Sound propagation over layered poro-elastic ground using a finite ...

Sound propagation over layered poro-elastic ground using a finite-difference model Hefeng Donga) Department of Telecommunications/Acoustics, Norwegian University of Science and Technology, Trondheim, Norway

Amir M. Kaynia and Christian Madshus Norwegian Geotechnical Institute, Oslo, Norway

Jens M. Hovem Department of Telecommunications/Acoustics, Norwegian University of Science and Technology, Trondheim, Norway

共Received 20 October 1999; revised 1 May 2000; accepted 15 May 2000兲 This article presents an axisymmetric pressure-velocity finite-difference formulation 共PV-FD兲 based on Biot’s poro-elastic theory for modeling sound propagation in a homogeneous atmosphere over layered poro-elastic ground. The formulation is coded in a computer program and a simulation of actual measurements from airblast tests is carried out. The article presents typical results of simulation comprising synthetic time histories of overpressure in the atmosphere and ground vibration as well as snapshots of the response of the atmosphere–ground system at selected times. Comparisons with the measurements during airblast tests performed in Haslemoen, Norway, as well as the simulations by a frequency-wave number FFP formulation are presented to confirm the soundness of the proposed model. In particular, the generation of Mach surfaces in the ground motion, which is the result of the sound speed being greater than the Rayleigh wave velocity in the ground, is demonstrated with the help of snapshot plots. © 2000 Acoustical Society of America. 关S0001-4966共00兲04008-X兴 PACS numbers: 43.20.Gp, 43.28.En, 43.28.Js 关DEC兴

INTRODUCTION

This article presents a new finite-difference calculation model for sound propagation in seismo-acoustic media. It also presents synthetic time histories of overpressure above ground and associated ground vibration for an instrumented airblast site in Haslemoen, Norway. The airblast tests, which were performed in June 1994, were part of an extensive field test program entitled Blast Propagation through Forest. One of the objectives of this program has been to use the collected data in order to verify existing noise/vibration prediction models and develop new calculation models. Over the years, researchers have used different approaches to study the propagation of airborne acoustic pulses above a uniform flat ground. The applied methods have included complex impedance ground representation 共e.g., Ref. 1兲, rigid-porous approximation,2,3 viscoelastic approach 共e.g., Ref. 4兲, and frequency-wave-number FFP 共Fast Field Program兲 method 共e.g., Ref. 5兲. The last model has been used by Chotiros6 for studying sound propagation in water-saturated sand in the 10–100-kHz frequency band and by Hole et al.7 for the simulation of low-frequency impulse noise and ground vibration in the Norwegian airblast tests in Haslemoen. A similar model has been used by Tooms et al.8 to predict sound propagation at single frequencies and to study a兲

Permanent address: Institute of Theoretical Physics Northeast Normal University, Changchum, People’s Republic of China. Electronic mail: [email protected]

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J. Acoust. Soc. Am. 108 (2), August 2000

the effect of ground elasticity on atmospheric sound propagation. A different computational technique based on pressurevelocity finite-difference formulation 共PV-FD兲 is used in this study to simulate the propagation of sound over a layered poro-elastic ground. This model, which is formulated in an axisymmetric coordinate system, is an extension of the model originally developed for simulating acoustic wave propagation in borehole logging and seismics.9 The model can handle a refracting atmosphere, although this is not considered in the simulations reported here. The advantage of this formulation is its flexibility in dealing with nonhomogeneities such as topographic features and range-dependent ground parameters. The present version of the model, however, does not consider body wave attenuation in the ground and the pore fluid. The preceding model is used to simulate the dynamic overpressures and ground vibration measured at the airblast test site in Haslemoen, Norway. The simulations are also compared with those by the frequency-wave-number FFP model OASES.7 In addition to presenting typical synthetic time histories of various response quantities, snapshots of the overpressure in the atmosphere and ground particle velocities are presented to highlight the overall features of sound propagation over porous media. In particular, the generation of Mach surfaces in the ground due to superseismic propagation of sound waves is demonstrated.

0001-4966/2000/108(2)/494/9/$17.00

© 2000 Acoustical Society of America

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sities of the solid grains and the pore fluid, respectively. The energy dissipation in the porous medium is reflected by the parameter b⫽ ␾ 2 ␩ / ␬ , where ␬ denotes the permeability and ␩ is the dynamic viscosity of the pore fluid. The stiffness attribute of the porous medium is represented by the shear modulus of the solid frame ␮ b , ␣ ⫽1⫺K b /K s and M ⫽( ␾ K f ⫹( ␣ ⫺ ␾ )/K s ) ⫺1 , where K b , K s and K f are the bulk moduli of the solid frame, solid grains and the pore fluid, ជ 1 and D ជ2 respectively. Finally, the differential operators D are defined by

ជ 1 共 ជf ;A 兲 ⫽Aⵜⵜ• ជf ⫹ⵜAⵜ• ជf , D

共3兲

ជ 2 共 ជf ;B 兲 ⫽Bⵜⵜ• ជf ⫹Bⵜ 2 ជf ⫹ⵜB⫻ 共 ⵜ⫻ ជf 兲 D ⫹2 共 ⵜB•ⵜ 兲 ជf .

