Image-source method and truncation of a series expansion of the integral solution—Case of an angular sector in two dimensions

Image-source method and truncation of a series expansion of the integral solution - Case of an angular sector in 2D Running title: Ray method compared to integral solution Vincent Martin CNRS - Laboratoire de Mécanique Physique, 2, place de la Gare de Ceinture, F-78210 St-Cyr l’Ecole, France, [email protected] Thomas Guignard EPFL – Laboratoire d’Electromagnétisme et d’Acoustique, STI-ITOP-LEMA, Station11, CH-1015 Lausanne, Switzerland, [email protected] Abstract - The acoustic ray method rests upon specular reflection, an intuition that gives access only to an approximation of the solution by not taking into account the parts of the field called diffusion and diffraction. In trying to understand rationally the roots of the approximation, it has appeared that the image source could be generalized and also that errors may be partially due to missing generalized sources, already in elementary geometries such as obtuse angles. Indeed, it is shown that the exact integral solution of a 2D acoustic problem, expressed as a series of terms, could be seen as the contribution of the different image sources, via a partial use of the Huygens’ Principle. With the correspondence between the terms and the image sources shown, the missing sources would appear and the method would thereby be refined. PACS numbers: 43.20.Dk I – INTRODUCTION In acoustic cavities such as concert halls or passengers’ spaces in vehicles, the numerical description of classical sound fields – those satisfying the Helmholtz equation in space-frequency domain with local boundary conditions – stems from various methods, the choice of which depends first of all on the ratio of the wavelength to a linear dimension of the considered cavity. The reasons for this choice are either of conceptual or practical nature and each method has its own advantages and drawbacks. For example, the boundary finite element method, developed from the exact integral representation of the Helmholtz operator solution, is impractical for high frequencies, as the necessary fine discretization of the boundaries would then lead to large and full matrices, taking a long time to build and inverse. In the adequate frequency range, the method cannot be extended to non-linear problems (at least not directly). It needs knowledge of the acoustic field everywhere on the boundaries before giving access to the field at the points of interest inside the domain. The finite element method (finite elements of volume), resting on the variational form stemming from the weak form of the equation under study, is also confined to sufficiently long wavelengths for the same practical reasons of discretization, this time of the domain, even if the matrices are more quickly built and inversed as they can be made with a large number of null terms. As an indication, it is not easy to describe sound fields in the audible medium frequency range (1kHz - 5kHz) in passengers’ space in aircrafts, helicopters, cars, etc... The method is appropriate for non-linear problems. It necessarily describes the field everywhere within the domain and on the boundaries (substructuration could lead directly to the boundary values but at the expense of supplementary calculation time). 2 As for the ray method, it is restricted to the description of fields arising from specular reflections and does not take diffraction into account. Diffusion effects can be inserted but require great precaution. However, it gives access to the medium frequency range mentioned above. Specular reflection – originating from geometrical optics concepts – applied to sound waves in air is quoted as early as the 1940s [1] if not before, with experimental validation. The principle of specular reflection on perfectly rigid walls is compatible with the modal theory in waveguides and in rectangular cavities [2]. It is also with specular reflection that it has been possible to obtain an understanding of some causality problems in the field of active acoustic control [3], as long as the geometrical configuration is very simple. In architectural acoustics, it is commonly accepted that the ray method is able to describe sound fields above 100Hz in large auditoriums [4]. Here, the calculation of the field at some particular point within the domain does not require that of the entire domain (contrarily to the finite element method) nor on all the boundaries (contrarily to the boundary element method). However, one has to remember that the method is not rigorous and that the reaction taken only at the impact point of reflection is an approximation of the more global reaction properly described by the integral representation; this is probably the reason why calculation on all the boundaries is not needed. The ray method, so widely known for room acoustics in the years around 1960/90 [5], has been revisited over the last ten years or so for its use in vehicle passengers’ space, with the sound field descriptions in the audible medium frequency range in view [6, 7]. Concerning the algorithmic procedure, improvements carried out by previous authors in two different directions are helpful [4, 8, 9]: one is called ray-tracing, the other virtual image sources, and their history shows that they were developed quite simultaneously. 3 In a ray-tracing algorithm, rays “leave” a point source emitting an impulse (in theory an infinity of rays) and, for each of them, the first point of impact on a wall is sought and from there on the next impact point on another wall, etc … Nothing prevents this method from being used within non convex cavities. A priori, the procedure goes on indefinitely for each ray. Given a receiver point R, rays originating from the source that, after a certain number of reflections, go through R, make up the sound history – called impulse response or histogram or echogram – at point R. In practice, the number of rays leaving the source is finite and the rays propagate in a divergent way with the consequence that the weaker the probability for the rays to go through the receiver R, the smaller the number of permitted reflections. This is the reason why the histogram is made for a neighbourhood of R rather than the point alone. In these conditions the procedure can be quite short but at the price of uncertainty. Nevertheless, this ray-tracing version has the great advantage of being able to insert diffusive walls (because of their geometry and not of their behaviour). The virtual image sources algorithm identifies the images of the real source by a mirror effect on each wall, then the images of the images are sought, etc …, a priori indefinitely. However all these images are only potentially