John Vanderkooy Audio Research Group, Department of Physics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Transcription

1 An analyti seondary soure model of edge diffration impulse responses U. Peter Svensson a) and Roger I. Fred b) Department of Applied Aoustis, Chalmers University of Tehnology, SE Göteborg, Sweden John Vanderkooy Audio Researh Group, Department of Physis, University of Waterloo, Waterloo, Ontario N2L 3G, Canada Reeived 4 July 997; revised 8 June 999; aepted 9 July 999 A new impulse-response model for the edge diffration from finite rigid or soft wedges is presented whih is based on the exat Biot Tolstoy solution. The new model is an extension of the work by Medwin et al. H. Medwin et al., J. Aoust. So. Am. 72, , in that the onept of seondary edge soures is used. It is shown that analytial diretivity funtions for suh edge soures an be derived and that they give the orret solution for the infinite wedge. These funtions support the assumption for the first-order diffration model suggested by Medwin et al. that the ontributions to the impulse response from the two sides around the apex point are exatly idential. The analytial funtions also indiate that Medwin s seond-order diffration model ontains approximations whih, however, might be of minor importane for most geometries. Aess to analytial diretivity funtions makes it possible to derive expliit expressions for the first- and even seond-order diffration for ertain geometries. An example of this is axisymmetri sattering from a thin irular rigid or soft dis, for whih the new model gives first-order diffration results within 0.20 db of published referene frequeny-domain results, and the seond-order diffration results also agree well with the referene results. Sattering from a retangular plate is studied as well, and omparisons with published numerial results show that the new model gives aurate results. It is shown that the diretivity funtions an lead to effiient and aurate numerial implementations for first- and seond-order diffration. 999 Aoustial Soiety of Ameria. S PACS numbers: Bi, Fn, Px DEC INTRODUCTION The lassi problem of edge diffration from an infinite wedge irradiated by a point soure has expliit impulseresponse IR solutions for the ases of a rigid wedge and a pressure-release wedge. These were presented by Biot and Tolstoy in 957, but few studies employed these solutions until Medwin applied them to underwater and noise-barrier ases. 2 Comparisons of his model with measurements show good auray for geometries with infinite edges 2 4 and Ouis has applied Biot Tolstoy s model in room aoustis to studies of a room with balonies, a ase whih an be modeled with infinite edges. 5 An earlier alternative to the Biot Tolstoy solution has been the Kirhhoff diffration approximation, whih an be used for both frequeny- and time-domain methods. In seismis, 6,7 as well as in room aoustis, 8 the Kirhhoff diffration approximation has ommonly been employed in time-domain methods for edge diffration. However, as has been shown using the Biot Tolstoy expressions, 9 and other aurate methods, 0 the Kirhhoff diffration approximation leads to large errors, not only for low frequenies but also for a Present address: Department of Teleommuniations, Norwegian University of Siene and Tehnology, N-749 Trondheim, Norway. b Present address: Akustik Forum AB, Stampgatan 5, SE Göteborg, Sweden. high frequenies in ertain diretions. Consequently, the Biot Tolstoy is the preferred method for impulse response models of edge diffration. The Biot Tolstoy expressions, expliit as they are, do not immediately suggest how the infinite wedge expressions an be interpreted to lead to expressions for finite wedges or for multiple diffration. Medwin, Childs, and Jebsen suggest an interpretation, a disrete Huygens interpretation, whih an be used for finite wedges and also be extended to handle multiple diffration. 3 This is the basis for numerial methods suh as the wedge assemblage WA method. Measurements of noise barriers with a finite thikness and omparisons with other alulation methods seem to support that model. Whereas Medwin s interpretation leads to numerial alulation methods, the urrent paper proposes an approah in whih diretivity funtions for the seondary edge soures á la Medwin are derived analytially from the IR solution for the infinite wedge. The expressions for analytial diretivities are diretly appliable to nonstraight edges and multiple diffration. It will be shown that these analytial expressions support Medwin s assumption for modeling first-order diffration, namely that the ontributions from the two sides of the edge around the apex point are exatly idential. On the other hand, the analytial funtions indiate approximations in Medwin s assumptions for seond- and higher-order diffration. These approximations, however, seem to be of minor importane for geometries 233 J. Aoust. So. Am. 06 (5), November /99/06(5)/233/4/$ Aoustial Soiety of Ameria 233

2 FIG.. The geometry of an infinite wedge irradiated by a point soure S. Cylindrial oordinates are used with the z-axis along the edge of the wedge. The soure has oordinates r S and S and is plaed at z S 0. The reeiver has the oordinates r R, R, and z R and the wedge has an open angle of w. where the seond-order sound paths pass edges with small angle deviations, as in most noise-barrier ases. Setion I of this paper reviews the Biot Tolstoy solution and Medwin s extension to this. A new derivation using analytial diretivity funtions is presented, and its extension to seond-order diffration is desribed. In Se. II the numerial alulation of edge diffration impulse responses is disussed. Setion III presents numerial alulations for finite edges and analytial derivations for a thin irular dis. The wealth of exat and asymptoti frequeny-domain solutions see Piere 2 for a review is addressed here only for omparisons with alulations where the uniform theory of diffration UTD by Kouyoumjian and Pathak is used. 3 I. THEORY The problem to be onsidered here is that of a point soure irradiating a rigid or soft, i.e., pressure-release objet, a speial ase of whih is the infinite wedge. Impulse responses will be used throughout as desriptors of the sound fields, with the sound pressure p(t) as output signal and a soure signal q(t) 0 A(t)/(4), where 0 is the density of the air and A(t) the volume aeleration of the point