共4兲

For a homogeneous ground, the wave equations reduce to

ជ⫹ ␳ f ⳵ tt uជ ⫹ ␳ c ⳵ tt W

␩ ជ ⫽ ␣ M ⵜ 共 ⵜ•uជ 兲 ⫹M ⵜ 共 ⵜ•W ជ 兲, ⳵W ␬ t

FIG. 1. Model geometry used for the finite-difference simulation. A point source is located at the z axis above ground.

共5兲 ជ ⫽ 共 H⫺ ␮ b 兲 ⵜ 共 ⵜ•uជ 兲 ⫹ ␮ b ⵜ 2 uជ ⫹ ␣ M ⵜ 共 ⵜ•W ជ 兲, ␳⳵ tt uជ ⫹ ␳ f ⳵ tt W

I. THEORY AND METHOD

where ␳ c ⫽e ␳ f / ␾ and H⫽K b ⫹4/3␮ b ⫹ ␣ 2 M . In the one-phase elastic limit, the porosity approaches zero and the coefficients in Biot’s equations reduce to K b ⫺2/3␮ b ⫽␭ b and K s ⫽K b where ␭ b and ␮ b are Lame´ constants for the solid. Furthermore ␳ 1 ⫽b⫽0, therefore the first part of Eq. 共2兲 becomes redundant and the second part becomes the wave equation for elastic media:

A. Theoretical formulations

Figure 1 shows a homogeneous atmospheric half-space above a layered poro-elastic ground which satisfies Biot’s theory.10,11 Under azimuth symmetry 共i.e., wave field being independent of ␪兲 the equations governing the pressure in the atmosphere, P(r,z,t), and the velocities in the poro-elastic ជ ␾ (r,z,t), due to a point source in the ground, uជ (r,z,t) and W atmosphere can be written as

ជ 1 共 uជ ;K b ⫺2 ␮ b /3兲 ⫹D ជ 2 共 uជ ; ␮ b 兲 ␳ s ⳵ tt uជ ⫽D ⫽ 共 ␭ b ⫹ ␮ b 兲 ⵜ 共 ⵜ•uជ 兲 ⫹ ␮ b ⵜ 2 uជ ⫹ⵜ␭ b 共 ⵜ•uជ 兲

⳵ tt P 共 r,z,t 兲 / 共 ␳ 0 V 20 兲 ⫽ⵜ• 共 ⵜ P 共 r,z,t 兲 / ␳ 0 兲

⫹ⵜ ␮ b ⫻ 共 ⵜ⫻uជ 兲 ⫹2 共 ⵜ ␮ b •ⵜ 兲 uជ .

⫹2g 共 t 兲 ␦ 共 r 兲 ␦ 共 z⫺z 0 兲 /r, 0⭐r⬍⫹⬁, h⬍z⬍⫹⬁, t⬎0, 共1兲

ជ ␾ ⫹b ⳵ t W ជ ␾ ⫽D ជ 1 共 uជ ; ␣ M ␾ 兲 ␳ 1 ⳵ tt uជ ⫹e ␳ 1 ⳵ tt W ជ 1共 W ជ ␾ ;M ␾ 2 兲 , ⫹D ជ ␾ ⫽D ជ 1 共 uជ ;K b ⫺2 ␮ b /3⫹ ␣ 2 M 兲 ␳⳵ tt uជ ⫹ ␳ 1 ⳵ tt W ជ 1共 W ជ ␾ ; ␣ M ␾ 兲 ⫹D ជ 2 共 uជ ; ␮ b 兲 , ⫹D 0⭐r⬍⫹⬁, ⫺⬁⬍z⬍h, t⬎0, where ␦ (r) is the Kronecker delta, g(t) defines the time variation of the source S, z 0 specifies the position of the source, and V 0 and ␳ 0 are the sound speed in the atmosphere and mass density of air, respectively. The vector uជ denotes ជ the velocity vector of the solid frame 共in the ground兲, W ជ ⫺uជ ) is the velocity vector of the pore fluid relative ⫽ ␾ •(U ជ ␾ ⫽W ជ (r,z,t)/ ␾ , U ជ represents the to the solid frame and W velocity vector of the pore fluid, ␾ is porosity and e denotes tortuosity. The poro-elastic mass parameters are ␳ 1 ⫽ ␾ ␳ f and ␳ ⫽ ␾ ␳ f ⫹(1⫺ ␾ ) ␳ s where ␳ s and ␳ f are the mass den495

J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000

This means that Eq. 共2兲 can be used for an elastic ground by setting porosity to zero and keeping the shear modulus finite. On the ground z⫽h, the interface conditions of pressure and velocities result in ⫺

共2兲

共6兲

1 ⳵ P⫽ ⳵ t u z ⫹ ⳵ t W z , ␳f z

ជ, ⫺ ⳵ t P⫽ ␣ M ⵜ•uជ ⫹M ⵜ•W ជ ⫹2 ␮ b 共 ⳵ z u z ⫺ⵜ•uជ 兲 , ⫺ ⳵ t P⫽Hⵜ•uជ ⫹ ␣ M ⵜ•W

共7兲

0⫽ ␮ 共 ⳵ z u r ⫹ ⳵ r u z 兲 . To simulate the infinite lateral and vertical extent of the media, the far-end boundary as well as the top and bottom boundaries of the model are equipped with dashpots, denoted as absorbing boundary in Fig. 1. More details on these boundaries are given in the next section. In a layered poroelastic ground one has to impose the conditions of stress equality 共both shear and normal兲 and velocity compatibility at each layer interface. The equations can be readily established from those listed above. Dong et al.: Sound propagation over ground

495

TABLE I. Geodynamic parameters of the soil profile at Haslemoen. The values in the parentheses are used in the finite difference models. Layer no.

