattered field” and “pure scattered field” formulations are rather similar in some regards to a technique called “Transparent sources” adapted to Acoustics by Schneider et al. [39] from previous versions developed in Electromagnetism. As commented earlier, one of the main advantages of the FDTD technique is that a wide band analysis can be performed in a single simulation. Therefore, the excitation signal needs to have a wide spectrum. However, mesh density places a high-frequency limit on propagation, which means an unfiltered Dirac pulse cannot be used. In the field of electromagnetism, Gaussian pulses are the most common whilst in the field of geophysics Ricker wavelets [40] are used by the majority of authors. In this work, Ricker wavelets are used because they do not introduce large-amplitude frequency components near D.C. which can cause resolution problems in the audible part of the spectrum. 2.4. Near Field to Far Field Transformation In the field of sound diffuser design and analysis, often the need arises to numerically extrapolate the far-field performance of a reflecting structure from its known (or assumed) near-field behaviour. Mathematical processes developed to accomplish such a task are commonly de-

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scribed in the literature as near-zone to far-zone transformations (NZFZT) or near-field far-field transformations (NFFFT). In general, most transformations make use of sampled near-field quantities along some particular closed contour that completely encloses the radiating or reflecting structure of interest. In our case, samples of the required near-field quantities are supplied via computational means. The far-field response of the radiating structure is then computed from the sampled near-field data using a numerical process. Throughout our discussion we shall refer to this virtual contour along which the near-field quantities are sampled, as the transformation contour. The comprehensive survey of articles by Johnson et al. [41] and Yaghjian [42] provides an excellent compendium of the many NFFFT methods with respect to their chronological development. The standard NFFFT formulation is based on the contour equivalence theorem. In this approach, the scattered pressure field along a closed virtual contour surrounding the structure of interest is computed via FDTD and integrated over the entire contour to provide the far-field response. Since the virtual surface is independent of the geometry of the scatterer which it encloses, it is usually chosen as a rectangular shape to conform to the standard Cartesian FDTD grid. Coupling the standard FDTD and NFFFT schemes, far-field scattering amplitudes and phases can be calculated for any scattering angle. Figure 1 shows a complete scheme of the problem. The NFFFT line represents the transformation contour Γ used in the equivalence theorem which encloses the scatterer. The scattered pressure and velocity fields, p(x, y, t) and u(x, y, t), computed via FDTD, can be integrated over Γ obtaining the pressure at a far point (x , y ) in a time domain formulation [43]: p(x , y , t) = − · +

∂ ∂t

np(x, y, t − Γ

Γ

R c)

dΓ 4πR ρ0 n · u(x, y, t − Rc ) 4πR

dΓ,

(8)

where R = (x − x )2 + (y − y )2 and n is the unit normal vector. In a frequency-domain formulation, the acoustic field equations (without sources) can be written as the Fourier transform of equations (1a) and (1b): ρ0

· uω (x, y) − iωρ0 pω (x, y) = 0,

(9a)

pω (x, y) − iωρ0 uω (x, y) = 0,

(9b)

or coupling equations (9a) and (9b) with the well known Helmholtz equation for the pressure: 2

pω (x, y) + kω2 pω (x, y) = 0,

(10)

where kω = ω/c. Due to the non existence of sources outside the Γ contour, the pressure at a far point (x , y ) can be computed



Vol. 93 (2007)

via the Helmholtz-Kirchoff integral: pω (x , y ) =

∂ (11) Gω (x , y ; x, y) ∂n Γ ∂ − Gω (x , y ; x, y) pω (x, y) dΓ, ∂n pω (x, y)

where Gω (x , y ; x, y) is the free-space Green function solution of the Helmholtz equation. In two dimensions it can be expressed with the help of the Hankel functions as Gω (x , y ; x, y) =

i (1) H (kω R). 4 0

(12)

Substituting equation (12) into (11), the pressure at a far point (x , y ) can be expressed as pω (x , y ) =

Γ

n·R ikω (1) pω (x, y)H1 (kω R) 4 R ωρ0 (1) H0 (kω R)n · uω (x, y) dΓ. − 4

(13)

In reference [44] a modified NFFFT is presented in which one of the faces of the transformation contour is not considered, producing more accurate results for surfaces that scatter the sound in a given direction. In the cases considered in this paper the back-scattered sound is much more important than the forward-scattered sound, so we will not consider the part of the transformation contour Γ to the left of the test sample, assuming that the incident plane wave arrives from the right as shown in the schematic representation in Figure 1.

