A Simple Decoupled Modal Calculation of Sound Transmission Between Volumes

Transcription

1 ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (22) A Simple Decoupled Modal Calculation of Sound Transmission Between Volumes P. Jean, J. F. Rondeau Centre Scientifique et Technique du Bâtiment, 24 rue Joseph Fourier, 384 Saint Martin d Hères, France Summary The transmission of sound through simple partitions is computed with a simplified modal approach where air loading is neglected. Using well known approximated formula, the case of plates either isotropic or orthotropic with various boundary conditions can be considered. Validation is obtained against finite element computations. This approach enables many simple situations to be rapidly computed. PACS no. 43..Jh 1. Introduction The computation of sound transmission through partitions is a key problem in acoustics. Many approaches can be employed depending on the aspect to be emphasised. Complex structures can be studied assuming that they are of infinite extend but with a precise description through their thickness [1]. Such approaches are well suited for parametric studies in order to optimise products. Sound reduction estimated with these models can be improved if filtering techniques are employed; by applying a windowing technique in the wave-number domain Villot and Guigou [2] showed that correct levels of attenuation can be obtained. This approach is very fast. However no modal behaviour can be recovered in this manner. Finite Element Methods (FEM) eventually coupled to Boundary Element Methods (BEM) [3] can be considered to be very precise if one can afford long computation times or if only low frequencies are of interest. Great efforts are being put into the development of dedicated finite elements for complex materials such as porous multi-layered structures [4]. An alternative to FEM is the use of a purely modal approach which in simple cases can give solutions with reasonable computation times. However, simple exact modal expressions only exist for simply supported plates and rigid volumes. In [, 6] Gagliardini presents a fully coupled modal evaluation of the transmission of sound between two rectangular rooms, through any number of parallel and simply supported panels. FEM is often employed to derive modal basis of uncoupled structures before doing modal resolutions. In many cases, the consideration of full coupling is not necessary. In [7, 8, 9] the transmission of sound through a wall is computed by means of a Rayleigh-like integral which makes use of decoupled plate velocities and Green Received 22 January 22, accepted 3 June 22. functions for isolated volumes. The velocities are computed by finite element techniques and the Green functions by means of a geometrical approach. This allows the consideration of complex partitions with various boundary conditions as well as non uniform absorption of the walls. This method has been named GRIM for Green Ray Integral Method, since in former studies the Green function has been computed by means of geometrical methods. A much faster alternative to these computations, also based on decoupled pressure and velocity fields and using a modal approach both for the plate and the Green functions of the volumes, is presented here. Various boundary conditions for the plate can be introduced without significant increase of computation time. A computer program named GAIA has been derived which enables the study of various situations. Finally this program can be interfaced with other models for complex multilayered structures provided that they can be described by equivalent data for simple panels. This paper is organised as follows. In paragraph 2, the decoupled formulation is described. As already mentioned, it stems from the GRIM method [7, 8, 9] which enables the use of uncoupled Green functions within an integral expression of the pressure field. A modal description of the plate can be used, either obtained by Finite Element calculations or from analytical expressions. In the case of rectangular plates, various boundary conditions have been introduced, simply by using Warburton s modal expressions as products of beam responses. The Green functions can be computed by several means, such as geometrical models as presented in previous papers [7, 8, 9] or using modal representations. Paragraphs 3 shows validation examples in the case of a baffled rectangular plate with various boundary conditions. Paragraph 4 considers sound transmission between two volumes. More complex partitions are mentioned in paragraph. Finally, paragraph 6 shows how sound transmission depends on the angle of incidence of incoming outdoor waves. 924 c S. Hirzel Verlag EAA