soure. Free-field radiation is then desribed by the IR, h FF () (R/)/R, where R is the soure-to-reeiver distane. The IR for plane-surfaed objets an be written as a sum of the geometrial aoustis IR, h GA, and diffration omponents h diffr. The diret sound and speular refletions of first and higher orders will be ontained in h GA, as long as their respetive validity riteria are fulfilled. In the ase of an objet with an entirely onvex geometry i.e., no indents, h diffr will onsist of only first- and higher-order edge diffration, whereas other geometries might ause ombinations of speular refletions and edge diffration. In this study, only onvex geometries will be onsidered, in partiular infinite and finite wedges, and irular and retangular plates. As a starting point, onsider an infinite rigid wedge with a geometry as indiated in Fig. where the ylindrial oordinates r S, S, 0 are used for the soure and r R, R, z R for the reeiver. The edge diffration IR an be written in a form whih is a ombination of the forms given in Refs. 3 and 4, and based on the solution presented in Ref., h diffr v 2 r S r R sinh H 0, FIG. 2. A plane view of the edge onstruted from the two half-planes ontaining the edge and the soure S, and the edge and the reeiver R, respetively. Two z-oordinates, z l and z u, are indiated for whih the two sound paths S z l R and S z u R have idential path lengths. Also indiated is the shortest distane, L 0, via the apex point, denoted A, of the edge. Angles are defined with signs so that sin z/m and sin (zz R )/l. where, sinv S R oshvosv S R, 2 3 osh 2 2 r S 2 r R 2 z R 2 2r S r R, 4 where is the speed of sound, the wedge index v equals / w, H( 0 ) in Eq. is Heaviside s unit step funtion, and the time 0 equals L 0 /. The distane L 0 (r S r R ) 2 z R 2 /2 is the shortest path from the soure to the reeiver via the edge of the wedge, passing through the so-alled apex point of the edge, indiated as A in Fig. 2. For an infinite wedge with soft pressure-release surfaes, the IR is given by using a modified version of the -expression 5 soft. 5 Wedges with a ombination of rigid and soft surfaes ould be studied using variants of the -expressions in Eqs. 2 or 5; however, in the following disussion, rigid surfaes will always be assumed exept for the example of a irular soft dis, studied in Se. III B. A. Models of finite wedge diffration Although the analytial solution in Eq. desribes expliitly the IR of an infinite wedge, it does not indiate what a solution for a finite wedge might look like. Medwin et al. suggest a disrete Huygens interpretation, 3 plaing a number of small seondary soures along the edge and adjusting their strengths so that together they give the known exat solution. Aording to this model, these seondary soures emit pulses when they are hit by the inident, impulsive, sound wave. Thus, the reation at the edge is assumed to be instantaneous. This leads to the onlusion that the value of the IR h diffr at time is aused by the reradiation from the two parts of the edge indiated in Fig. 2, a lower 2332 J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 2332

3 FIG. 3. Illustration of a diffration impulse response for a finite wedge, h finite solid line and the orresponding infinite wedge, h infinite dotted line, indiating the three parts where i h finite h infinite, ii h finite is a saled version of h infinite,andiii where h finite 0. The initial part of the impulse responses has been trunated. branh indiated by the subsript l, and an upper branh indiated by a subsript u. The two branhes must obey the ondition m l l l m u l u. The two edge portions at positions z l and z u will ause two radiated sound-field omponents, h l and h u, whih summed together equal the known amplitude h diffr, h l h u h diffr. As indiated in Fig. 3, an IR for a single finite wedge will then have three parts. In the initial part, i, the IR is idential to the infinite wedge response. In the final part, iii, the IR is zero, after the sound wave has reahed the furthest end of the edge. The intermediate part, ii, where only h l or h u in Eq. 7 is present, is then a saled version of the infinite wedge IR. The saling for the intermediate part was suggested in Ref. 2, and is based on the finite wedge IR having half the amplitude of the infinite wedge IR, i.e., h l h u.ina later paper, 3 it was instead suggested that this fator should be h finite h infinite h l h l h u m ul u, 8 m u l u m l l l where it is assumed that the lower branh of the edge has the longest extension. It will be shown below that a theoretial derivation supports the relation h l h u. For multiple sattering, Medwin s model and the WA method assume that the seondary edge soures radiate as point soures, with soure strength modifiations based on Eq. 8. Seond-order diffration is then alulated by having all these seondary edge soures along the first edge generate individual diffration ontributions via the seond edge. Medwin et al. present omparisons between measurements and alulations for noise barriers of finite widths where seond-order diffration must be taken into aount, and the agreement seems to be quite good. 3 The agreement is also good for the sattering from a irular dis. As will be shown in Se. I C, however, the derived analytial diretivity funtions indiate that having the seondary soures along the first edge use the ordinary diffration expression to generate seond-order diffration via the seond edge, as in Eq., is only an approximation as far as we an determine. 