H 共m兲

␳s (kg/m3)

1 2 3 4 5 6

1.0 1.5 2.5 5.0 10.0 ⬁

1500 1500 1500 1600 1700 1800

Vp 共m/s兲 250 260 280 300 1500 共400兲 1500 共500兲

Vs 共m/s兲 130 140 150 160 180 共200兲 250

B. Finite-difference formulation

The finite-difference approximations of Eq. 共1兲 and the treatment of point source have been given by Dong et al.9 and are incorporated in Appendix A for completeness. In a layered poro-elastic ground, the finite difference solution of Eq. 共2兲 can be simplified by using the finite-difference operators defined with the following form,

ជ j 共 ជf ;A 兲 ⫽F d • 兵 D ជ j 共 ជf ;A 兲其 , ⍀

共8兲

where j⫽1,2 and F d • 兵其 indicates a finite-difference representation of the variable inside the brackets. Using this notation, Eq. 共2兲 can be expressed as

再

ជ ␾⫹ ␾ 2 F d ␳ 1 ⳵ tt uជ ⫹e ␳ 1 ⳵ tt W

␩ ជ ⳵W ␬ t ␾

冎

ជ 1 共 uជ ; ␣ M ␾ 兲 ⫹⍀ ជ 1共 W ជ ␾ ;M ␾ 2 兲 ⫽⌽ ជ k1 , ⫽⍀ ជ ␾其 F d 兵 ␳⳵ tt uជ ⫹ ␳ 1 ⳵ tt W

共9兲

ជ 1 共 uជ ;␭ b ⫹ ␣ 2 M 兲 ⫹⍀ ជ 1共 W ជ ␾ ; ␣ M ␾ 兲 ⫹⍀ ជ 2 共 uជ ; ␮ b 兲 ⫽⍀ ជ k2 , ⫽⌽ ជ k1 and ⌽ ជ k2 contain only spatial derivatives that need where ⌽ to be evaluated at the current time k⌬t in the context of a time marching algorithm. Using the standard second-order explicit finite difference approximations, du u 共 x⫹⌬x 兲 ⫺u 共 x⫺⌬x 兲 ⫽ , dx 2⌬x d 2 u u 共 x⫹⌬x 兲 ⫺2u 共 x 兲 ⫹u 共 x⫺⌬x 兲 ⫽ , dx 2 ⌬x 2

共10兲

ជ k1 and ⌽ ជ k2 . one can derive the finite-difference schemes for ⌽ Using the central finite difference representation in time on the left-hand side of Eq. 共9兲 and rearranging the terms, one can write ជ ␾k⫹1 ⫽bជ 1 ⫹ 共 ⌬t 兲 2 ⌽ ជ k1 , a 11uជ k⫹1 ⫹a 12W ជ ␾k⫹1 ⫽bជ 2 ⫹ 共 ⌬t 兲 2 ⌽ ជ k2 . a 21uជ k⫹1 ⫹a 22W

共11兲

FIG. 2. Source pulse and its Fourier amplitude spectra used in present study. Central frequency is 30 Hz.

ជ k⫹1 ⫽uជ k⫹1 ⫹W ជ ␾k⫹1 . U

共13兲

For the points on the ground surface, the interface equations between pressure and velocities were treated by Dong et al.12 with a stable integration technique. The procedure is given in Appendix C. For the points on the artificial boundaries 共r⫽d, z⫽0 and z⫽H兲 the absorbing boundary conditions are used. Since the slow P-wave in a poro-elastic ground is strongly attenuated, its reflection from the artificial boundaries can be neglected. Therefore, only the reflections of the fast P-wave and S-wave from the artificial boundaries are considered. In the present study the Reynolds’ absorbing boundary condition13 was used. Numerical experiments showed that this method provides a good approximation. The system of finite-difference equations is stable provided that the selected time step satisfies the following:14 ⌬t⭐

⌬x &V max

共14兲

,

where V max is the maximum body wave velocity on the grid and ⌬x is the minimum grid size in both the r and z directions. In addition, ⌬x should not be taken greater than the following value, ⌬x⫽

␭ min , G

共15兲

where ␭ min⫽Vmin /f up is the minimum wavelength in the model, V min is the minimum body wave velocity on the grid and f up is the upper half-power frequency of the source;15 this frequency is 2.5 times the central frequency of the source in this article. G is the number of grid points per minimum wavelength and should be taken at least equal to

ជ k⫹1 can be obtained from The vectors uជ k⫹1 and W ជ 1 /⌬, uជ k⫹1 ⫽⌬

ជ ␾k⫹1 ⫽⌬ ជ 2 /⌬. W

共12兲

ជ i (i, j⫽1,2) and ⌬ are given in Expressions for a i j , bជ i , ⌬ ជ k⫹1 at the next Appendix B. The velocity of the pore fluid U time step can be obtained from Eq. 共12兲: 496