Figure 1. General geometry of the simulation with FDTD. The recordings at the 19 microphones, 50 m away from the test specimen are calculated after Near to Far Field Transformation from the values of pressure and particle velocity obtained at the NFFF transformation line (see Figure 2).

3. Numerical setup In this sub-section the details of the simulations carried out for this work (FDTD and FEM) are briefly outlined. Concerning the simulations with FDTD technique a small area around the test specimen has been simulated. All the calculations have been carried out in a mesh comprising approximately 400 by 700 elements, each with an approximate size of about 1 cm. The unit cell dimensions have been chosen in order to better simulate the particular test specimen. For instance, for the modelled diffuser, the width of the wells is related to the size of the unit cell by an integer multiplier. In order to operate with a Courant number as close to 1 as possible, the sampling frequency is about 10 KHz. For frequencies above 10KHz numerical dispersion should be significant enough to mask the reflected sound. The numerical scheme is excited by a line source placed at the right hand side of the integration area (see Figures 1 and 2). The time domain signals corresponding to the pressure and to the particle velocity at the Near Field Far Field Transformation line are recorded in order to obtain the sound pressure at the 19 Far Field locations (see Figure 1) at a distance of 50 meters from the test specimen with angles between 90 and -90 degrees with a step of 5 degrees, following the AES standard for diffuser characterization [3].

Figure 2. Detailed view of Figure 1a that corresponds to the calculation area with FDTD. The different zones and contours are schematically represented: PML (Absorbing boundary), Scattered Field Zone (Only reflected sound propagates in this area), Total Field Zone (Incident and reflected sound propagate in this area), NFFT Tras. line (Near field to far field transformation line, and Test specimen. (Sample under analysis). The well separator is not plotted for better visualization.

As our main purpose is to check if the FDTD technique allows researchers to study the time spreading provided by diffusers, it is extremely important to use excitations signals as short in time as possible. Figure 3 illustrates the two different signals used for this work. Both are Ricker wavelets [40] with central frequencies at 250 Hz and at 2 kHz, covering all the frequency bands of interest in diffuser characterization. We might note that the whole calculation, including the propagation of the waves in the vicinity of the test specimen, the Fourier transform of the sound

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1

0

p (Pa)

125 Hz

-10

SPL

-20

0 250 Hz 2000 Hz

-1 48

t (ms)

0

-30 -40

0 -10

SPL(dB)

500 Hz

-20 -30

-200

-40

f (Hz) 0

Figure 3. Upper plot: time histories of the two Ricker wavelets used for model excitation (Central frequencies 250 Hz and 2000 Hz). Lower plot: corresponding frequency content (dB) referred to the maximum signal level.

2000 Hz

-10 -20 -30 -40 T

Figure 5. One-third octave band sound pressure level (sound pressure normalised to the total reflected sound pressure) versus angle at different frequencies for a flat panel (3.6 m wide) in the far field. Continuous line: FDTD, dashed line BEM, dotted line: FEM. 1/3 OB centre frequencies are given in the upper left-hand corner of each plot.

Figure 4. Detailed view of the calculation area corresponding to the FEM simulation. The Boundary conditions, distances from specimen to boundaries and the NFFF transformation line are represented.

pressure and particle velocity, and the transformation to the 19 Far Field positions, takes a few minutes using a standard personal computer. In the FEM technique used for model comparison, the simulation domain is defined as a small rectangular area whose boundaries are spaced approximately 2 m from the specimen, as showed in Figure 4. The right-hand boundary is excited with a peak amplitude of 1Pa, and a plane wave radiation condition is imposed at the remaining boundaries. The incident field is obtained from a reference simulation without the specimen. Simulations have been carried out in the time-harmonic regime. The mean size of the elements of the shell depends on frequency, and is estimated as one-twentieth of the wavelength at the highest frequency of interest. Smaller elements have been chosen at the edges and boundaries of the specimen.