6 7 However, the errors in Medwin s model do not appear to show up for geometries with symmetrial situations, suh as for noise barriers with parallel edges. The seond-order diffration paths whih will be of highest amplitude are those passing the apex points, that is, with as little angular deviation as possible during the passage of the edge. The methods in Refs. 2, 3, and orretly predit these high-amplitude parts, and thus predit the seond-order diffration quite well, as long as the geometry is suh that sound paths with little angular deviation are possible. A ritial benhmark ase would then be one where all sound paths aross edges experiene large angular deviation. For suh ases, an aurate seondary soure model would be important. B. Derivation of analytial diretivity funtions for the seondary edge soure A derivation of analytial diretivity funtion for the seondary edge soures starts by assuming the existene of a diretivity funtion for the seondary edge soures whih depends only on the angles of the inident sound path, and s, and of the reradiated sound path, and r in Figs. and 2. This diretivity funtion must be independent of the distanes m and l, and must be symmetri so that the soure and reeiver positions an be interhanged with idential results. Thus, the reiproity priniple is always fulfilled. A onsequene of suh diretivity funtions is that the inident wavefront is split up when it hits a point of the edge, and the reradiated wavefront spreads in all diretions with different amplitudes. To derive suh a diretivity funtion, the method of retarded potentials is employed as in Ref. 7, and a prototype solution an be formulated for the edge diffration. The prototype solution is then mathed to the known solution for the infinite wedge. Thus, the diffrated pressure p diffr (t) ould be written as an integral over ontributions from the entire wedge P diffr t q t mzlz Dz,z, S, R dz, mzlz 9 where D(z),(z), S, R is the unknown diretivity funtion. The position along the wedge is given by the z-oordinate, and this auses a retardation of the soure signal q(t), an amplitude attenuation aused by the ray paths m and l, and a further amplitude attenuation by the diretivity funtion D. The variable z an be substituted for a variable (ml)/, the time delay. There will, however, be two values of z giving the same value of, orresponding to the upper and lower branhes of the edge. Thus, the integral in Eq. 9 is first rewritten as a sum of the upper and lower branh integrals, as p diffr t z apexq zapex t mzlz q t mzlz Dz,z, S, R mzlz dz Dz,z, S, R dz, mzlz J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 2333

4 where z apex orresponds to the apex point on the edge, whih is given by z apex z R r S /(r S r R ). Now, a variable substitution is arried out so that p diffr t 0qt D l m l l l 0 0 D u qt m u l u qt D l m l l l D u m u l u dz u d d, dz l d d dz u d d dz l d where the subsripts l and u, respetively, indiate that for eah value of, the solution of D(), m(), l(), and z() is hosen to orrespond to the lower and upper branhes, respetively. Also, in Eq. the diretivity funtion is written as D l () rather than D l (), l (), S, R. The quantities dz/d an then be defined as being zero before 0, the first arrival time of the diffrated sound-field omponent, so that the lower integration limit an be expanded to. From this it beomes lear that the integral in Eq. is a onvolution integral of the soure signal q(t) with an IR whih must be the diffration IR, sine p diffr t qthdiffr d, and h diffr () an be identified as h diffr D l dz l m l l l d D u m u l u dz u d. 2 3 To find a diretivity funtion D whih satisfies Eq. 3, the observations dz l m l l l d dz u m u l u d H 0 r S r R sinh, 4 whih are proven in Appendix A, will be used. The relations in Eq. 4 an be inserted into Eq. 3, whih leads to v 2 D l, l, S, R D u, u, S, R. 5 Furthermore, it is possible to express as funtion of the angles and. To do this, another relation shown in Appendix B, sin sin osh, 6 os os an be used. It should be noted that and an be exhanged without affeting the result, and thus reiproity is assured. This relation is valid regardless of whether one hooses the lower or upper branh ombination of the and angles, an observation whih ould be formulated as l (), l (), S, R u (), u (), S, R and onsequently, l (), l (), S, R u (), u (), S, R, so it an be stated that 2 l, l, S, R u, u, S, R. 7 This expression an finally be inserted into Eq. 5, and the unknown diretivity funtion D an be identified as D,, S, R v,, S, R, 8 4 where either l () and l (), or u () and u (), ould be hosen as solutions for and. These derivations lead to the onlusion that the branhes of the edge on eah side of the apex point ontribute with exatly equal amounts to the diffration IR h diffr (). This has two onsequenes. First, one ould hoose either the upper or lower branh solutions for,, z(), m(), and l() to formulate the total diffration IR in this new form, h diffr v,, S, R dz 4 ml d, 9 where is as in Eqs. 2 and 3, and uses from Eq. 6. Seond, for a finite wedge, h l (t)h u (t) in Eq. 7, i.e., the part of the IR whih is a saled version of the infinite wedge IR, illustrated as part ii in Fig. 3, has the simple saling fator /2, h finite h infinite This saling fator supports the suggestion by Medwin in Ref. 2, but ontradits the suggested relationship in Ref. 3, as disussed in Se. I A. Deomposition of the infinite wedge diffration into loal edge ontributions also makes it possible to derive diffration from nonstraight edges. As is shown below among the examples in Se. III, the first- and seond-order diffration omponents of a irular thin dis an be derived analytially. Suh derivations an employ a rewritten version of the original expression in Eq. 9, using the funtion, as FIG. 4. The geometry of a trunated infinite wedge, with two parallel infinite edges, a point soure S and a reeiver R. One sound path, via the two points z and z 2 on the two edges, is marked. For the first edge with the open wedge angle w, the soure S has the ylindrial oordinates r S, S, z S, and the point at z 2 along edge 2, whih ats as a reeiver, has the oordinates w, w,z 2. Relative to the seond edge with the open wedge angle w2, the point z along edge ats as a soure and has the oordinates w, S2 0, z, and the reeiver R has the oordinates r R2, w2 R2, z R2.It is assumed that the two z-axes are aligned with respet to eah other. The angles and are as defined in Fig J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 2334

