On the use of leaky modes in open waveguides for the sound propagation modeling in street canyons

Transcription

1 On the use of leaky moes in open waveguies for the soun propagation moeling in street canyons Arien Pelat, a Simon Félix, an Vincent Pagneux LAUM, CNRS, Université u Maine, avenue Olivier Messiaen, 7285 Le Mans, France Receive 3 January 29; revise September 29; accepte 2 September 29 An urban, U-shape, street canyon being consiere as an open waveguie in which the soun may propagate, one is intereste in a multimoal approach to escribe the soun propagation within. The key point in such a multimoal formalism is the choice of the basis of local transversal moes on which the acoustic fiel is ecompose. For a classical waveguie, with a simple an boune cross-section, a complete orthogonal basis can be analytically obtaine. The case of an open waveguie is more ifficult, since no such a basis can be exhibite. However, an open resonator, as isplays, for example, the U-shape cross-section of a street, presents resonant moes with complex eigenfrequencies, owing to raiative losses. This work first presents how to numerically obtain these moes. Results of the transverse problem are also compare with solutions obtaine by the finite element metho with perfectly mathe layers. Then, examples are treate to show how these leaky moes can be use as a basis for the moal ecomposition of the soun fiel in a street canyon. 29 Acoustical Society of America. DOI:.2/ PACS numbers: 43.2.Mv, Js, 43.5.Vt LLT Pages: I. INTRODUCTION The aim of the present work is to solve the wave equation in a long open rectangular enclosure Fig.. This open waveguie is an iealize omain of propagation moeling a street canyon. This iealize omain being a very simple representation of a real street canyon, competitive effects between guie waves between street facaes, an leaky waves through the opening on the sky occurring along the propagation can be more easily apprehene. In this paper, a moal base metho is propose to escribe acoustic propagation in rectangular straight open waveguies with a particular attention pai to wave raiation in the free space above. Then, in other future stuies in the fiel of urban acoustics, one coul exten the principles of this metho to treat more realistic street geometries. The investigation of soun propagation in urban environments an streets has been the subject of extensive researches in the past 4 ecaes, as the response to a growing social eman. After experimental observations in the 96s, 3 the earliest theoretical works on this topic were conucte in the 97s. Davies, 4 Lee an Davies, 5 Stenackers et al., 6 Lyon 7 use image sources to stuy multiple soun reflections in a street consiere as a channel between two infinite walls. Later, image source metho has been improve consiering scattering at facae irregularities, 8 iffusely reflecting facaes, 9 or coherent image sources. Other energetic approaches were also use in urban acoustics. Kang evelope a raiosity base moel, Braley 2 use ray tracing metho, an Picaut 3 propose a metho base on a iffusion equation governing soun particles propagation. These energetic approaches give statistical escription of soun fiels in urban environments an are able to moel more or less accurately numerous phenomena occurring in streets. Since these approaches assume high frequency hypothesis, they cannot escribe soun fiels when the wavelength is in the range of street with. Furthermore, computation costs strongly increase for complex geometries or for threeimensional 3D problems. As pointe out by Lu an Walerian, 4 interference effects are significant for relatively narrow street canyons. Hence, wave methos present real interests for soun propagation moeling. Bullen an Fricke, 5,6 3 years ago, stuie the wave propagation in two-imensional 2D street using a moal approach, notably to moel junctions of streets. To solve the wave equation in a 2D street canyon, finite element metho FEM or bounary element metho 7 coul as well give solutions to the wave equation for realistic geometries, but the high computation costs restrict their use at low frequencies or for 2D problems. The equivalent sources approach 8 2 is fiel-base rather than ray-base, an apprehens the resonant behavior of a city canyon. In this approach, a set of equivalent sources is use to couple the free half space above the canyon to the cavity insie the canyon. The finite ifference time omain FDTD metho escribes the soun fiel in 2D even 3D problems 2 an can moel a priori a very large number of phenomena. The parabolic equation couple with FDTD metho can also be useful to take into account meteorological effects. 22 So, even if these wave methos can moel soun propagation in realistic situations, the necessary numerical resolution of the wave equation oes not provie explicitly links between solution behaviors an omain geometry of the stuie problem. The aim of the present paper is to establish a multimoal escription of the wave propagation in a 3D street canyon, regare as a straight open waveguie. In the classical case of close waveguies having simple an boune cross-section, a complete orthogonal moal baa Author to whom corresponence shoul be aresse. Electronic mail: arien.pelat.etu@univ-lemans.fr 2864 J. Acoust. Soc. Am. 26 6, December /29/266/2864/9/$ Acoustical Society of America