TABLE II. Parameters of the poro-elastic layer. Layer H ␳f ␳s ␮b Kf Ks Kb no. 共m兲 (kg/m3 ) (106 Pa) (kg/m3) (106 Pa) (106 Pa) (106 Pa) 1

0.5

1.2

0.13

2700

9060

60.7

␾

25.5

0.44

Dong et al.: Sound propagation over ground

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FIG. 3. Comparison between poroelastic finite-difference model PORAC 共solid line兲, OASES 共dashed line兲 and experimental data 共dotted line兲. The upper two panels for pressure and lower panels for vertical velocity. 共a兲 Overpressure at 195 m, 共b兲 overpressure at 260 m, 共c兲 vertical particle velocity of ground surface at 195 m, and 共d兲 vertical particle velocity of ground surface at 260 m.

10 in order to reduce the grid dispersion. A value of G ⫽15 was used in this study. II. NUMERICAL RESULTS

The preceding pressure-velocity finite-difference formulation was coded in the computer program PORAC. This code was then used to simulate the overpressure in the air and the corresponding particle velocity in the ground that was measured at the Norwegian test site in Haslemoen, Norway in June 1994. The same measurements were also used by Hole et al.7 in assessing the performance of the frequency-wave-number FFP code OASES.5 The results from the present work are compared with both the measurements and the simulations by Hole et al.7 The Norwegian airblast tests were carried out in the forests of Haslemoen in June of 1994. The measurement setup comprised five microphones for registration of the impulse noise and three seismometers 共one horizontal and two vertical兲 for recording ground vibration. The microphones were installed on two masts at elevations 2, 4, 8, 16 and 30 m

above ground and the seismometers were positioned on the ground surface close to the masts. A total of 35 airblasts were carried out in which charges of 1, 8 and 64 kg were detonated at 2 m above ground at various distances ranging from 190 to 1400 m from the recording station. In addition, extensive meteorological and geotechnical/seismic tests were performed to characterize the atmospheric and ground conditions. A detailed account of the test program and measurements has been reported by Kerry16 and Hole et al.7 Based on these tests it was believed that an average uniform atmosphere with a sound speed of 340 m/s could provide an adequate model for numerical simulations.7 In situ tests by the Spectral Analysis of Surface Waves method17 and 共Troxler兲 radioactive sound resulted in the soil profile given in Table I. The time function describing the acoustic source is the same as that used by Hole et al.,7 g 共 t 兲 ⫽A 关 sin共 2 ␲ f c t 兲 ⫺0.5sin共 4 ␲ f c t 兲兴 ,

0⬍t⬍1/f c , 共16兲

where A is a constant with dimension (s⫺2 ), t denotes time

FIG. 4. Stacked plots of overpressure at 2 m above ground 共a兲 and vertical particle velocity on the ground surface 共b兲 at different distances from the source ranging from 130 to 230 m with spacing of 10 m.

497



497

and f c is the central frequency. The above source and its Fourier spectrum are shown in Fig. 2. The central frequency of the sources is 30 Hz and f up⫽75 Hz. The source positioned at the symmetry axis and 2 m above the ground is in accordance with the location of the C4 explosive charges in the tests. Examination of noise propagation with distance had shown a marked decrease in the dominant frequency of the noise with distance, which was stronger than could be explained by damping alone. This observation led Hole et al.7 to conclude that the lengthening of the noise signals was due to the presence of a top porous soil layer. An extensive simulation study by Hole et al.,7 using various combinations of permeability and thickness in the assumed surficial layer, revealed that a poro-elastic layer with a thickness of 0.5 m and permeability of ␬ ⫽10⫺8 m2 gave a good match with the measured noise over a wide range. In this study the same thickness and permeability of the poro-elastic layer are used. The dynamic viscosity of the pore air ␩ ⫽1.74 ⫻10⫺5 kg/ms and tortuosity e⫽1.25 were used; the other parameters of the porous layer are listed in Table II. In order to reduce grid dispersion and CPU runtime the parameters V p ⫽400 and 500 and V s ⫽200 and 250 m/s were used in layers 5 and 6 instead of those in Table I. The density and sound speed in the atmosphere were taken equal to ␳ 0 ⫽1.2 kg/m3 and V 0 ⫽340 m/s, respectively. For the parameters of the system, Eqs. 共14兲 and 共15兲 give ⌬x⫽0.115 m and ⌬t⫽1.33⫻10⫺4 s. Figure 3 displays synthetic time histories of overpressure at 2 m above ground and associated particle velocity on the ground surface computed by PORAC at distances of 195 and 260 m from the blast. The figure also displays the corresponding results by the frequency-wave-number FFP model OASES7 and the experimental data. Both the present results and OASES results were scaled to produce the same peak overpressure at 195 m. Examination of the plots for the various pressure data shows a fairly good accord in the amplitudes and arrival times between the cases. Moreover, the simulations reveal a decrease in the dominant frequency of the sound with distance as observed in the experimental data. For the ground response, the arrivals and amplitudes of the first peaks are almost the same in the two calculation models. However, the amplitudes of the subsequent peaks 共resulting from reflected and refracted waves in the ground兲 are different. The difference could be partly due to absence of body wave attenuation in the ground. Figure 4共a兲 displays a stacked plot of the simulated overpressures at different distances from the source ranging from 130 to 230 m with 10-m spacing. Figure 4共b兲 shows the corresponding ground motion stacked seismograms for the same receiver locations as in Fig. 4共a兲. The direct wave and surface wave can be recognized in this figure. The velocities of these waves can be calculated from the distance between the receivers and the corresponding difference of the arrival times 共straight lines drawn through these arrivals provide a convenient way to calculate the velocities兲. The figure also displays the oscillatory nature of the ground motion that is a result of waves generated by air and refraction and multiple reflections at layer interfaces. A global picture of the sound propagation and its inter498