5 p diffr t v z 2q 4 z t mzlz z,z, S, R dz, 2 mzlz where z and z 2 are the two endpoints of the finite or infinite wedge. C. Seond-order edge diffration To derive expressions for seond-order diffration, the ase of a trunated infinite wedge with two parallel, infinitely long edges as in Fig. 4 is studied. Expanding the formulation in Eq. 2, and using the sound-path designations in Fig. 4, the seond-order diffrated pressure an be written p diffr t v v q t m z m 2 z,z 2 l 2 z 2 z, z,z 2, S,0 2 z,z 2, 2 z 2,0, R2 dz 2m z m 2 z,z 2 l 2 z 2 dz 2, 22 where it has been assumed that the path from edge to edge 2 runs along a plane onneting the two edges. This is indiated by the values R 0 and S2 0 in the two fators and the fator of 2 in the denominator of the integral. This fator of 2 ompensates for the doubling in pressure generated by an aousti soure when it is mounted on a baffle. It should have the value of if the ray from edge to edge 2 does not run along a plane. In the double integral in Eq. 22, the fator m and the delay orresponding to m an be moved out from the integral over dz 2, so that p diffr t v 4 tm z / *I m z 2dz, 23 where* indiates a onvolution and I 2 v 2 q 2 4 t m 2z,z 2 l 2 z 2 z, z,z 2, S,0 2 z,z 2, 2 z 2,0, R2 m 2 z,z 2 l 2 z 2 dz For eah value of z, there is a z 2,apex around whih the integral I 2 an be split into the upper and lower branhes, as was desribed before. Then, a variable substitution is possible with (m 2 l 2 )/, and the fators dz 2 /d an be defined as being zero before 02. Here, 02 L 02 / where L 02 (wr R2 ) 2 (z R2 z ) 2 /2. Sine the geometry is basially the same as for the diffration for a single wedge, the left-hand relation in Eq. 4 holds here, too. The integral I 2 an then be written I 2 v 2 l 2l u 2u dz 2 qt 2 4 m 2 l 2 d d qt*h diffr,z R, 25 where the values of m 2, l 2, and dz 2 /d an be hosen as the upper or lower branh solutions for a ertain value of. If one studies the impulse response h diffr,z R () further, h diffr,z R 2 v 2 dz 2 4 m 2 l 2 d, l, S,0 2l, 2l,0, R2, u, S,0 2u, 2u,0, R2. 26 Equation 7 an be applied, implying that 2l, 2l,0, R2 2u, 2u,0, R2 and h diffr,z R 2 v 2 2u, 2u,0, R2 dz 2 4 m 2 l 2 d, l, S,0, u, S,0 h diffr,st,edge2, l, S,0, u, S,0/2. 27 This means that the IR h diffr,z R () is a first-order edge-diffration IR from the position z on edge to the reeiver position R via edge 2, saled by /2, and multiplied with a weighting funtion whih is the sum of the two funtions in Eq. 27. The approah for seond-order diffration in Ref. 3 is to plae disrete edge soures along edge, and to use an ordinary first-order diffration IR to alulate their ontributions at the reeiver point. However, the fator with two funtions in Eq. 27 will modify the first-order diffration IR h diffr,st,edge2, whih in priniple makes it impossible to let the edge soures on edge irradiate edge 2 via ordinary first-order diffration IRs. The method used in Ref. 3 probably does alulate the onset of h diffr,z R () orretly and, sine the high-frequeny properties are determined to a large degree by the transient onset, the overall seond-order edge diffration is predited rather well. Employing the IR h diffr,z R, the seond-order diffrated pressure beomes 2335 J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 2335

6 p diffr t v 4 tm z / m z dz qt* v dz. 4 tm z / m z *qt*h diffr,z Rz,t *h diffr,z Rz,t 28 This expression means that the seond-order diffration IR, h diffr,2nd (), an be found from a single integral, h diffr,2nd v 4 m z / m z *h diffr,z Rz,dz v h diffr,z Rm z / dz 4 m z. 29 However, sine the IR h diffr,z R() is zero for 02, the infinite integration limits in Eq. 29 an be replaed by the values z, and z,2, whih both satisfy 02 m /. Expliit solutions to Eq. 29 might be possible for ertain geometries, for instane the axisymmetri refletion from a irular dis, as shown in Se. III B. II. NUMERICAL IMPLEMENTATION DISCRETE-TIME IMPULSE RESPONSES The default solution for numerial alulations is to use a disrete-time IR whih, for instane, an be given by area sampling of the ontinuous-time expression in Eqs. or 9. A value of the disrete-time diffration IR h diffr (n) ata ertain disrete time n/ f s, where f s is the sampling frequeny and n is a sample number, is thus given by n/2/ f shdiffr h diffr d, nn/2/ f s 30 where h diffr () is the ontinuous-time diffration IR, suh as in Eq.. The singularity at 0 in Eq. deserves speial attention, and the original funtion an be approximated by an analytially integrable funtion for values of lose to 0. 4 The integration in Eq. 30 orresponds to filtering the ontinuous-time signal h diffr () prior to sampling, employing a low-pass filter with an IR in the form of a retangular pulse of width /f s. This is quite a rude low-pass filter, and Clay and Kinney suggest the use of a wider pulse of width 4/f s. 4 The sampling frequeny an, however, always be raised high enough to give a low enough aliasing error at the highest frequeny of interest. In the numerial examples in this study, numerial integration as in Eq. 30 has been used throughout with rather high sampling frequenies 20 to 40 times the highest frequeny of interest to get aurate results. Formulating the diffration as a sum of ontributions from along the edge, as in Eq. 9, it an be seen that the integration of h diffr () in Eq. 30 over a time segment d is equivalent to the integration over a segment of the edge whih orresponds to this partiular d. To do this, another variable substitution an be done so that hnn0.5/ f s n0.5/ f shdiffr d zn z n2hdiffr z d dz dz, 3 where z n and z n2 are the z-oordinates that orrespond to (n)/(2 f s ), respetively. As disussed in onjuntion with Eq. 9, one ould hoose either the lower or upper branh z-oordinates that orrespond to. Sine Eq. 9 an be used to find that h diffr z d dz v z,z, S, R, 32 4 mzlz the integration in Eq. 3 an be written hn v z n2 z,z, S, R dz zn mzlz This integration an be approximated by the midpoint value so that hn v z n,z n, S, R z 4 mz n lz n n, 34 where z n is the midpoint between z n and z n2. If, for a simplified numerial implementation, one hooses to divide the edge into equally sized elements of size z, then an element i plaed at z i will give a ontribution h i to the IR h i v z i,z i, S, R z mz i lz i This should be added to a single time sample n f s (m l)/ or divided among two or more onseutive time samples. This division is determined by the edge element s position and size relative to the time positions for sample n. The ontribution h i should be viewed as a retangular pulse with a width t as given by the element size z and its position z i. If the edge element size is hosen so that z / f s, the ontribution from eah element will never spread out over more than two time samples. For effiient numerial implementations, the funtion an be written in a muh more ompat form than in Eqs. 2, 3, and 6 if one uses the relation oshva v A v /2, 36 where A(y 2 ) /2 y and ymlz(zz r )/(r s r r ), whih is taken from Eq. A7, with the same variables as before. With this formulation, only basi mathematial omputations need to be realulated for eah edge element and very effiient implementations are possible. Seond-order diffration is straightforward to implement using a formulation whih is based on Eq. 22. If two finite edges are divided into equally sized edge elements z and z 2, the seond-order ontribution h ij to the IR h diffr,2nd, of edge element i at z i of edge and edge element j at z j of edge 2, an be written 2336 J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 2336