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The results obtained are compared in the next section with respective theoretical and BEM numerical data to provide additional validation of our NFFFT process. We also have combined the FEM with the NFFFT to obtain a better validation of the overall process.

4. Comparison of Results In order to validate the FDTD scheme outlined in the previous section, several surfaces have been simulated using FEM, and FDTD. BEM simulations of diffusing surfaces are provided by Cox and D’Antonio [45]. A detailed description of the BEM technique can be found at [46]. It must be pointed out that these BEM simulations have been compared with experimental data in a number of papers (see for example [8]) showing, in general, an excellent agreement. After the simulation, and for the sake of brevity, we have chosen three examples sufficiently representative to cover the range of possible cases. Specifically, the selected cases are: a) a flat panel, b) six periods of a Quadratic Residue (QR) diffuser with a maximum depth of 20 centimetres, c) a set of nine triangles.



0

Vol. 93 (2007)

0

125 Hz

-10

-10

SPL

SPL

-20

-20

-30

-30

-40

-40

0

0

500 Hz

-10

-10

-20

-20

-30

-30

-40

-40

0

125 Hz

500 Hz

0

2000 Hz

-10

-10

-20

-20

-30

-30

2000 Hz

-40

-40

T

T

Figure 6. One-third octave band sound pressure level (sound pressure normalised to the total reflected sound pressure) versus angle at different frequencies for a QR diffuser (3.6 m wide) in the far field. Continuous line: FDTD, dashed line BEM, dotted line: FEM. 1/3 OB centre frequencies are given in the upper left-hand corner of each plot.

Figure 7. One-third octave band sound pressure level (sound pressure normalised to the total reflected sound pressure) versus angle at different frequencies for nine triangles (3.6 m wide) in the far field. Continuous line: FDTD, dashed line BEM, dotted line: FEM. 1/3 OB centre frequencies are given in the upper left-hand corner of each plot.

All the samples are 3.6 m wide. The flat surface has been chosen as a reference, and helps in determining whether the observed spreading is due to the finite size of the samples under test or is caused by their detail structure. The diffuser is the main surface of interest to this paper. As it consist of a series of wells of differing depth, several semi-closed volumes are defined causing a time spreading effect. The selection of the set of triangles has been made mainly to consider a situation in which some spatial spreading is produced, but, since the structure does not define any semi-closed volumes, any resulting time spreading should be relatively small. As a further aid to brevity we will present here only the results obtained for normal incidence. Figures 5 to 7 illustrate the far field polar patterns obtained with the different numerical techniques described above. Sound pressure level versus reflection angle is plotted for several representative third octave bands. It must be pointed out that the frequency range over which the chosen sound diffuser might be expected to be effective extends from approximately 550 Hz to 2000 Hz. In fact, the maximum frequency is not a real limit on performance (as will be shown), but only a limit of validity of the theory used for its design.

In the case of the flat panel (Figure 5) the main differences between BEM, FEM and FDTD simulations appear at angles far from the specular direction (0◦ ), especially at low frequencies. However, generally a good agreement is observed between all the chosen techniques. The same applies for the cases of the diffuser (Figure 6) and the set of triangles (Figure 7). Additionally some small differences are found at medium and high frequencies concerning the level of the minima generated using the FEM technique. In the case of the flat panel we have also calculated the reflected sound pressure using the Fraunhofer approximation which can be written as follows ps (r) ≈

−jkab e−jk(r0 +r) x x0 sinc k a − 2 r r r r0 2π 0 y y0 · sinc k b , − r r0

(14)

where ps is the scattered sound pressure, k is the wave number, a and b are the dimension of the flat panel, sinc is the sine cardinal (sinc(x) = sin(x)/x), x, y and r correspond to the coordinates of the measurement position and x0 , y0 and r0 correspond to the sound source position. Results using this approximation are presented in Figures 8 to 10.