2 y x Ω 2 L Ω z y sis i can be analytically obtaine. The case of open waveguies, as the one shown in Fig., with a partially boune cross-section, is more ifficult, since no such a basis can be exhibite. However, such a cross-section being regare as an open resonator also isplays eigenmoes with complex eigenfrequencies, owing to the raiation losses. 5,23 25 In this paper, we propose to escribe how the resonant moes of the open cross-section of an open waveguie Fig. can be use to give a multimoal formulation of the soun propagation in long open enclosures. A general metho to compute the resonant frequencies an moe shapes in the transversal open cross-section of the uct is escribe. Results of the transverse problem are also compare with FEM computations using perfectly matche layer PML. Then, the multimoal propagation in a straight open waveguie is formulate an numerical examples are given an iscusse. II. EIGENMODES OF THE TRANSVERSE PROBLEM A. Theory an formulation The wave equation to solve is 2 z FIG.. A straight open waveguie as moel of a street canyon. 2 2 c t 2p =, with 2 the Laplacian operator, c the wave spee, an p the pressure. In a moal approach within a uniform waveguie, elementary solutions for the pressure are written as p i x,y,z = e jk i x t i y,z, 2 where k 2 i =k 2 2 i, Rk i, Ik i, k=/c, an i, i are the eigensolutions of the transverse eigenproblem 2 = 2, 3 with proper bounary conitions, in the cross-section of the waveguie. Then, a solution in the waveguie can be built as a sum on these elementary solutions the time epenence exp jt is omitte: px,y,z, = a i e jk i x + b i e jk i x i y,z, 4 i with i an integer number an the coefficients a i an b i are foun as functions of the bounary conitions efine at the FIG. 2. The cross-section of a street canyon seen as a 2D open rectangular cavity. waveguie extremities. The transverse moes of the open waveguie shown in Fig. are written as the solutions, of the eigenproblem 3 with bounary conitions z = ifz = an y, 5a z = ifz = an y, 5b y = ify = an z 5c in the cross-section of the waveguie, regare as a 2D rectangular cavity of with L an epth, open on the semiinfinite space 2 Fig. 2. As we present below, the moes i y,z can be foun by solving the continuity equation at the interface between an 2. A similar problem, with elastic waves, has been treate by Marauin an Ryan. 23 In their work, authors have calculate the iscrete frequencies of the elastic vibration moes of a 2D rectangular rige fabricate from one material that is bone on the planar surface of a substrate of a secon material. In the following, part of the equations are erive from this work to be aapte in the case of a flui resonant open cavity. In the cavity z, the general solution satisfying bounary conitions can be written as a iscrete sum of functions, y,z = A n cos n z n y n with the epth of the cavity, n 2 = 2 n/l 2 with n an integer number an n y = 2 n cos n L y L 2, where mn is the Kronecker symbol. Note that a finite impeance at the walls absorbing material coul be consiere by, e.g., moifying expression 7, although it coul result in a less straightforwar formulation. An alternative, then, woul be to solve the transverse problem using a FEM. This woul be also particularly aapte in the case of more complex cavity shapes, moeling facaes irregularities. Above the cavity z, the general solution is written as the spatial Fourier transform 6 7 J. Acoust. Soc. Am., Vol. 26, No. 6, December 29 Pelat et al.: Multimoal soun propagation in street canyons 2865