2 Jean, Rondeau: Sound transmission through panels ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (22) 2. Formulation This paragraph gives a short summary of the GRIM approach, as described in [7, 8, 9] first for a radiation problem and then for a sound transmission problem. A modal description of a rectangular plate with various boundary conditions is proposed. The calculation of the uncoupled Green functions can be modelled by various means such as geometrical or modal approaches. Ω 1 E V 1 S 1 S 2 V 2 *R Ω The GRIM approach: a decoupled method GRIM is the acronym of Green Ray Integral Method and has been fully detailed in [7, 8, 9]. It expresses the acoustic pressure P at a point R, caused by a vibrating plate S 2 of known velocity V,as P (R) =;j! G 2 (N R)V 2 (N )ds 2 (N ) (1) S 2 where G 2 is the Green elementary solution on the receiving side, between receiver R and running point N on S 2, when S 2 is kept rigid. The harmonic dependency e i!t is implicit. This equation can be considered as a generalisation of the Rayleigh integral written for a baffled plate. It is exact when the velocity and the pressure are considered to be related but it is a decoupled approximation when V 2 is evaluated for an isolated plate. In [7] this expression has been employed in the case of a plate excited by a unit force, radiating in an elongated volume with dimensions close to those of a train and validated against a fully coupled modal computation. This approach has been extended to the study of transmission of sound between two rooms 1 and 2 (Figure 1) [9] as P (R) = ;j! S 2 (2) S 1 G 1 (Q R)Y (Q N )ds 1 (Q) ds 2 (N ) where G 1 and G 2 are, similarly to equation (1), the Green functions respectively in volumes 1 and 2; Y (N Q) is the cross mobility velocity at N due to a unit force at Q between two points on both sides S 1 and S 2 of the partition. In [9], an optimised Finite Element calculation of Y is used. This approach permits the consideration of complex partitions provided that the function Y can be computed Modal response of a plate As mentioned in [9] the mobility Y in equation (2) can be computed not only using FEM but also using the modal approach. The velocity V of a plate of dimensions L x, L y can be expressed [1] as V (x y) = X l m j! 4F lm ML x L y H lm (x y)! 2 lm ;!2 +2j lm! lm! (3) Figure 1. Sound transmission between two volumes 1 and 2, separated by a partition of boundaries S 1 and S 2. Source at E and receiver at R. where F lm is the generalised force expressed as F lm = F H lm (x y ) (4) for a normal force of amplitude F placed at (x y ); H lm is the mode shape function of mode lm; M is the mass of the plate; lm is the modal loss factor;! lm =2f lm, where f lm is the eigen frequency of mode lm. The summation over the modes lm must in theory be carried up from 1 unto infinity. In practice a limited number of modes will suffice for convergence. The consideration of the physical significance of these modes must be used for their selection []. In the case of a simply supported plate H lm (x y) = sin l m Lx x sin Ly y : () This expression is easily programmed and its evaluation can be optimised to lead to very short computation times if compared to finite element computations. The direct application of (3) in (2) has been programmed Y is obtained by taking F =1. A much faster alternative consists in using equation (1) where the velocity V is computed by (3) with F lm = G 1 (M )H lm (M )ds: (6) S 1 This last expression consists in an estimation of the generalised forces by the integration of the Green function G1 in the source volume for a blocked partition, which corresponds to the same assumptions done in (3). Indeed, both computations have been tested and compared in different situations with exactly the same results. Therefore, the combined use of (1), (3) and (6) will be used throughout the remaining examples Other plate boundary conditions A well known approximation by Warburton [11, 12, 13] of the response of single plates with various boundary conditions has been introduced. First, the mode shape H of a mode is approximated by the product of the response 92