7 h ij v v 2 z, z,z 2, S, R 2 z,z 2, 2 z 2, S2, R z m z m 2 z,z 2 l 2 z 2,z This expression is valid for both parallel and nonparallel edges, as long as the appropriate expressions for the involved,, and angles, and m and l distanes, are used. Note that a seond-order ontribution via two edge elements should be viewed as a triangular pulse, and a third-order ontribution appears as a pulse with seond-order polynomial shape, et. This affets how eah ontribution should be divided among time samples. A simplified implementation, giving the orret low-frequeny response, was used by Vanderkooy. 6 The ontribution h an be distributed over the two time samples adjaent to the arrival time of the edge element enter, with a linear weighting that depends on the enter arrival time s position relative to the two sample times. Other models have used time-domain formulations for edge diffration, the most ommon ones being based on the Kirhhoff diffration approximation, as mentioned in the Introdution. Another model, suggested by Vanderkooy, has an interesting equivalene to the new model suggested here. 6 Vanderkooy s model started from a frequeny-domain highfrequeny asymptoti expression for the infinite wedge. 7 The model uses disrete edge soures, eah of whih gives a ontribution h v S, R 4 ml z, 38 whih is delayed by the time (ml)/. Vanderkooy used a parameter v Vand whih is the reiproal of the wedge index v used here, but for larity the parameter v Vand was replaed by v in Eq. 38 so that the similarity with Eq. 35 is lear. The diretivity funtion is S, R sin v os vosv R S, os v osv2 R S 39 learly due to using the onstant value in Eq. 4 instead of the monotonially inreasing funtion osh v. III. EXAMPLES In this setion, the new model employing analytial diretivity funtion for the seondary edge soures is ompared with other methods. Only IRs have been disussed so far, but results are often more interesting when presented as transfer funtions. These are alulated from the IRs using a disrete Fourier transform DFT, and it is then possible to hek if a suffiiently high sampling frequeny has been used. The sampling frequeny should be inreased until the results at the highest frequenies of interest have stabilized. In all alulations, unless otherwise stated, a value of 344 m/s has been used for the speed of sound. A. Infinite wedges Two different infinite and rigid wedges were studied as illustrated in Fig. 5. Figure 5a shows a thin plate edge, with one fixed soure position S with ylindrial oordinates r S 25 m, S 5, and z S 0) and with two different reeiver positions, R and R2 at r R 25 m, z R 0, and R 300 for R, and R 96 for R2. The position R2 is into the shadow zone, whih is ritial from a numerial point of view due to the singularity of the diffration IR at the boundary of shadow zones. In Fig. 5b, another ase is shown with a plate of thikness w, an example whih is dupliated from Ref. 3. In this ase, the soure is at ylindrial oordinates r S 25 m, S 5, and the reeiver is at r R 25 m, R 300, z R 0. It should be noted that beause of the width w of the plate, the reeiver oordinates are relative to the seond edge, losest to the reeiver. Calulations of IRs were made with the new model using the disrete edge deomposition desribed by Eq. 35 for the single edge, and by Eq. 37 for the double edge. IRs whih an be rewritten into the form S, R Vand Vand Vand Vand, 40 where Vand sinv S R osv S R. 4 If osh(v), whih happens for the time 0 i.e., at the onset of the diffration, these terms are idential with the analytial ones in Eqs. 2 and 3. Sine rapid variations in the IR always our at the onset, the high-frequeny part of the response is determined entirely by this initial part of the diffrated signal. Thus, it is not surprising that Vanderkooy s model orretly reprodues the initial part of the IR and onsequently the high-frequeny response. The model does, however, give large errors at low frequenies, whih is FIG. 5. The two infinite wedge situations that have been studied. a shows a thin plate edge. b shows a plate of thikness w, where the values w 5, 22, and 35 mm have been studied J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 2337

8 FIG. 6. Transfer funtion relative to free-field propagation at 50 m distane for the two reeiver points at the infinite wedge in Fig. 5a. The frequeny marked, f low,utd, is the frequeny above whih the UTD method should give less than 0.5 db error aording to Ref. 6. were also alulated with the Biot Tolstoy model from Ref. 3, here denoted BTM, and transfer funtions were alulated for the single edge with the uniform theory of diffration UTD by Kouyoumjian and Pathak. 3 The lowest frequeny for whih the UTD method gives aurate results that is, within 0.5 db of exat solutions was found to be f low,utd 40 Hz based on Fig. 4 in Ref. 8. Sampling frequenies of 800 khz were used for the IRs sine, ompared with a sampling frequeny of 400 khz, the differene was less than 0. db at 20 khz. The edge element sizes were 0.43 mm for all ases, and the wedge had an extension of 5 m relative to the soure. Results for the wedge in Fig. 5a are presented in Fig. 6, where the results for the new model and the BTM model ome within 0.00 db. It an also be seen that the UTD method is indeed aurate down to f low,utd 40 Hz. Figure 7 shows results for the double-edge wedge in Fig. 5b, together with single-edge diffration for the orresponding infinitely thin wedge. Comparing these results with Fig. 9 in Ref. 3, it an be seen that the results with the new method and the BTM results are very lose to eah other. As is also observable for the double-edge plates in Ref. 3, the differenes between the thin plate limit and the low-frequeny results derease with dereasing thiknesses, yet a small but signifiant differene always remains. This remaining differene is probably aused by the lak of higher-order diffration omponents in these alulations. It an also be noted FIG. 8. The geometry for the axisymmetri sattering from a thin, rigid irular dis of radius a, with oloated soure and reeiver at height d. that this new model and the BTM model give