3 y,z = + 2 Be j y y z z y, where z 2 = 2 y 2 with R z an I z. Continuity equations in the interface plane z=, y are an y,z = + = y,z = z y,z = + = z y,z =. 8 9 A first relation between the A n an B is foun by substituting Eqs. 6 an 8 for in combine Eqs. 5a an : to fin where A n n sin n n y = j + B z e j y y y, n 2 B = j A n n sin n S n y, 2 n z S n y n ye = j y y y. 3 A secon relation is foun by using the continuity equation 9: A n cos n n y = + Be j y y y ; n 2 whence it follows that A m cos m = + 2 BS m * y y, 4 5 where m is an integer. Then, Eqs. 2 an 5 lea to the set of linear, homogeneous equations 6 for the A n : min, A m cos m = j mn n A n sin n, 6 n where mn = 2 + S * m S n y. 7 z It is easily shown that, for real values of y, mn vanishes unless m an n have the same parity. 23 Then, Eq. 6 breaks up into two sets of linear equations, one governing the symmetrical eigenmoes even functions of y with even values of m an n, an the other governing antisymmetrical eigenmoes o functions of y with o values of m an n. The following equation gives a general expression of mn for both even or o inices m an n: αl I (2, ).2 (, ).3 (, ) (a) mn = 3m+n/2 2 m 2 n n + y 2 z y + m L y + n L sinc m y L sinc n y L y Finally, Eq. 6 can be written in the matrix form DA =, (2, ) (, ) (2, 2) 9 where the components of vector A are A n =A n sin n an terms of matrix D are D mn = cot m mn j mn n. 2 Then, the eigenvalues of the transverse eigenproblem are the values i of for which etd=, an the eigenfunctions i are given by the corresponing set of coefficients A n i, satisfying Eq. 9 with D=D i. B. Numerical resolution L =.4 αl I (b) L =2 αl R αl R FIG. 3. Color online Spectrums of complex eigenvalues of the transverse problem for two ifferent values of the aspect ratio: a /L=.4; b /L =2. : symmetric moes; : antisymmetric moes; : eigenvalues compute with FEM. After truncation of Eq. 9 at a finite-size matrix problem, zeros of the eterminant of D are numerically locate in the complex -plane to compute eigenvalues i of the transverse problem Fig J. Acoust. Soc. Am., Vol. 26, No. 6, December 29 Pelat et al.: Multimoal soun propagation in street canyons

4 R φ (2,) I φ (2,) φ (2,).4.2 Owing to the raiation losses in the infinite space 2 above the waveguie, eigenvalues are complex, lying in the lower half-plane. 5,23 25 The spectrum isplays families of eigenvalues corresponing to either symmetric blue circles or antisymmetric moes re crosses. An analogy with the classical, real, moes of the simple problem with a Dirichlet conition = at z= instea of the exact raiating conition written in Sec. II A, allows us to label the complex moes i at least the moes which eigenvalue is locate close enough to the real axis in the complex -plane with a couple of integers enoting the number of vertical an horizontal noal lines Fig. 4. Following this terminology, the families isplaye in Fig. 3 are the p,q with p constant. Figure 3 shows the spectrum of eigenvalues for two ifferent values of the aspect ratio of the cavity: a /L=.4 an b /L=2. The pattern in both plots is similar, exhibiting the families of moes p,q with p constant. However, in the eeper cavity, with the aspect ratio /L=2, the confinement of the moes is more important; thus, eigenvalues i have a smaller imaginary part than in the cavity with aspect ratio.4. For comparison, a finite element metho is use to solve the transverse eigenproblem 3. The semi-infinite space above the cavity is boune with PMLs, as use by Hein et al. 26 an Koch 5 in a similar problem Fig. 5. The results shown in this paper Figs. 3b an 6 have been obtaine with parameters =, l=l, PML =, an y = z =+j see Appenix. A Dirichlet conition = is impose on the outer bounaries of the PML. Moreover, as the geometry of the cross-section is symmetric about the z FIG. 4. Symmetric moe 2, real part, imaginary part, an moulus. The inices 2, are chosen following an analogy with the classical eigenmoes in a close cavity with a Dirichlet conition = on the upper bounary: these inices enote the number of vertical an horizontal noal lines. PML PML l z FIG. 5. Geometry of the omain with PML in the FEM computations. y axis, one-half only of the omain is meshe, with the appropriate symmetry or antisymmetry conition impose at y =. Computations have been performe using the partial ifferential equation toolbox from MATLAB. The results of the two compare methos the resolution of Eq. 9 an the FEM are in goo agreement Fig. 3b. The iscrepancy between the results increases for larger values of the imaginary part I i. However, eigenvalues that are less well-estimate are associate with moes that will be strongly attenuate when propagate in the waveguie. They are then of seconary importance when consiering the transport of energy on a sufficiently long istance. Moreover, it will be shown in the following that the contribution of these moes in the etermination of the sets of coefficients a i an b i is almost negligible. A comparison between the eigenfunctions i euce from the metho etaile in Sec. II A an from the FEM computation shows also a goo agreement Fig. 6. Both methos give very similar results, even when the error on the imaginary part of i becomes more significant Fig. 6, bottom. The figure