3 ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (22) Jean, Rondeau: Sound transmission through panels -8 RMS velocity level in db RMS velocity level in db Figure 2. RMS velocity of a 16 cm thick, CCCC 32. m 2 concrete plate. Unit force at (.9,.7). FEM, Modal. Figure 3. RMS velocity of a cm thick, 32. m 2 plaster plate. Unit force at (.9,.7). FEM, Modal. of beams which have the same boundary conditions than two perpendicular edges of the plate. Six beam conditions are proposed by Warburton: SS, CC, SC, SF, CF, FF where S, C and F stand respectively for Simply supported, Clamped and Free end conditions. For instance, for a clamped-clamped beam (CC) of length L H lm (x y) =X l (x)x m (y) a) b) c) with if X l (x) = cos l x L ; 1 2 l =1 3 ::: + sin( l=2) sinh( l =2) cosh l X l (x) = sin l x L ; sin( l=2) sinh( l =2) sinh l x L ; 1 2 x L ; 1 2 if l =2 4 6 :::, where l are the solutions of cos( l L) cosh( l L)=1: H lm forms a set of orthogonal functions which approximate the real mode shapes of a plate clamped on all its edges. They can therefore be used as a set of shape functions, with associated eigen frequencies f lm to describe the behaviour of the plate. Reporting (3) with (7,8) in the differential equation governing the thin plate behaviour leads to a set of equations in terms of f lm. The resolution of this matrix system is time-consuming and an alternative consists in using the approximate solution derived by Warburton and which was obtained by a Rayleigh-Ritz computation. The expressions of f lm for a plate clamped on its four sides (noted CCCC) are given in the appendix. Similar expressions for the other conditions are given in [11]. These formula are reputed to be reliable only for the first modes. Its application has therefore been tested against a finite element computation using DKT elements [14]. In Figure 2 the mean velocity (referenced to 1 m/s) of (7) (8) d) Figure 4. Several simple situations which can be considered by a modal approach in GAIA, a) baffled plate, b) baffled plate with source or receiver volume, c) as b) with an infinite flat ground. d) plate between two volumes. a rectangular 16 cm-thick concrete plate of dimensions 32. m 2 excited by a unit normal force at (.9,.7) computed both by finite elements and by the above approximation are compared. The plate is clamped on its four sides (CCCC). See Table I for material properties. A good agreement can been seen. Similarly, Figure 3 shows a similar comparison for a -cm thick 32. m 2 CCCC plaster plate. Again a good agreement can be found. In both cases the FEM computation is made with a regular meshing of DKT elements which for the concrete plate corresponds to 31 elements per bending wavelength at 3 Hz and in the plaster case corresponds to 1 elements per wavelength at 2 Hz Green functions The Green functions G 1 and G 2 in (1), (2) and (6) can be computed in different manners. Green functions computed with geometrical models such as image sources [14] or beam tracing [1] can be employed. Simple cases have been directly incorporated into the program. Figure 4 shows several possible situations which have been programmed. In the presence of a rigid baffle the free field Green function is simply multiplied by two. When an in- 926

4 Jean, Rondeau: Sound transmission through panels ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (22) finite ground is introduced the image source is added. More complex situations are handled by the more general ray tracing program ICARE [1] which includes multiple diffraction and curved surfaces Modal approach The case where the volumes are of rectangular shape has been programmed and included in GAIA using the modal approach. In a volume of dimensions L x L y L z, the pressure is expressed as G(M Q) = X lmn L lmn (M )L lmn (Q) N lmn ;(k 2 ; klmn 2 )+j lmn N lmn = " l " m " n V=8 (9) " i =2if i = =1otherwise! x y z lmn = + + 2c L x " l L y " m L z " n L lmn (M ) =cos lxm L x cos mym L y cos nzm L z : L lmn is the modal response of mode lmn for rigid boundaries. Pairs of parallel plates are supposed to have an identical and constant diffuse field absorption coefficient d. In practice, for parallel plates having different d, their mean value is