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Additionally we have calculated the diffusion coefficient defined as follows [47, 48]: n

δ=

i=1

10Li/10 (n − 1)

2

− n i=1

n i=1

10

,

(15)

where δ is the diffusion coefficient, Li is the is the Sound Pressure Level of the reflected sound at the i-th measurement position, and n is the number of measurement positions. In Figures 8 to 10 the diffusion coefficient is plotted against frequency for each considered surface. In the case of the flat panel (Figure 5) there is a good agreement among all the techniques. However small differences appear when comparing FEM results with the remaining techniques at low frequency due to reflections close to the boundaries of the simulation space. To expand - using a similar approach to that reported for FDTD simulations (in order to reduce the computational cost of the FEM simulation) the FEM domain size has been reduced, meaning that the boundaries are close to the diffuser. The physical behaviour imposed at the domain edges is a radiation boundary condition, simulating plane wave propagation away from the edges. The intention is to create a similar boundary to the PML in the FDTD simulations, where the domain appears to extend beyond the boundary. When the boundary is extremely close to the source, as in our case, the pressure field at the boundary is strongly influenced by the surface under test, especially at low frequencies. The imposition of a plane wave radiation condition at the boundary may therefore not be appropriate, and at low frequencies the simulation result may be biased by the limited dimensions of the domain. In the case of the diffuser (Figure 9), results provided by BEM and FDTD show a good degree of similarity, but differences persist between these and the FEM simulation, again at low frequency. Particularly, at 300 Hz a spurious maximum in the diffusion coefficient is observed in the FEM result. This can again be attributed to reflections at the boundary of the simulation area. The FDTD results underestimate the diffusion coefficient somewhat when compared to the BEM solution, but to a lesser degree than the FEM case. Agreement is encouraging between approximately 500 and 2000 Hz - the designed frequency range of the diffuser. Similar observations can be made in the case of the set of triangles. Additionally, the divergence of FEM results at high frequencies (commented above) is more evident. These results give some weight to the argument that FDTD modelling can provide an acceptable alternative to BEM or FEM approaches for frequency-domain diffuser modelling, with a much-reduced computation time overhead. However, the FDTD technique also allows the explicit calculation of the time-domain behaviour of a scattering surface, and it is to this that we next turn our attention.

618

FDTD BEM FEM Fraunhofer

0.8

Li/10 2

2 10Li/10

1.0

0.6 0.4 0.2 0.0 100

500

1000

f [Hz]

5000

Figure 8. Diffusion coefficient versus frequency for a flat panel (3.6 m wide). Continuous line: FDTD, dashed line: BEM, dotted line: FEM. Fraunhofer approximation for a flat panel also shown for comparison.

1.0

FDTD BEM FEM Flat panel

0.8 0.6 0.4 0.2 0.0 100

500

1000

f [Hz]

5000

Figure 9. Diffusion coefficient versus frequency for a QR diffuser (3.6 m wide). Continuous line: FDTD, dashed line: BEM, dotted line: FEM. Fraunhofer approximation for a flat panel also shown for comparison.

1.0

FDTD BEM FEM Flat panel

0.8 0.6 0.4 0.2 0.0 100

500

1000

f [Hz]

5000

Figure 10. Diffusion coefficient versus frequency for a set of nine triangles (3.6 m wide). Continuous line: FDTD, dashed line: BEM, dotted line: FEM. Fraunhofer approximation for a flat panel also shown for comparison.

5. Time spreading It is well-known that the classic Schröeder diffuser provides not only spatial but also time spreading of the re-



flat panel diffuser

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0.005 flat panel diffuser

p [Pa]

p [Pa] 0.0

0.0

-0.01

0.00

0.02

0.04

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Figure 11. Sound pressure reflected at 0 degrees versus time: flat panel (continuous line), diffuser (dashed line).