also clearly shows that low orer moes weakly raiate in the infinite space 2 : the effect of the opening of the waveguie appears as a small perturbation on the classical, nonraiating, solutions that woul be obtaine by applying a homogeneous Dirichlet conition at the top of the waveguie z=. For higher orer moes, however, eigenfunctions i iffer more an more from the real, Dirichlet, solutions. Patterns of noal lines are more complex. Consequently, the inexation with inices p,q become less relevant, an the confinement, which was strong for low orer moes, becomes weaker, with the energy increasing near the interface z=. Now that the transverse eigenmoes are etermine, they can be use to write a multimoal formulation of the soun propagation within the open waveguie that isplays a street canyon. III. PROPAGATION ALONG THE STREET As explaine in Sec. I, the transverse moes i are use, for given source an raiation conitions at the ens of the waveguie, to built a solution of the wave equation, as written in Eq. 4. Because the transverse eigenvalues i are complex with I i, the propagation constants k i are also complex, with Ik i, even for real source frequency. Then, all the moes i exp jk i x in the waveguie ecrease exponentially while propagating, reflecting the raiation losses uring the propagation along the open waveguie. This correspons to leaky moes. Then, the two sets of coefficients a i an b i must be foun, as functions of the en conitions in the waveguie. At the input en of the waveguie, a source conition is efine as a given acoustic pressure istribution in a plane x=const, with frequency. For example, at x=: p,y,z,= p y,zexp jt. At the output en of the waveguie is given a raiation conition. J. Acoust. Soc. Am., Vol. 26, No. 6, December 29 Pelat et al.: Multimoal soun propagation in street canyons 2867

5 φ (3,) φ FEM (3,) φ(3,) φ FEM (3,) αl =.55.i φ(3,2) α FEM L =.55.i φ FEM (3,2).54.4i φ(3,3).54.5i φ FEM (3,3) i.6.i.75.9i.78.2i φ(3,4) φ FEM (3,4) φ(3,5) φ FEM (3,5) i.84.32i 2.5.7i i φ(3,6) φ FEM (3,6) φ(3,7) φ FEM (3,7) i i i i FIG. 6. Antisymmetric eigenfunctions 3,q, q, an comparison with FEM computations. The PML omain is not shown in the FEM results. A. Input conition Let us call P i x=a i expjk i x+b i exp jk i x the coefficients in the evelopment in series 4. Since the moes i are not orthogonal, the initial fiel p y,z, cannot be projecte on the i as it is classically mae to fin the P i. Thus, after truncation at a finite number N of terms in the evelopment 4, a least squares metho is thus use to fin these coefficients: 27 P = p, where the ith component of P is P i an ij = i j, p i = i p, with the prouct 2 22a 22b 2868 J. Acoust. Soc. Am., Vol. 26, No. 6, December 29 Pelat et al.: Multimoal soun propagation in street canyons

6 fg =f g. 23 Practically, we consier source conitions p y,z, with a compact support, inclue in, i.e., insie the street canyon. Thus, for convenience when etermining P, we consier, as the eigenfunctions i, their restriction to the omain, that is, y,z L 2, L 2,, i y,z = A i n cos i n z n y, n an the prouct above is 24 fg f gyz. 25 = Note that in the case of real orthogonal moes, as in classical close waveguies, the least squares metho gives the usual projection coefficients P m = m p. B. Output conition Let Q i x= jk i a i expjk i x b i exp jk i x be the coefficients in the evelopment of the x-component of p. One assumes that at the output en of the waveguie, say, at x =x en, the conition is given as an amittance matrix Y en fulfilling Q x en =Y en P x en. Again, as for the formulation at the input en, an ue to the non-orthogonality of the eigenmoes, the matrix Y en, for some complex en conitions, may not be straightforwarly calculate. However, usual en conitions rigi en, nonraiating open en, anechoic termination can be easily formulate with this type of amittance matrix, generalization for all moes of the usual amittance for the plane wave. 28 C. Solutions for ˆa i an ˆb i Now that an input conition P O an an output conition Y en are known, the vectors a an b of the a i an b i in Eq. 4 can be calculate: 29,28 a = P, b = P, where =D Y en +Y c Y en Y c D, D is iagonal with terms D i =expjk i x en, Y c is iagonal with terms Y ci = jk i. Thus, with these solutions for a an b, the pressure fiel in the open waveguie can be calculate. However, terms exp jk i x can be the source of numerical problems of convergence. Then, efining b =D b, the pressure fiel is written as p N x,y,z, = a i e jk i x + b ie jk i x en x i y,z. 28 i This new formulation epens only on D, not on D, an on exponentials with positive arguments x or L x. 3 IV. RESULTS In the following, for simplicity, we will consier the wave fiel