retained. The above modal formulation can only be applied [] for values of d < :. For higher absorption the volume can no longer be described with this modal approach Image sources In the case of a rectangular volume the image sources method can easily be implemented [16]. It consists in summing the contributions of image sources obtained by successive symmetries of the real source with respect to the six walls of the volume. Reflection coefficients can be computed with plane wave or spherical approximations. It is particularly fast for absorbent conditions and can therefore be seen as complementary to the modal approach which becomes more precise for rigid boundary conditions. It must be noted that the source image approach describes the walls in terms of their normalised impedance n, whereas the modal description of the volumes defines the acoustic properties of the walls in terms of d. Both approaches assume locally reacting conditions and n and d can be related by the approximate formula given by Morse and Ingard [17] and represented in [9]. In order to verify the validity of this expression simple rectangular volumes have been treated with the modal and the image source techniques. A given n is chosen for the image source computations and a best fit with the modal results is carried out in order to determine the corresponding value d. This method has been used for four different volume sizes. Figure compares the relation between n and d obtained by Morse and Ingard and the results of this best fit. The computations have only been made for values of n higher than 1 since lower values are not suited for modal computations. It shows that for low values of n Diffuse alpha coefficient Normalized impedance Figure. Relation between normalised impedance n and diffuse field absorption coefficient d. Best fit between image source and modal calculations. Sound pressure level in db Figure 6. Comparison of the acoustic pressure inside a m 3 rectangular volume. Walls with n = 38. Source at (.2,.2,.2) receiver at (.2,.1,.1). BEM, Image sources, Modal. (softer surfaces) the expression given in [17] must be taken with caution whereas there is a good agreement for harder walls. The dashed curve corresponds to the first term in [17] which is d 8= n and appears to be more satisfactory than the results obtained with the whole expression (solid curve). Figure 6 shows a comparison of computations made with three different methods: finite elements, modal and image source. A small m 3 rectangular volume with walls having a normalised impedance n =38is considered. A unit point source giving a pressure of 1=4 N/m 2 (.2,.2,.2) and the acoustic pressure is computed at (.2,.2,.1). In the modal calculation the walls have an absorption d =8= n. Good agreement is obtained between the three models. 3. Baffled plate The case of the baffled plate is often considered because either it is faster to compute or because it enables the study of a partition independently of source and receiver volumes. A cm-thick plaster 32. m 2 partition (see Ta- 927

5 ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (22) Jean, Rondeau: Sound transmission through panels Table I. Material properties. Kevlar. All materials isotropic except material E x MN/m 2 E y MN/m 2 G xy MN/m 2 steel concrete plaster Kevlar glass material kg/m 3 v xy steel concrete plaster Kevlar glass ble I for material properties) with different boundary conditions is considered. The plate is excited by a unit plane wave having an inclination of 3 degrees relative to the normal, with respect to both x and y axis on the plate. Three different methods have been employed. The results denoted FEM correspond to the application of the GRIM equation (2) with the mobility Y computed as in [9] using FEM. The results noted AMOTRA have been obtained using the program developed by I. Bosmans at CSTB [18]; it is based on a combination of a simply supported plate solution and a Lévy-type solution. It enables the study of rectangular plates with various boundary conditions provided that one pair of parallel sides is simply supported. The third results named GAIA have been obtained using the present approach. Figures 7 show the mean velocity level (referenced to 1 m/s) for a plate with boundary conditions SSSS, SSCC, SSFF and CCCC. In the CCCC case there is no result from AMOTRA which can