FIG. 5. Snapshots of overpressure in the atmosphere and vertical particle velocity of the ground normalized by pressure at t⫽0.1397 s 共a兲, 0.1929 s 共b兲 and 0.2293 s 共c兲.

action with the ground can be obtained by snapshot plots of the spatial variation of overpressure in the atmosphere and particle velocity in the ground. Figures 5共a兲–5共c兲 present a set of such plots normalized by overpressures at times Dong et al.: Sound propagation over ground

498

FIG. 6. Comparison between the results with different permeability 共solid line for ␬ ⫽10⫺10 m2 and dashed line for ␬ ⫽10⫺8 m2 兲 at 195 m from airblast position: 共a兲 overpressure at 2 m above the ground and 共b兲 vertical particle velocity of the ground surface.

0.1397, 0.1929 and 0.2293 s, respectively. The snapshots show a complicated deformation pattern in the ground, which results from the reflection and refraction of P- and S-waves as well as the surface waves. A careful examination of these plots shows the presence of Mach surfaces in the ground 共although not quite straight due to ground layering兲. Because the velocities of P- and S-waves in the upper layers of the ground are lower than the sound speed, a superseismic condition, characterized by two Mach surfaces associated with the two body waves, is realized. This visual feature is one of the most powerful attributes of the finite difference model PORAC that allows one to gain insight into the mechanism of generation and propagation of waves in the ground. To discern the influence of permeability of the porous layer on the sound absorption and the induced ground vibration, different simulations were carried out by decreasing the permeability by 100 times, that is to ␬ ⫽10⫺10 m2. Figure 6 shows a comparison between the results of overpressure at 2 m above ground 关Fig. 6共a兲兴 and vertical vibration on the ground surface 关Fig. 6共b兲兴 for different permeabilities. These results corroborate the conclusions by Hole et al.7 in that the amplitude of overpressure and the peak particle velocity decrease with permeability. Furthermore, the dominant frequencies in both overpressure and ground vibration decrease as permeability increases. Figures 7共a兲 and 7共b兲 show synthetic time histories by PORAC of the vertical velocities of pore air 共solid line兲 and solid frame 共dashed line兲 on the ground surface at 195 m from the source for different permeabilities 关␬ ⫽10⫺8 m2 in Fig. 7共a兲 and ␬ ⫽10⫺10 m2 in Fig. 7共b兲兴. All of the plots are normalized by the magnitude of the velocity of pore air with

permeability ␬ ⫽10⫺8 m2 关solid line in Fig. 7共a兲兴. In Fig. 7共a兲 the particle velocity of the solid frame is scaled up by a factor of 250 relative to the velocity of pore air. In Fig. 7共b兲 the velocities of pore air and solid frame are scaled up by factors of 4.2 and 110 relative to the velocity of pore air with ␬ ⫽10⫺8 m2. Because there is only one poroelastic layer in the ground there is no refraction or reflection in the velocity of the pore air and the arrivals occur simultaneously with the sound wave. The multiple reflections can, however, be seen in the plots of the solid frame velocity. As expected, the amplitude of the velocity of pore air reduces as permeability decreases. Figure 8 presents the snapshots of overpressure in the atmosphere and vertical particle velocity in the ground normalized by the pressure computed with the lower permeability of ␬ ⫽10⫺10 m2. For direct comparison, Figs. 8共a兲 and 8共b兲 are plotted for the same times as in Figs. 6共a兲 and 6共b兲. The same general patterns are displayed in Figs. 8 and 6. III. CONCLUSION

In this article a pressure-velocity finite-difference technique 共PORAC兲 was developed for the simulation of sound propagation over layered porous media. Numerical simulations were carried out for the Norwegian airblast tests in Haslemoen. The top layer was taken as a poro-elastic material. Typical results of the simulations of the overpressure in the atmosphere and the velocities of the solid frame and pore air were presented. The results for the former two parameters were compared with the measurements and the results calculated by Hole et al. using a frequency-wave-number formulation. In addition, a number of snapshots of overpressure

FIG. 7. Comparison of vertical velocities on the ground surface at 195 m from the airblast position 共solid line for the velocity of pore air and dashed line for the velocity of solid frame兲. 共a兲 ␬ ⫽10⫺8 m2 共magnitude of the velocity of solid frame is amplified 250 times relative to that of pore air兲. 共b兲 ␬ ⫽10⫺10 m2 关magnitudes of the velocity of pore air and solid frame are amplified 4.2 and 110 times relative to the magnitude of the velocity of pore air in 共a兲兴.