very similar results for double-edge diffration for a geometry with parallel long edges, a happenstane that does not display well the signifiant differenes in the two theories. B. On-axis sattering from a rigid irular dis The ases of on-axis sattering from rigid and soft irular diss was studied in Ref., employing both an aurate, frequeny-domain T-matrix formalism, and timedomain expressions using the WA method. The aurate frequeny-domain expressions were transformed into IRs using the inverse disrete Fourier transform, and adjusting the sampling frequenies to the geometry so that delays orresponded to exat integer sample numbers, whih should ensure that aurate time-domain expressions were retrieved. Thus, these ould be seen as aurate referene ases against whih the present method an be ompared. Nevertheless, although the possible effets of windowing on the time signal were mentioned in Ref., no estimate was given. The first-order diffration is a single pulse, for whih it should in priniple be possible to aurately find the amplitude. Sine seond- and higher-order diffration omponents are ontinuous-time signals with onsets exatly at the arrival of the first-order impulse, these higher-order omponents will inevitably influene the sample in whih the impulse arrives. This effet is small sine the onset amplitude of the higherorder diffration is very small, but the aumulated effet is not lear. To derive analytial expressions for the diffration IR, the formulation in Eq. 2 is used. The funtion must be found for the ase with the soure and reeiver oloated, as illustrated in Fig. 8. In this ase, the angles S and R are equal and onsequently, both will be referred to as. The funtion an be derived using the form of osh(v) whih is valid for the wedge index v0.5, as shown in Appendix C, osh 2 os/2 os os /2, 42 FIG. 7. Transfer funtion relative to free-field propagation at 50 mw for the infinite plate in Fig. 5b, alulated with the new method. The dashed line indiates the single-edge ase, that is, the thin plate limit. Solid lines give results for a double wedge with w5, 22, and 35 mm from the top. whih, as is also shown in Appendix C, leads to 2 osh os 2 S R /2 osh 2 /2sin 2 S R /2 os S R /2 osh 2 /2sin S R /2 When S equals R and both are denoted, an be written 2338 J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 2338

9 TABLE I. Amplitude of the first-order diffration omponent for axisymmetri sattering from a irular rigid dis of radius m, normalized to the speular refletion. Results are either alulated using the new method, Eq. 47, or taken from Ref.. The errors are relative to the T-matrix solution whih is onsidered as the referene result. Height m T matrix Ref. WA Ref. Rel. error % Eq. 47 Rel. error % osh/2 os osh/2sin 44 osh/2sin. Furthermore, the osh(/2) fator is simplified for the axisymmetri ase that is studied here. For the entire irular edge, the inident ray angles and the reradiated ray angles will have the value 0, and then the quantity osh(/2), as is lear from Eq. 42. The funtion in Eq. 44 an onsequently be further simplified as 2os sin sin 2 2 os al a, 45 where a is the radius of the dis and l(a 2 d 2 ) /2, d being the height of the soure and reeiver above the dis. Inserting this onstant value of the term into Eq. 2, together with the onstant values of m and l, it is found that if the z-oordinate runs along the dis perimeter, p diffr t al 2a q 4al 2 0 t 2l dz al 2l 2 q t 2l, 46 where it an be noted that p diffr (t) is nothing but a saled and delayed version of the soure signal. In other words, h diffr al 2l 2 2l. 47 The first-order diffration for a soft dis an be found by employing the expression in Eq. 5 and repeating the derivations above, whih yields h diffr,soft al 2l 2 2l. 48 With Eqs. 47 and 48, the value of the first-order diffration omponent an be alulated diretly as a Dira impulse amplitude. In Tables I and II, the values given by Eqs. 47 and 48 are, respetively, ompared with orresponding values from Ref., when normalized to the speular refletion for the dis, /(2d). The results, while not idential, differ at most by 2.3%, or 0.20 db, from the referene T-matrix results. The sign of the error is always negative for the rigid dis and positive for the soft dis so that, if the seond-order diffration omponent ontributes within the same sample as the first-order omponent, the error would derease, if just marginally. The results with the WA method, as reprodued from Ref., give errors of the same magnitude as those given by Eqs. 47 and 48 but with a larger spread, beause of varying signs of the errors. An expliit expression for seond-order diffration an be derived as well, and it is done here for the rigid dis only. The expression in Eq. 22 an be used if the two z-oordinates run along the dis perimeter, from z0 toz 2a, so that 2a p diffr t2 q t m m 2 z,z 2 l 2, z,z 2, S,0 2 z,z 2, 2,0, R2 2m m 2 z,z 2 l 2 dz dz Here, the parameters m and l 2 are onstant and will be denoted l; S and R2 are idential and will be denoted ; and and 2 are both equal to zero. The initial fator 2 reflets the fat that for a thin plate, there will always be idential diffration omponents via the rear side of the plate whih ontribute to double the seond-order diffration amplitude. Furthermore, beause of rotational symmetry, one of the integrations an be replaed by a fator 2a, and any fixed value of z an be used in evaluating the z 2 -integration. Then, with z 0, TABLE II. Amplitude of the first-order diffration omponent for axisymmetri sattering from a irular soft dis of radius m, normalized to the speular refletion. Results are either alulated using the new method, Eq. 48, or taken from Ref.. The errors are relative to the T-matrix solution whih is onsidered as the referene result. Height m T matrix Ref. WA Ref. Rel. error % Eq. 48 Rel. error % J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 2339