not solve this situation. FEM computations were made using a meshing of 8722 triangles. A very good agreement is found between FEM and AMOTRA. The GAIA results are also very close except for the SSFF case where a progressive frequency shift can be seen. The slight frequency shift between GAIA and FEM in the CCCC case, similar to the shift between AMOTRA and FEM in the SSFF case, must be attributed to an insufficient FEM meshing Radiated power The radiated power is defined as W r = <e S R P (Q)V (Q)dS which using expression (1) can be rewritten as W r = <e ( ; j! S V S R V (Q) V (M )G(M Q)dS! ds (a) (b) (c) (d) Figure 7. RMS velocity of a cm thick, 32. m 2 baffled plaster plate excited by a plane wave ( =3, ' =3 ) with different boundary conditions, a) SSSS, b) SSCC, c) SSFF, d) CCCC. GAIA, AMOTRA, FEM. where S R is any surface surrounding the vibrating surface S V ; it can be directly S V or, in the case of a baffled plate, an hemisphere surrounding the plate. This last solution takes longer computation times than a direct inte- 928

6 Jean, Rondeau: Sound transmission through panels ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (22) (a) Radiation loss factor in db Radiation loss factor in db Radiation loss factor in db Radiation loss factor in db (b) (c) (d) Figure 8. Radiation loss factor of a cm thick, 32. m 2 baffled plaster plate excited by a plane wave ( =3, ' =3 ) with different boundary conditions, a) SSSS, b) SSCC, c) SSFF, d) CCCC. GAIA, AMOTRA, FEM. gration over the plate (a factor of for a square plate) but it avoids singularity problems. Both approaches have been programmed. In the case of an hemisphere, the velocity V on S R is computed with the pressure gradient on two Radiation loss factor in db Figure 9. Radiation loss factor of a baffled plaster plate. FFFF, SSFF, 2 SSSS, 4 SSCC, 3 CCCC. Radiation factor in db Figure 1. Radiation loss factor of a baffled concrete plate. FFFF, SSFF, SSSS, SSCC, -- CCCC. close concentric hemispheres. The radiation loss factor is defined as = W r csv 2 where V 2 is the mean square velocity of the plate. Figures 8 show the radiation loss factor for the SSSS, SSCC, SSFF and CCCC boundary conditions respectively. Excellent agreement is obtained between GAIA and AMOTRA except in the SSFF case. Above Hz the FEM results are slightly different from those obtained by GAIA and AMOTRA. The integration of Wr has been done either on an hemisphere or directly on the plate (S R = S V ) and identical results have been obtained. However the direct integration on the plate leads to longer computation times for a given precision. The radiation loss factor obtained with GAIA for different boundary conditions is showed in Figure 9 in third octaves with significant differences below the critical frequency (789 Hz). In Figure 1 is represented for a 16 cmthick 32. m 2 concrete plate. Below the plate critical frequency (f c = 12 Hz) the values of from lowest to highest for the different BC can be ordered as FFFF, SSFF, SSSS, SSCC, CCCC. For the plates with free boundaries the values of are less regular. Note, that in the case of the concrete plate, a smooth low frequency response is ob- 929

7 ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (22) Jean, Rondeau: Sound transmission through panels 1-4 Vlocity level in db Power radiated in db Figure cm-thick, SSSS 11m 2 steel plate between two volumes (1 and 1.2 m 3 ). d =:2. Unit source near left corner at (.1,.1,.1). Plate RMS velocity. ARGOS, GAIA. Figure 13. Computation of the acoustic power radiated by a 11m 2, 1 cm-thick steel plate, inside a 222m 3 volume. Source side baffled. Integration, P 2 A=4c. RMS pressure level in db Figure 12. Same as Figure 11. RMS sound pressure level in receiver volume. tained since, as can be seen in Figure 2, the first resonant frequency only occurs around 13 Hz. 4. Sound transmission between two rooms Although the case of a baffled panel offers a means of studying the sound isolation devoid of room effects, real situations of insulation between rooms are strongly affected by the modal behaviour of the volumes. The validity of the present decoupled modal approach must therefore also be assessed in the case of a partition between two rooms Simply supported partition A simple 1 cm-thick 11m 2 SSSS steel panel is placed between two rooms of volumes 1 m 3 and 1.2 m 3. In each