499



499

APPENDIX A: TREATMENT OF THE SOURCE AND FINITE-DIFFERENCE SCHEMES OF EQ. „1…

Integrating Eq. 共1兲 over the domain G containing the singular point source in cylindrical coordinate system one gets

冕

G

关 ⳵ tt P/ 共 ␳ 0 V 20 兲兴 dV

⫽

冕

G

ⵜ• 共 ⵜ P/ ␳ 0 兲 dV⫹

冕

G

关 2g 共 t 兲 ␦ 共 r 兲

⫻ ␦ 共 z⫺z 0 兲 /r 兴 dV.

共A1兲

With the integral theorem of mean value, Green’s formula, and the property of ␦ (rជ ) function in Eq. 共A1兲, one can easily obtain

⳵ tt P/ 共 ␳ 0 V 20 兲 ⌬V⫽

冖

⌺

共 ⳵ n P/ ␳ 0 兲 dS n ⫹4 ␲ g 共 t 兲 ␦ 0r ␦ z 0 z ,

共A2兲

where ⌺ is the surface of the volume G. The finite-difference approximation of Eq. 共A2兲 can be expressed as P 共 0,j,k⫹1 兲 ⫽2 P 共 0,j,k 兲 ⫺ P 共 0,j,k⫺1 兲 ⫹ ␥ V 20 兵 4 关 P 共 1,j,k 兲 ⫺ P 共 0,j,k 兲兴 ⫹ P 共 0,j⫹1,k 兲 ⫺2 P 共 0,j,k 兲 ⫹ P 共 0,j⫺1,k 兲 ⫹ 共 16␳ 0 g 共 k 兲 /⌬r ␦ 0i ␦ j 0 j 兲其 , J⬍ j⬍N,

0⬍k⬍L.

共A3兲

The finite-difference formulation of Eq. 共1兲 without source is P 共 i, j,k⫹1 兲 ⫽2 P 共 i, j,k 兲 ⫺ P 共 i, j,k⫺1 兲 ⫹ ␥ 2 V 20 FIG. 8. Snapshots of overpressure in the atmosphere and vertical particle velocity of the ground normalized by pressure with permeability ␬ ⫽10⫺10 m2 and the rest of the parameters are same as those in Fig. 6 at t ⫽0.1397 s 共a兲 and 0.1929 s 共b兲.

⫻ 兵 P 共 i⫹1,j,k 兲 ⫺ P 共 i, j,k 兲 ⫹ P 共 i⫺1,j,k 兲 ⫹ 关 P 共 i⫹1,j,k 兲 ⫺ P 共 i⫺1,j,k 兲兴 / 共 2i 兲 ⫹ P 共 i, j⫹1,k 兲 ⫺2 P 共 i, j,k 兲 ⫹ P 共 i, j⫺1,k 兲其 ,

and solid frame velocity were presented at different time steps. Among the attractive features of the presented model are its capability to present the results visually together with its versatility in dealing with arbitrary terrain conditions and range-dependent properties. Work is underway to extend PORAC to incorporate these features.

ACKNOWLEDGMENTS

The authors wish to thank Dr. Lars R. Hole of the Norwegian Defense Construction Services 共NDCS兲 and Professor Ulf Kristiansen of the Norwegian University of Science and Technology for their helpful discussions and suggestions. Partial financial support from Norwegian Defense Construction Service 共NDCS兲 and the Norwegian Research Council under the program SIP-4 at Norwegian Geotechnical Institute 共NGI兲 are gratefully acknowledged. 500


0⬍i⬍M , J⬍ j⬍N, 0⬍k⬍L,

共A4兲

where the space grids are given by r⫽i⌬r and z⫽ j⌬z and the time grid is given by t⫽i⌬t; P(r,z,t) is expressed by P(i, j,k); and M ⫽a/⌬r, J⫽h/⌬z, N⫽H/⌬z, and L ⫽T/⌬t where h is the ground surface and T the maximum measure time. In this article, we assume that ⌬z⫽⌬r, and ␥ ⫽⌬t/⌬r. ᠬi, ⌬ ᠬ i AND ⌬ APPENDIX B: EXPRESSIONS FOR a ij , b IN EQS. „10… AND „11…

a 11⫽ ␳ 1 , a 12⫽E ␳ 1 共 1⫹b⌬t/ 共 2E ␳ 1 兲兲 , a 21⫽ ␳ , a 22⫽ ␳ 1 , bជ 1 ⫽ ␳ 1 共 2uជ k ⫺uជ k⫺1 兲

共B1兲

ជ ␾k ⫺ 共 1⫺b⌬t/ 共 2E ␳ 1 兲兲 W ជ ␾k⫺1 兴 , ⫹E ␳ 1 关 2W ជ ␾k ⫺W ជ ␾k⫺1 兲 , bជ 2 ⫽ ␳ 共 2uជ k ⫺uជ k⫺1 兲 ⫹ ␳ 1 共 2W Dong et al.: Sound propagation over ground

500

d 2 ⫽ ␥␾ P 共 i,J,k⫺1 兲 ⫹ ␳ 1 关 2u z 共 i,J,k 兲 ⫺u z 共 i,J,k⫺1 兲兴

ជ 1 ⫽a 22关共 ⌬t 兲 2 ⌽ ជ k1 ⫹bជ 1 兴 ⫺a 12关共 ⌬t 兲 2 ⌽ ជ k2 ⫹bជ 2 兴 , ⌬ ជ 2 ⫽a 11关共 ⌬t 兲 2 ⌽ ជ k2 ⫹bជ 2 兴 ⫺a 21关共 ⌬t 兲 2 ⌽ ជ k1 ⫹bជ 1 兴 , ⌬

共B2兲

⫹e ␳ 1 关 2W ␾ z 共 i,J,k 兲 ⫺W ␾ z 共 i,J,k⫺1 兲兴 ⫹

⌬⫽a 11a 22⫺a 12a 21 .