10 FIG. 9. Sound paths for seond-order diffration for the irular dis, indiating that the inident sound path to the first edge has a onstant length l and a onstant inidene angle 0. Also, the reradiated sound path from the seond edge point has a onstant length l and a onstant angle 2 0. The intermediate sound path has the length m 2, and the two angles and 2 have the same value. p diffr t a 2a q 32l 2 0 t m 20,z 2 2l 0, 0,z 2,,0 2 0,z 2,0,0, dz m 2 0,z Beause of symmetry around z 2 a, the upper integration limit an be halved, and the result doubled instead. Furthermore, a variable substitution with (m 2 2l)/ leads to p diffr t a a qt 6l 2 0 0,,,0 2,0,0, m 2 dz 2 d d, 5 and, as before, the fator dz 2 /d an be defined as being zero before 2l/ and after 2(la)/, so that the integration limits an be expanded to, and a onvolution integral an be identified. The impulse response is then h diffr,2nd a 0,,,0 2,0,0, 6l 2 m 2 dz 2 d. 52 The values of m 2, the angles and 2, and dz 2 /d an be found by inspeting Fig. 9. Thus, m 2 2l, os os 2 m 2 2a, z 2 a2 a 2 os m 2 2a dz 2 d. 55 sin By employing the expression in Eq. 43, the produt of the funtions in Eq. 52 an be molded into the form 0,,,0 2,0,0, 6 os 2 2 osh2 2 osh sin2 2, 56 where the fat that osh(/2) is idential for the two edges has been used. Using Eq. 42 and the fat that 2 0, the funtion osh 2 (/2) an be simplified to FIG. 0. Transfer funtion for the ase in Fig. 8, with a m and d 3 m, alulated with the new method. The speular refletion plus firstand seond-order diffration is inluded, and the amplitude is normalized to the speular refletion. The fine struture in this graph agrees with the details in Fig. 6 of Ref. 7. osh 2 2 os, 57 2 os whih is valid for both the first and seond edges, as 2. If the expression in Eq. 57, together with expressions for sin 2 (/2) and os 2 (/2), are inserted into Eq. 56, whih is then used in Eq. 52, an expression for h diffr,2nd () results h diffr,2nd os os 2l 2 sin os os 2 w os 2l 2 os os os 2 w, /2 os 58 where os () an be found from Eqs. 53 and 54, os a/l, and w()h(2l/)h(2(al)/). Inluding the first-order diffration, Eq. 47, and seond-order diffration, Eq. 58, IRs were alulated and transformed into transfer funtions. Figure 0 presents the resulting transfer funtion for the ase with a m and d3 m. Calulation parameters were f s Hz and a DFT size of 2048 was used. These results are very similar to those in Fig. 6 in Ref. 7. Figure a shows the seond-order diffration IR when normalized to the speular refletion amplitude /(2d). In Ref., the amplitude of the seond-order diffration was presented in terms of the peak value of the funtion illustrated in Fig. a. This peak value is ritially dependent on the sampling frequeny and the low-pass filtering tehnique used. For purposes of omparison, the proess in Ref. was reprodued as losely as possible. The sampling frequeny was hosen so that the delay between the speular refletion and the first-order diffration pulse was an integer number of steps, using a speed of sound of 500 m/s. Furthermore, an oversampling by a fator of 8 was used here and, after transforming the IR into a transfer funtion using a DFT size of 6 384, only the first 024 frequenies were kept before transforming bak to an IR using a DFT size of J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 2340

11 We believe that this is a mistake for three reasons. First, the values in Table IV of Ref. are positive rather than negative. Seond, it is stated earlier in Ref. that the speular refletion was used for normalization for all the results presented, even for off-axis geometries. The third argument is that our results are very lose to the referene results when the speular normalization is assumed, and espeially after the frequeny-domain low-pass filtering was used. The results with our new method are lose enough to the referene results for the irular dis to generally support the relations derived here, but more omparisons with referene alulations should be made. Comparisons of the frequenydomain values might then be easier to arry out, avoiding ambiguities involving filtering effets as disussed above. FIG.. Seond-order diffration impulse response for the ase in Fig. 8, with a m and d3 m, alulated with the new method. The amplitude has been normalized to the speular refletion. a A sampling frequeny of Hz was used. b An oversampling by a fator of 8, relative to a, was used and a frequeny-domain low-pass filtering as desribed in the text gave the same final sampling frequeny as in a. This frequeny-domain low-pass filtering should be equivalent to the tehnique used in Ref.. Figure b shows an example of one suh seond-order diffration IR, where the small ripple indiates the filtering effets. Table III gives the results for the peak amplitude of the seond-order diffration IR, both when alulated using edge soures as in Eq. 50, and after the oversampling/frequenydomain filtering desribed above. Although the results from Ref. are reprodued here, it should be noted that it is stated in Ref. that the values in their Table IV are normalized to the first-order diffration pulse rather than to the speular refletion, whih is used elsewhere in that paper. C. Sattering from a retangular plate The last example is sattering from a retangular plate. Cox and Lam present an example of a retangular plate of size m and with alulated diretivity plots at two single frequenies, 202 and 3995 Hz. 9 The soure was positioned at a height of d3.96 m, right above the enter point of the plate. The reeiver was moved along an ar at a onstant radius of R.78 m, from the enter points of the plate. The ar was in the diretion of the short length of the plate. The total refletion strength was alulated using the boundary element method both for a threedimensional model and for a thin plate limit model of the plate. It was normalized to the diret sound amplitude. Figure 2 shows results using Eqs. 35 and 37 to alulate the first- and seond-order diffration IRs for an infinitely thin model of the plate. A sampling frequeny of Hz was used, orresponding to 28 times oversampling with respet to the target frequeny of 202 Hz. The four edges were divided into 0.67-mm elements for the first-order diffration, and twie that size for the seond-order diffration. The reeiver was moved along the ar in steps of deg. A DFT size of 892 was used, to get the transfer funtion values exatly at 202 Hz. The sound speed 346 m/s was used in Ref. 9 and here, too. If one ompares the level of the total field in Fig. 2 with the results for the thin plate limit in Fig. 4 of Ref. 9, one an see good agreement for angles between 0 and a. 80 deg. The larger deviations above 80 deg are probably due to the need for higher-order diffration omponents. This is TABLE III. Peak amplitude of the seond-order diffration omponent for axisymmetri sattering from a irular rigid dis of radius m, normalized to the speular refletion. Results are either alulated using the new method based on Eq. 50 or taken from Ref.. The results denoted LPF have been low-pass filtered as desribed in the text. The errors are relative to the T-matrix solution whih is onsidered as the referene results. The T matrix and WA results were speified in Ref. as being relative to the first-order diffration strength, but it has been assumed here that the speular refletion was used for the normalization as disussed in the text. Height m T matrix Ref. WA Ref. Rel. error % Eq. 50 Rel. error % Eq. 50 LPF Rel. error % J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 234