0

0.00

0.02

0.04

t [s]

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Figure 13. Sound pressure reflected at 45 degrees versus time: flat panel (continuous line), diffuser (dashed line).

flat panel diffuser triangles

0

flat panel diffuser triangles

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-20

-40

-40 0.00

0.02

0.04

t [s]

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0.00

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Figure 12. Normalized back-integrated time responses with Ricker wavelet excitation (only reflected sound) at 0 degrees versus time: flat panel (continuous line), diffuser (dashed line) and triangles (doted line). Crossed line corresponds to the reference (50 mfree propagation).

Figure 14. Normalized back-integrated time responses with Ricker wavelet excitation (only reflected sound) at 45 degrees versus time: flat panel (continuous line), diffuser (dashed line) and triangles (doted line). Crossed line corresponds to the reference (50 mfree propagation).

flected sound. Using the FDTD models described above, time spreading has been evaluated for each of the three surfaces considered in this paper. Preliminary results and discussions are now presented. For the sake of brevity only far field simulation results are presented here. Figures 11–14 illustrate the time spreading of the considered surfaces for a reflection direction of 0o (specular zone) (Figures 11 and 12) and 45o (outside the specular zone) (Figures 13 and 14). In Figures 12 and 14, in order to present a complete analysis of the results, we have performed a back-integration of the impulse responses (Schröeder integrals [49]) to obtain characteristic decay times of the reflected signals. These are analogous to reverberation times in room acoustics. In these figures a reference case has been included for better comparison of the results, corresponding to a 50 meter free propagation of the sound wave in which only the typical 2D dispersion can be observed. An evaluation of results using the three test surfaces suggests an important result; that without considering spectral variations too closely, we may generally hold that surfaces which are known to be effective in the spatial dispersion of reflected energy, are also associated with a greater degree of spread of reflected energy in time. Only specular and 45◦ reflection angles are shown; more widely,

this relationship was seen to depend strongly on the angles of incidence and reflection, and the distance of a chosen observation point or receiver location to the reflecting surface. In the case of a flat surface, time spreading only appeared at angles outside the specular reflection zone, where less energy was reflected. For the sound diffuser, time spreading was much more similar for all the observed reflection directions. The reflecting triangles present an intermediate case – here, the time taken for the sound to decay after the first arrival is much smaller than in the case of the diffuser but, outside the specular zone, considerably longer than in the case of the flat surface. In any case, the absence of semi-closed volumes (and hence classical resonant behaviour) causes small time spreading in comparison with that observed for the diffuser. It must be pointed out that these differences are not so evident in the near field. The results described in the previous paragraph are general in that no attempt has yet been made to comment on any spectral variation of time spreading provided by all simulated surfaces. In fact, simulations show that for low frequencies all the tested surfaces behave in a similar way, i.e. time spreading only appears at angles outside of the specular reflection zone. In the ‘design’ frequency range for which the diffuser might be expected to be most ef-