⫻W ␾ z 共 i,J,k⫺1 兲 ⫺ ␥ 2 兵关 QR 共 i,J 兲 ⫹QR 共 i,J⫺1 兲兴 Z 1

APPENDIX C: TREATMENT OF BOUNDARY CONDITIONS ON GROUND SURFACE z Ä h

⫹ 关 QM 共 i,J 兲 ⫹QM 共 i,J⫺1 兲兴 Z 2 其 ,

Let us consider a symmetrical grid block containing the interface 共ground surface z⫽h⫽J⌬z兲. Integrating Eqs. 共1兲 and 共2兲 over domains V 1 and V 2 , respectively, we get

冕冕

V1

V2

V2

冊

1 ⳵ PdV⫽ ⵜ• ⵜ P dV, ␳ ␳ 0 V 20 tt V1 0

⫹ ␳ 1 共 2W ␾ z 共 i,J,k 兲 ⫺W ␾ z 共 i,J,k⫺1 兲兲 ⫺ ␥ 2 兵关 QA 共 i,J 兲 ⫹QA 共 i,J⫺1 兲兴 Z 1 ⫹ 关 QR 共 i,J 兲

共C1兲

⫹QR 共 i,J⫺1 兲兴 Z 2 其 ⫺2 ␥ 2 关 ␮ b 共 i,J 兲 ⫹ ␮ b 共 i,J⫺1 兲兴 ⫻关 u z 共 i,J,k 兲 ⫺u z 共 i,J⫺1,k 兲兴 ,

冕

V2

冕

V2

⫹

ជ ␾ ;M ␾ 2 兲 ␾ 兴 dV, 关 D 1 j 共 uជ ; ␣ M ␾ 兲 ⫹D 1 j 共 W

共C2兲

共 2i 兲 ⫹u z 共 i,J,k 兲 ⫺u z 共 i,J⫺1,k 兲 ,

关 D 1 j 共 uជ ;K b ⫺2 ␮ b /3⫹ ␣ 2 M 兲兴 dV

冕

V2

再

for j⫽1,

uz ,

for j⫽2,

W␾ j⫽

再

共C3兲

冉

m 12 m 13

m 21 m 22

m 23

m 31 m 32

m 33

⫺W r 共 i⫺1,J⫺1,k 兲兴 /4⫹ 关 W r 共 i,J,k 兲 ⫹W r 共 i,J⫺1,k 兲兴 / 共 2i 兲 ⫹W z 共 i,J,k 兲 ⫺W z 共 i,J⫺1,k 兲 .

j⫽1,

W ␾z ,

j⫽2.

共C4兲

冊冉冊

冉冊

P d1 u • 共 i,J,k⫹1 兲 ⫽ d 2 , 共C5兲 z W ␾z d3

冉

n 11 n 12 n 21 n 22

冊冉冊 •

m 11⫽

␳ 0 V 20

m 21⫽ ␾␥ , m 31⫽ ␥ , d 1⫽

,

m 12⫽⫺ ␥ ,

n 11⫽ ␳ 1 , n 12⫽e ␳ 1 ⫹

b⌬t , 2

m 22⫽ ␳ 1 , m 32⫽ ␳ ,

共C6兲

m 33⫽ ␳ 1 ,

f 1 ⫽ ␳ 1 关 2u r 共 i,J,k 兲 ⫺u r 共 i,J,k⫺1 兲兴 ⫹e ␳ 1 关 2W ␾ r 共 i,J,k 兲

共 2 P 共 i,J,k 兲 ⫺ P 共 i,J,k⫺1 兲兲 ⫺ ␥ 关 u z 共 i,J,k⫺1 兲 ␳ 0 V 20

⫺ P 共 i,J,k 兲兲 ⫹

共C14兲

f 2 ⫽ ␳ 关 2u r 共 i,J,k 兲 ⫺u r 共 i,J,k⫺1 兲兴 ⫹ ␳ 1 关 2W ␾ r 共 i,J,k 兲 ⫺W ␾ r 共 i,J,k⫺1 兲兴 ⫹ ␥ 2 关 QA 共 i,J 兲 Z 3 ⫹QR 共 i,J 兲 Z 4 ⫹Z 5 兴 ,

冋

1 ⫺ ␾ W ␾ z 共 i,J,k⫺1 兲兴 ⫹ ␥ 2 共 P 共 i,J⫹1,k 兲 ␳0 P 共 i⫹1,J,k 兲 ⫺ P 共 i⫺1,J,k 兲 2i