volume, the four sides perpendicular to the separating wall have an absorption d =:2, the end walls being rigid. A unit point source is placed in a corner of the source room. A reference computation is made with a fully coupled modal approach (program ARGOS [19] written after []). Figures 11 and 12 compare the rms velocity of the partition and the rms pressure level (referenced to 1 N/m 2 ) in the receiver volume computed both with ARGOS and GAIA. Very little differences can be observed between both computations which shows that in such situations there is no need to carry fully coupled calculations. The evaluation of the sound pressure level at a few positions is very rapid with the decoupled approach but may become significant if a precise mean pressure level is required whereas the fully coupled approach gives direct access to this value []. When there is a volume on the receiver side the radiated power can be approximated using the assumption of a diffuse field as W r = P A 2 4c (1) where P 2 is the mean square pressure, A is the total area of absorption (sum of surface areas multiplied by d ). Figure 13 shows a comparison of the power radiated (referenced to 1 W) in this manner with the previous approach. This figure corresponds to the case of a 1-cm thick steel plate of dimensions.. m 2 centred on one side of a 222m 3 volume with walls having an absorption coefficient d =.2 on five sides. A normal incident plane is considered. A good agreement is obtained. This validates the use of equation (1) Clamped partition A cm-thick 32. m 2 CCCC plaster panel separating two rooms is considered. The source and receiver rooms have depths respectively of 3 m 2 and 4 m 2 and all walls except the partition have an impedance n = 9. The source is positioned near one corner. In [9] this case has been treated using equation (2) with a finite element computation of the plate mobility Y and a source image evaluation of the Green functions G 1 and G 2. These results have been used as a reference and in order to do a fair comparison the same evaluations of the Green functions have been employed. Figure 14 compares the RMS velocity of the plate obtained with both computations. A 1/12th octave representation is chosen. Good agreement can be observed. Figure 1 compares the rms sound pressure level 93

8 Jean, Rondeau: Sound transmission through panels ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (22) Figure 14. cm-thick, CCCC 32. m 2 plaster plate between two volumes (22. and 3 m 3 ). n =9. Unit source near end corner at (.2,.2,.2). Plate RMS velocity. ARGOS, GAIA. Figure 16. RMS velocity of a 2 mm thick, 1.21m 2 baffled SSCC orthotropic kevlar plate excited by a plane wave ( =3, ' = ). AMOTRA, GAIA. Pressure level in db Figure 1. Same as Figure 13. Acoustic pressure level in receiver volume at (2.,1.,1.). averaged over 4 positions in the receiver volume. Good agreement is again obtained.. Complex partitions The proposed decoupled method can not be generalised directly to double partitions since the coupling between two panels separated by a narrow air gap can not be neglected. For instance, tests have shown that for a double 1 cm-thick glazing with a separation inferior to 1 cm, it does not give acceptable results. However, any model which can deal with the panel / air gap /panel system can be used with the help of equation (2) provided that transfer Y between both sides can be computed. A modal description of this double partition can also directly be used in the same way as previously done for the simple panel. Many authors have modelled complex structures by equivalent properties which vary with frequency and which can be used in a simple plate model. For instance, in a recent presentation Nilsson [2] has used the Hamilton principle to show that a complex sandwich structure can be reasonably well approximated by a simple partition with Figure 17. Transmission directivity for a.91.4 m 2, 1 cmthick window pane in a m 2 facade. 