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fective (500Hz-3200Hz), time spreading is more or less consistent for all reflection angles, while the flat surface continues to behave as for low frequencies. For the diffuser, time spreading shows a much more complicated behaviour as a function of frequency than spatial spreading, and both vary considerably over the operating bandwidth. This means that although we may state that in general, surfaces exhibiting strong spatial spreading also show strong time spreading, it is possible to find particular frequency bands away from the ‘design frequencies’ within which the diffuser shows similar spatial spreading to the flat panel, whilst still retaining much stronger time spreading features. Therefore over frequency, the time-space spreading relationship cannot be considered too simplistically. In any case, significant time spreading is expected to result in a decrease in the coherence of the reflected signal, and therefore to lead to large values of the scattering coefficient measured using the ISO standard. The AES diffusion coefficient is less obviously related directly to time behaviour, albeit that it has been observed above that surfaces which exhibit significant time spreading may well also show strong spatial diffusion, and large values of the AES diffusion coefficient can be found if the spatial spreading is large. We must note however that the frequency variability of the time-space spreading relationship implies that any projected relationship between ISO and AES results is likely to be complex (see for instance [50]), and that the main reason for differences in results using the two standards is that the diffusion coefficient (see equation (15)) has a tendency to underestimate the spatial spreading. It is interesting to speculate about the practical applications of an understanding of the time, as well as spatial behaviour of sound diffusing surfaces. Not least, the crosscorrelation of signals measured at closely adjacent points in space might be expected to be modified by reflecting surfaces with complex space-time spreading characteristics. The inter-aural cross correlation coefficient (IACC) is known to be strongly associated with the subjective perception of ‘spaciousness’ [51, 52] - we might note that this term often suffers from similar abuse to the terms ‘diffusion’ and ‘scattering’ as regards rather casual usage without due care with precise definitions! However, rigorous studies have shown that it is possible to correlate IACC measures with subjective test results for Apparent Source Width [53], which suggests that further work to establish a link between time spreading and subjective discrimination and perhaps preference of diffusing/scattering surfaces may yield useful results. It is even possible that modifications to the time-structure of decaying reverberant energy may have implications for the modulation transfer function and hence speech intelligibility in non-diffuse spaces – so it can be seen that several avenues for interesting further work remain open.

6. Conclusion and further research A Finite Difference scheme has been employed to simulate diffuse reflections as an alternative to previously used

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methods. It has been shown that there is a good agreement between FDTD and BEM simulations when compared in the frequency domain. Further, the use of Near-field Farfield Transformations has been shown to be effective in an acoustical application, to generate rapid results for diffuser modelling using the FDTD technique. Additionally, preliminary results concerning the time domain features of the sound reflected by three different surfaces have indicated that FDTD can be useful for the evaluation of sound diffusers in terms of ‘time-spreading’. Further research must be carried out in this area, to build a knowledge (objective and subjective) of the ability of sound diffusers to spread sound in time in addition to their well-known spatial spreading, towards a global time-frequency parameter to quantify scattering. In future work the authors plan to include the vibration of well separators, in order to estimate its significance on the scattering provided by diffusers. Additionally, active elements will be introduced in the simulation in order to evaluate directly the performance of an active diffuser. Acknowledgements We acknowledge fruitful discussions with Prof. Trevor J. Cox from Salford University. We would like to thank Fernando Hernandez for helping us with the configuration and maintenance of computers used for the simulations presented here. Finally, we appreciate Dr. M. Ferri’s help with the optimization of the code. The work was financially supported by the Spanish Ministerio de Educación y Ciencia, and European Union FEDER Project No. FIS200507931-C03-02. References [1] M. R. Schroeder: Diffuse sound reflection by maximum length sequences. J. Acoust. Soc. Am. 57 (1975) 149–150. [2] M. R. Schroeder: Binaural dissimilarity and optimum ceilings for concert halls: More lateral sound diffusion. J. Acoust. Soc. Am. 65 (1979) 958–963. [3] AES information document for room acoustics and sound reinforcement systems - Characterization and measurement of surface scattering uniformity. Audio Engineering Society, 2001. [4] M. Vorländer, E. Mommertz: Definition and measurement of random-incidence scattering coefficients. Applied Acoustics 60 (2000) 187–199. [5] ISO 17497-1: Acoustics - Sound-scattering properties of surfaces - Part 1: Measurement of the random-incidence scattering coefficient in a reverberation room. 2004. [6] T. J. Cox: Predicting the scattering from reflectors and diffusers using 2d boundary elements methods. J. Acoust. Soc. Am. 96 (1994) 874–878. [7] T. J. Cox, Y. W. Lam: Evaluation of methods for predicting the scattering from simple rigid panels. Applied Acoustics 40 (1993) 123–140. [8] T. J. Cox, Y. W. Lam: Prediction and evaluation of the scattering from quadratic residue diffusers. J. Acoust. Soc. Am. 95 (1994) 297–305.


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