共C15兲 Z 3 ⫽ 关 u r 共 i⫹1,J,k 兲 ⫺2u r 共 i,J,k 兲 ⫹u r 共 i⫺1,J,k 兲兴 ⫹ 关 u r 共 i⫹1,J,k 兲 ⫹u r 共 i,J,k 兲兴 / 共 2i⫹1 兲 ⫺ 关 u r 共 i⫺1,J,k 兲 ⫹u r 共 i,J,k 兲兴 / 共 2i⫺1 兲

册

⫹ P 共 i⫹1,J,k 兲 ⫺2 P 共 i,J,k 兲 ⫹ P 共 i⫺1,J,k 兲 , 501

b⌬t W ␾ r 共 i,J,k⫺1 兲 2

1

2

共C13兲

n 21⫽ ␳ , n 22⫽ ␳ 1 ,

⫹ ␥ 2 关 QR 共 i,J 兲 Z 3 ⫹QM 共 i,J 兲 Z 4 兴 ,

b⌬t m 23⫽e ␳ 1 ⫹ , 2

共C12兲

where

⫺W ␾ r 共 i,J,k⫺1 兲兴 ⫹

m 13⫽⫺ ␾␥ ,

冉冊

ur f1 , 共 i,J,k⫹1 兲 ⫽ W ␾r f2

where 1

共C11兲

For j⫽1, W ␾r ,

With the integral theorem of the mean value, the divergence theorem and the boundary conditions in Eq. 共7兲 we can get a set of equations for P(r,z,t), u j (r,z,t) and W ␾ j (r,z,t). The difference schemes can be obtained by center difference and one-side difference. For j⫽2, m 11

共C10兲

Z 2 ⫽ 关 W r 共 i⫹1,J,k 兲 ⫹W r 共 i⫹1,J⫺1,k 兲 ⫺W r 共 i⫺1,J,k 兲

ជ ␾ ; ␣ M ␾ 兲 ⫹D 2 j 共 uជ ; ␮ b 兲兴 dV, 关 D 1 j共 W

ur ,

Z 1 ⫽ 关 u r 共 i⫹1,J,k 兲 ⫹u r 共 i⫹1,J⫺1,k 兲 ⫺u r 共 i⫺1,J,k 兲 ⫺u r 共 i⫺1,J⫺1,k 兲兴 /4⫹ 关 u r 共 i,J,k 兲 ⫹u r 共 i,J⫺1,k 兲兴 /

where j⫽1,2 and u j⫽

共C9兲

where QR⫽ ␣␾ M , QM ⫽ ␾ 2 M , QA⫽K b ⫺ 23 ␮ b ⫹ ␣ 2 M and

关 ␳⳵ tt u j ⫹ ␳ 1 ⳵ tt W ␾ j 兴 dV

⫽

共C8兲

d 3 ⫽ ␥ P 共 i,J,k⫺1 兲 ⫹ ␳ 共 2u z 共 i,J,k 兲 ⫺u z 共 i,J,k⫺1 兲兲

关 ␳ 1 ⳵ tt u j ⫹e ␳ 1 ⳵ tt W ␾ j ⫹b ⳵ t W ␾ j 兴 dV

⫽

冕

冕冉

1

b⌬t 2


⫹ 关 u z 共 i⫹1,J,k 兲 ⫺u z 共 i⫹1,J⫺1,k 兲 ⫺u z 共 i⫺1,J,k 兲共C7兲

⫹u z 共 i⫺1,J⫺1,k 兲兴 /2, Dong et al.: Sound propagation over ground

共C16兲 501

Z 4 ⫽ 关 W r 共 i⫹1,J,k 兲 ⫺2W r 共 i,J,k 兲 ⫹W r 共 i⫺1,J,k 兲兴

5

⫹ 关 W r 共 i⫹1,J,k 兲 ⫹W r 共 i,J,k 兲兴 / 共 2i⫹1 兲 ⫺ 关 W r 共 i⫺1,J,k 兲 ⫹W r 共 i,J,k 兲兴 / 共 2i⫺1 兲 ⫹ 关 W z 共 i⫹1,J,k 兲 ⫺W z 共 i⫹1,J⫺1,k 兲 ⫺W z 共 i⫺1,J,k 兲 ⫹W z 共 i⫺1,J⫺1,k 兲兴 /2,

共C17兲

Z 5 ⫽2 ␮ b 共 i,J 兲关 u r 共 i⫹1,J,k 兲 ⫺2u r 共 i,J,k 兲 ⫹u r 共 i⫺1,J,k 兲兴 ⫹2 ␮ b 共 i,J 兲

u r 共 i⫹1,J,k 兲 ⫺u r 共 i⫺1,J,k 兲 2i

⫺2 ␮ b 共 i,J 兲

u r 共 i,J,k 兲 ⫹ 关 ␮ b 共 i,J 兲 ⫹ ␮ b 共 i,J⫺1 兲兴 i2

⫻兵 u r 共 i,J,k 兲 ⫺u r 共 i,J⫺1,k 兲 ⫹ 关 u z 共 i⫹1,J,k 兲 ⫹u z ⫻ 共 i⫹1,J⫺1,k 兲 ⫺u z 共 i⫺1,J,k 兲 ⫺u z 共 i⫺1,J⫺1,k 兲兴 /4其 . 共C18兲 1

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502