12 FIG. 2. Sattering from a thin retangular plate, desribed in the text, alulated with the new method. Shown are levels of the total field, of the speular refletion only, of the first-order diffration only, and of the seond-order diffration only. The frequeny was 202 Hz; the soure was fixed symmetrially above the plate and the reeiver position was varied along an ar. also indiated in Fig. 2 by the large amplitude of the seond-order diffration omponent for large reeiver angles. The deviation at the dip around 30 deg might be aused by a less dense sampling in Ref. 9, or small differenes in the speed of sound. Furthermore, the smoothness of the total field around 0 2 deg, where the speular refletion disappears, indiates that the numerial method used, Eq. 35, is aurate enough up to within half a deg from the transition zone where the speular refletion disappears. This is ompensated by the orresponding jump in value for the firstorder diffration. Very high oversampling was used, together with a very fine division of the edge into elements. This is needed only for the positions lose to the transition zone and for the largest reeiver angles. For the larger reeiver angles, higher-order diffration is needed anyway, so the total field might be alulated more aurately by using fewer edge elements but higher orders of diffration. A three-dimensional model of the plate was tested as well, but it was then lear that it was neessary to inlude higher-order diffration sine the results differed signifiantly from the thin plate limit model, when seond-order edge diffration was inluded. In onlusion, the results with the new model and the results in Ref. 9 seem to agree well enough to support the new model as long as it is realized that it may be neessary to inlude higher-order diffration omponents for some situations. IV. DISCUSSION AND CONCLUSIONS The interpretation of the exat Biot Tolstoy solution for the infinite wedge diffration that was presented here has not been proven to be true per se. It has, however, been shown that if the existene of analytial diretivity funtions for the seondary edge soures is assumed, suh funtions an indeed be derived and yield the exat solution for an infinite wedge. Sine this also leads to a possible appliation to nonstraight edges, and the omparison with the result for the irular rigid dis was aurate within 0.20 db, it is onluded that the suggested interpretation is generally valid. It should then, in priniple, be possible to show that the total field, speular refletions plus diffration omponents, satisfies the wave equation. The omplexity of the higher-order diffration omponents might, however, prevent this possibility for ases other than the infinite wedge. The proposed model gives results that, for first-order diffration, should be idential to those by the WA model. For seond-order diffration, the results by the proposed model are very similar to results by methods whih are based on Medwin s model. 3, Coneptually, however, the differene between Medwin s model and the proposed model is signifiant sine the proposed model gives a omplete motivation and mathematial proof for the diretivity funtions of the seondary edge soures. As for the numerial implementation, very high sampling frequenies are often needed. This is beause the rude low-pass antialiasing filter, whih is equivalent to the single time sample integration, has a gentle roll-off harateristi. Refined integration tehniques ould be tested, and the diretivity funtions might lead to somewhat simpler funtions to handle, ompared to the original Biot Tolstoy solution in Eqs. 4. Also, sine eah higher order of diffration auses a response that basially falls 3 db/otave more quikly than the previous order, lower and lower sampling frequenies ould be used for eah new order. This derease ours, however, above a utoff frequeny whih is given by the size of the individual planes of the objet, and the proximity to the various shadow zones. Below those frequenies, the response dereases in a way whih is more diffiult to predit. Higher-order diffration IRs should tend towards Gaussian-shaped pulses, aording to the entral limit theorem in statistis, sine it is equivalent to onvolving any funtion with itself many times, resulting essentially in a Gaussian funtion. Thus, properly time-aligned Gaussian pulses with the right areas, widths, and polarities ould serve as replaement funtions for higher-order diffration. It was shown that the ase of axisymmetri baksattering from a irular dis ould yield expliit expressions for the first- and seond-order diffrations. It is probably possible to find suh expliit expressions for several other geometries too. Further developments ould lead to solutions whih satisfy more general boundary onditions. The proposed model is relevant for all ases where sattering/diffration from idealized, rigid, or soft objets is studied, suh as when the WA is applied to underwater aoustis ases, noise barriers, et. In arhitetural aoustis, most IR predition models are based on geometrial aoustis, possibly handling surfae sattering, but without the possibility to handle edge diffration aurately. 20 The Kirhhoff diffration approximation has been tested before, but this new model should be muh more aurate sine it is valid at all frequenies and for all soure and reeiver positions. 0 In eletroaoustis, edge diffration has important appliations suh as the effet of the loudspeaker enlosure on the radiated sound; many simpler edge diffration models have been tested, 6 but the proposed model should, here too, prove to be more aurate. In onlusion, deriving analytial diretivity funtions for the edge soures both supports, and takes a step beyond, 2342 J. Aoust. So. Am., Vol. 06, No. 5, November 999 Svensson et al.: Seondary soure model of edge diffration 2342