1/3 octave frequencies, db per graduation. + baffle, m 3 reverberant volume, volume with T R =:16 sec. an equivalent stiffness which varies with frequency. Complex partitions with porous material, such as those used in cars or in airplanes can then be modelled by equivalent properties. The case of orthotropic rectangular plates can be easily introduced by applying the formula proposed by Dickinson [21] which is a generalisation of Waburton s expression for the eigenfrequencies of an isotropic plate. It assumes that the axis of orthotropy are parallel to the sides of the rectangular plate. The same shape functions (equation 7) are employed. The generalised expression for f lm is given in the appendix. The case of a rectangular plate made of Kevlar 2 mm-thick, of dimensions 1.21m 2 is placed in a baffle and excited by a plane wave of incidence ( =3, ' = ). It has boundary conditions SSCC. Figure 16 compares the velocity computed both with AMO- TRA and GAIA. Very good agreement is obtained. More complex geometries and structures can also be considered by using the modes of the isolated structures computed by a finite element code. The same decoupled 931

9 ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (22) Jean, Rondeau: Sound transmission through panels 1 db/grad 1 db/grad 1 db/grad 1 db/grad (a) (b) (c) (d) Figure 18. Sound pressure level behind a baffled window excited by a plane wave with different angles of incidence. a) deg (normal incidence), b) 3 deg., c) 6 deg., d) 9 deg. (grazing wave). Different boundary conditions: SSFF, SSSS, 2 SSCC, 4 CCCC. approach can then be applied as for the flat rectangular plate. First the plate s velocity is obtained through a modal computation of the plate where the generalised forces are computed as in (6) by integrating the product of mode shapes and incident Green functions. Finally the transmitted acoustic pressure and power are computed using the GRIM methodology..1. Directivity A last example corresponding to the third case of Figure 4 is now treated. More detailed numerical results as well as results from measured velocities on real windows, using a laser velocimeter, will be given in an other paper [19]. Vertical directivity diagrams for incoming plane waves are given in three cases (i) a baffled window pane with a rigid horizontal ground (ii) a reverberant volume is added behind the window (iii) the volume has a revereberation time of.16 sec (walls with d =1). The window consists of a 1 cm-thick glass of dimensions.91.4m 2 (see Table I) and is placed 1 m above ground. The volume has dimensions m 3. The average pressure level is computed inside the volume at 1/3 octave frequencies. The three different computations are compared in Figure 17. Above 12 Hz, the three cases give very similar directivities (the critical frequency f c of the panel is 1 Hz) meaning that a computation carried with a simple baffled plate suffices to characterise the directivity of transmitted waves above f c. However significant differences can be noticed at lower frequencies showing that the effect of the volume is essential. Last, the effect of the plate s boundary conditions is considered. Figure 18 shows results, in the case of the baffled plate of Figure 17. The sound pressure level behind the partition is computed for various boundary conditions and is plotted for different vertical angles of incidence. Differences in the order of to 1 db can be seen between the pressure obtained for various boundary conditions which shows that simplified energy-like methods would be difficult to derive. 6. Conclusion Computations based on the modal approach can often be carried without considering full coupling, at least for simple panels. Based on this remark a computer program has been written in order to rapidly compute sound transmission through homogeneous rectangular plates both isotropic or orthotropic- with academic boundary conditions. The underlying assumption of this work seems to be valid in many situations. The interfacing of this program with geometrical models in order to compute complex Green functions offers a wide range of applications. The consideration of more complex structures will be tested in the future by using equivalent properties obtained by programs which consider complex stratified plates. Appendix Eigenfrequencies of a rectangular plate Isotropic plate In the case of a plate of dimensions L x L y and thickness h, the eigen frequencies are approximated [1, 11] as f 2 lm = 2 D 4hL 4 x ( A 4 l + A4 m Lx L y 4 +2 Lx L y 2 vbl B m +(1; )C l C m ) (A1) where the A, B, C coefficients are given in [11] for different boundary conditions, combination of Free, Simply 932