ownstream from a source in an infinite waveguie. Then, there are no back propagate waves: b i =. A. Input conition The initial conition at x= is the pressure istribution shown in Fig. 7 left part an given by 3 p y,z = e k= k y y k 2 z zk 2 2 /2 k, 29 2 where k R + an y k,z k,,. The associate imensionless frequency is k=.2. This input conition is chosen as a nontrivial solution for the moal formulation. Moreover, it can be seen as a simple an general way to escribe several, spatially istribute, sources. Then, the a i are foun by substituting Eq. 29 for p y,z in Eqs. 2 an 22. The moal reconstruction is shown in Fig. 7. Using a basis of N=N=3 moes we recall that N is the number of terms in the series 28 an N is the size of the linear problem 2 in the least square estimation of the a i, so that NN, the input pressure conition is well reprouce with a resiual error of 3.6%. This error is ue to the high orer epth moes of the first families that have eliberately not been consiere in the moal basis because of their weak of relevance in the propagation. Furthermore, it will be shown in the following that omitting these moes oes not affect significantly the estimation of the a i for the moes taken into account. To evaluate the convergence of the metho when increasing the number N of moes taken into account in Eq. 28, from a N=3 moes basis, an error is efine as = Input source conition p (y, z) Moal reconstruction with 3 moes p (3) (,y,z) R{p (3) (,y,z)} I{p (3) (,y,z)}.5 p N p 2 yz p 2 yz FIG. 7. Top: Initial conition at abscissa x=. The parameters of the three Gaussian functions in Eq. 29 are y,z, =.8L,.42,.24, y 2,z 2, 2 =.3L,.26,.22, an y 3,z 3, 3 =.4L,.7,.24. Bottom: Moal reconstruction of the initial conition p y,z, using N=N =3 moes., 3 where p N is the moal solution obtaine with N moes Eq. 28, an p the reference fiel. The moes are sorte by.5.5 J. Acoust. Soc. Am., Vol. 26, No. 6, December 29 Pelat et al.: Multimoal soun propagation in street canyons 2869

7 TABLE I. Classification of the moes by increasing value of Ik i at frequency k=.2. Classification st 2n 3r 4th 5th 6th 7th 8th 9th th increasing value of the imaginary part of their propagation constant k i, that is, from the least ampe to the most ampe leaky moe propagating along the canyon Table I. This type of classification epens irectly on each eigenvalue an the source frequency. The convergence of the reconstruction of the initial conition p y,z is shown in Fig. 8 circles. Naturally in such a moal approach, epening on the source istribution, each moe introuce in the computation contributes ifferently to the reconstruction of p y,z. B. Propagation within the waveguie Couple m,n,, 2,, 2,,,2 2,2,2,3 Assuming that the convergence is reache for N=3, the fiel p 3 x,y,z, is now taken as the reference fiel to compute an estimation error at abscissa x=l triangles in Fig. 8. The variability, epening on the initial conition p y,z, of the contributions of the moes to the convergence is still visible, but, moreover, it clearly appears that only a few moes the less ampe moes still contribute to the transport of energy at that istance from the source. Practically, sorting the moes as one in Table I is thus a goo choice to increase the convergence, as soon as one is intereste by the wave fiel in the street canyon at a sufficient istance from the source. Figure 9 shows the fiel in the cross-section of the waveguie at abscissa x=l, L the with of the waveguie. The left plot is obtaine using Eq. 28 an N=3 the total number of moes use to perform the least squares estimation, while the right plot is obtaine using only the first six ǫ (%) FIG. 8. Evolution of the error inicator with the number of moes N taken into account in the computation of the solution. : the error is compute at x=, with given in Eq. 3. : the error is compute at x=l, with p 3 L,y,z, as the reference fiel. N p (3) (L, y, z).5.5 p (6) (L, y, z) FIG. 9. Moal fiel at x=l using N=3 moes left an N=6 moes right. moes, with the orering efine above. Both results are very close: the relative error between them, efine as in Eq. 3, is less than.35%. Since only a few moes are necessary to escribe the fiel at certain istance from the source, it woul be avantageous to use a reuce moal basis in the computation. Since moes are not orthogonal, the value of each moal coefficient a i epens on the size of the basis use in the least squares estimation. One shows, however, that this epenency is rapily weak, in particular, for the first moes, that is, the less ampe Fig.. The notation a i N is use to enote the number of moes N taken into account for the least squares estimation of a i. To evaluate the relevance of using a reuce moal basis, two moal solutions of the wave equation in the infinite street canyon with the initial conition 29 at x= are compare: the solution with N=N=, an the solution with N a (N ) (a) 3 /a (3) 3 a (N ) (b) /a (3) N N FIG.. Evolution of the moal coefficients a 3 a an a 6 b with the number of moes N use for the least squares estimation J. Acoust. Soc. Am., Vol. 26, No. 6, December 29 Pelat et al.: Multimoal soun propagation in street canyons