10 Jean, Rondeau: Sound transmission through panels ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (22) supported and Clamped. D = Eh 3 =12(1;v 2 ) is the bending stiffness, is the plate s density. In the case of a plate clamped on its four edges, with axis x and y, one has on the y-side. A m =1:6 for m =1, A m = J for l =2 3 ::: with J =(m ; :). B m =1:248 for m =1, B m = J 2 (1 ; 2=J) for m = 2 3 :::C m = B m, with similar expressions for A l, B l, C l on the x-axis. In [1, 11] expressions of A, B, C s for one side are given for the six boundary conditions (BC) SS, CC, SC, SF,CF,FF where S, C, F stand for simply supported, clamped and free, so that plates with combinations of these BC are possible. Orthotropic plate For a rectangular plate with Young s moduli E x and E y in the x and y directions, v xy and v yx are Poisson s ratios and a shear modulus G xy one has [21] f lm = 2 4hL 2 xl 2 y " D x A 4 l L 2 y L 2 x L 2 + D y A 4 x m L 2 y +2; Bl B m H +2D x (C l C m ; B l B m ) # where D x = E x h 3 =12(1 ; v xy v yx ), D y = D x E y =E x, D xy = G xy h 3 =12, H = v xy D y +2D xy, v xy E y = v yx E x. References [1] M. L. Munjal: Response of a multi-layered infinite plate on an oblique plane wave by means of transfer matrices. Journal of Sound and Vibration 162 (1993) [2] M. Villot, C. Guigou, L. Gagliardini: Predicting the acoustical radiation of finite size multi-layered structures by applying spatial windowing in infinite structures. Journal of Sound and Vibration 24 (21) [3] P. Jean: Une méthode variationnelle par équations intégrales pour la résolution numérique de problèmes intérieurs et extérieurs de couplage élasto-acoustique. [a variational formalism using integral equations for the numerical resolution of interior and exterior elasto-acoustical coupled problems (english translation available upon request)]. Thèse de doctorat, Université de Technologie de Compiègne, 198. [4] N. Atalla, R. Panneton, P. Debergue: A mixed displacement-pressure formulation for poroelastic materials. Journal of the Acoustical Society of America 14 (1998) [] L. Gagliardini: Simulation numérique de la transmission acoustique par les parois simples et multiples. Thèse, Institut National des Sciences Appliquées de Lyon (32 pp), [6] L. Gagliardini, J. Roland, J. L. Guyader: The use of a functional basis to calculate acoustic transmission between rooms. Journal of Sound and Vibration 14 (1991) [7] P. Jean: Coupling integral and geometrical representations for vibro-acoustical problems. Journal of Sound and Vibration 224 (1999) [8] P. Jean: Coupling geometrical and integral methods for indoor and outdoor sound propagation validation examples. Acta Acustica 87 (21) [9] P. Jean, J. Roland: Application of the Green Ray Integral Method (GRIM) to sound transmission problems. Journal of Building Acoustics 8 (21) [1] C. Lesueur: Rayonnement acoustique des structures. Editions Eyrolles, [11] G. B. Warburton: The vibration of rectangular plates. Proceedings of the Institution of Mechanical Engineers 168 (194) [12] A. Leissa: Vibration of plates. Chapter 4. Acoustical Society of America, [13] G. B. Warburton: The dynamical behaviour of structures. Pergamon International Library, [14] J.-L. Bathoz: A study of three node triangular plate bending elements. Int. J. for Numerical Methods in Engineering (198). [1] F. Gaudaire, N. Noe, J. Martin, P. Jean, D. van Maercke: Une méthode de tir de rayon pour caractériser la propagation sonore dans les volumes complexes. Congrès SIA, Le Mans, 1-16 Nov. 2. [16] M. Gensane, F. Santon: Prediction of sound fields in rooms of arbitrary shape: validity of the image sources method. Journal of Sound and Vibration 63 (1979) [17] P. M. Morse, K. U. Ingard: Theoretical acoustics. p.8. Mc Graw-Hill, New-York, [18] I. Bosmans: Numerical and experimental characterisation of structure-borne sound transmission between highly damped plates. Theoretical aspects. CSTB Report ER , Marie Curie sholarship, 21. [19] J. F. Rondeau, P. Jean: Application of the grim approach to the sound transmission of aircraft noise inside dwellings. Submitted for publication. [2] A. C. Nilsson: Wave propagation and sound transmission in sandwich composite plates. The 8th International Congress on Sound and Vibration, Hong-Kong, 2-6 July 21. [21] S. M. Dickinson: The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh s method. Journal of the Acoustical Society of America 61 (1978)