8 ǫ (%) =N=3. The relative error between these two solutions, as efine in Eq. 3, is plotte in Fig. as function of the istance from the source plane. Naturally, the source conition p y,z is baly reconstructe with a limite number of moes, an the relative error near the plane x= is thus significant. But it ecreases rapily, to be uner % after less than two withs. It follows that the acoustic fiel, rapily, is carrie by a small number of moes, the less ampe moes, that are therefore well confine an guie in the open geometry of the street canyon. This points out a ouble interest of the moal formulation using leaky moes in a street canyon: first, as a physically relevant approach, escribing the competitive effects of confinement an raiation in such an open geometry, secon, as an efficient numerical metho, since a few moes only is use to accurately moel the wave propagation in the waveguie. V. CONCLUSIONS p () (5L, y, z) p () (2L, y, z) p () (,y,z) x/l p (3) (5L, y, z) p (3) (2L, y, z) p (3) (,y,z) FIG.. Relative error as efine by Eq. 3 between the moal solution 28 with N=N= an the same moal solution with N=N=3, as function of the istance from the source plane. The problem of soun propagation in an iealize urban street canyon is solve using a multimoal formulation that gives the acoustic fiel as a sum on the leaky moes of the open waveguie that isplays the canyon. The leaky moes that naturally reflects the competitive effects of confinement an raiation of the wave in such partially boune geometries can be numerically etermine. As these are complex moes that ecay exponentially while propagating, the number of moes that effectively carry the wave fiel emitte by some source in the waveguie ecreases rapily, so that only a few moes, at a sufficiently large istance from the source, is necessary to accurately moel the wave propagation. This gives this approach a real interest for numerical computations, in aition to its interest as a physically meaningful escription of the street as a partially confining an guiing meium for the acoustic waves. As a first step an to clearly point out the principles an interests of our approach, only a uniform open waveguie was consiere in this paper. Following, cross-section iscontinuities, as moel of the junctions between builings, can be consiere by using moe matching techniques. APPENDIX: FORMULATION IN THE PERFECTLY MATCHED LAYERS PML are use as a to avoi non-physical reflections at the bounaries of a necessarily finite omain in a numerical computation. The metho works as follows: the solution y,z of the eigenproblem 3 above the cavity is analytically continue in the PML with respect to variables y,z to complex variables ŷ,ẑ. The extene solution ˆ satisfies 2 ŷ ˆ =. A ẑ Complex variables ŷ,ẑ are now written as y ŷy y yy, ẑz z zz A2 = = with R y,z, I y,z, an y yl=, z z= Fig. 5. The results in this paper have been obtaine with y yl= z z=+j. F. M. Wiener, C. I. Malme, an C. M. Gogos, Soun propagation in urban areas, J. Acoust. Soc. Am. 37, W. R. Schlatter, Soun power measurements in a semi-confine space, MS thesis, MIT, Cambrige, MA P. R. Donovan, Soun propagation in urban spaces, Ph.D. thesis, MIT, Cambrige, MA H. G. Davies, Noise propagation in corriors, J. Acoust. Soc. Am. 53, W. Koch, Acoustic resonances in rectangular open cavities, AIAA J. 43, P. Stenackers, H. Myncke, an A. Cops, Reverberation in town streets, Acustica 4, R. H. Lyon, Role of multiple reflections an reverberation in urban noise propagation, J. Acoust. Soc. Am. 55, D. J. Olham an M. M. Rawan, Soun propagation in streets, Buil. Acoust., J. Kang, Soun propagation in street canyons: Comparison between iffusely an geometrically reflecting bounaries, J. Acoust. Soc. Am. 7, K. K. Iu an K. M. Li, The propagation of soun in narrow street canyons, J. Acoust. Soc. Am. 2, J. Kang, Numerical moelling of the soun fiels in urban streets with iffusely reflecting bounaries, J. Soun Vib. 258, J. S. Braley, A stuy of traffic noise aroun builings, Acustica 38, J. Picaut, L. Simon, an J. Hary, Fiel moelling in a street with a iffusion equation, J. Acoust. Soc. Am. 6, z J. Acoust. Soc. Am., Vol. 26, No. 6, December 29 Pelat et al.: Multimoal soun propagation in street canyons 287

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