useful for calculating the acoustic field (except for rectangular enclosures) and only a small number of them actually “light” the domain, while still less are “seen” by the given receiver point. Validity (for “lighting” the domain) and visibility (of the receiver point) tests reduce hugely the number of images and a proximity test restricts their number by limiting the acceptable distance between the images and the receiver point. Nonconvex domains need an obstruction test [9] or call for another approach [7, 10]. The algorithm is precise in giving the rays leaving the source and propagating to the receiver, but distinguishing the useful sources from the potential ones is a heavy task and moreover it is difficult here to take 4 diffusion into account. However, this procedure is chosen in this paper for its precision. It has to be mentioned that the virtual sources procedure can also be understood as that of virtual receivers. Indeed, by determining the receiver images it is possible to retro-propagate rays issued from R until the real source is reached [9]. At this stage, it must be noted that the definition of image sources makes the problem independent of the type of signal emitted by the primary source and of the usage of the signal at the reception point. For example, in [4, 8, 9], rays are energy carriers and are used to assess sound intensity (under consideration of the form of the sound field). Here, phased (therefore in terms of complex amplitudes) sound fields in the frequency domain are considered, so as to observe systems of standing waves forming resonances and anti-resonances. In that sense, comparison between acoustic fields calculated by the finite element method and the ray method (with the image source algorithm) has shown differences which constitute a handicap for going further in small enclosures with the latter method [7]. What ought then to be done in order to reduce the differences? It is known qualitatively that the ray method does not take into account diffraction and/or diffusion and that the solution obtained cannot in general be exact. But even by dealing only with the part of the acoustic field made up of specular reflections, what do we know quantitatively about the ability of the image sources to reveal the field? In trying to answer this question, it would first be necessary to sustain the intuitive notion of image sources by a rational formulation and, in doing so, to have a tool to master their contribution to the sound field. Looking in that direction in the framework of a very elementary geometrical configuration, it has been found that sources said to be invalid by the current sourcechoosing algorithm could improve the description of the acoustic field. This being said, it has not yet been possible to know if this improvement resulted from a better description of diffraction, or 5 of the reflected field, or of both. Thus, the work presented here has the form of a theoretical investigation in a simple configuration, an investigation not yet found in the acoustic literature. The beginning of this paper is a recall of one of the algorithms of the ray method that defines the image sources, and also emphasizes two figures of an angular sector in the plane which motivate the study. After a first premiss that sets out a particular presentation of the acoustic field in presence of a reflective wall in a 2D half-space, the exact solution of the angular sector arises from the integral representation. The solution thus obtained on the walls is liable to be developed in series, and arguments associated with the Huygens’ Principle lead us to think that each term of the series could reveal the contribution of an image source. The same formalism is then extended to the case of walls with damping material. Then, the transformation of the pressure on the walls into the pressure inside the sector shows the possible contribution of image sources inside the domain. Numerical experiments in the third section of this paper support broadly the hypothesis of a correspondence between terms of the series and image source contributions, opening a door towards an improvement of the current algorithms for identifying the useful virtual sources. At this point it is necessary to cite in more detail the work of Mechel [10], presenting a comprehensive overview of the image sources method. The reassembling of sources in a 2D angular sector and the development of the exact solution into a modal series (different from the series development presented in this paper) are of particular interest here. The reassembling of sources leads to the definition and the insertion of a “corner source”, along with a particular directivity, and to an algorithm to compute the validity of images. The modal series suggests the idea of inserting the exact solution into the image sources method, thereby resulting in a mixed analytical-image source method. Although the approach presented here has not been inspired by 6 Mechel’s work and follows a different path, a certain relation between the objectives of both approaches must be assessed. In fact, these two approaches could converge by extrapolation of the fact that the elementary solution classically associated with each image source could be replaced by a more complete solution (including diffraction) associated with a certain set of images. The present text develops, extends and explores in greater depth the subject of a relatively short communication given recently at a congress [11]. II – FORMALISM ON THE BOUNDARIES AND IN THE DOMAIN A. Preamble, configurations and premisses Any comparison between acoustic fields obtained by the image source method and by the boundary integral method needs, as a preliminary, to speak of the algorithm which usually chooses the image sources. The image from a wall numbered n originates by a mirror effect on that wall from a source, that could itself be the image from wall numbered . It is convenient to write it as to signify that it will give rise to l reflections from the actual source, the last one on wall n, the previous one on wall , etc... For example, the source denoted is the image of the real source through wall 5, and source is the image of source through wall 3. Its presence will show two reflections. This can happen only if the last reflection is able to reach a point inside the domain. To clarify, Figure 1 presents a 2D domain made up of an angular sector defined by two semi-infinite straight lines, in fact two segments (of finite length). Six image sources are liable to reveal reflections. However to reach point , only four image-sources k N .......kn indices S l k5 S 53 S5 S P 7 are useful; only source would give a reflected ray reaching point ; for , three image sources intervene. In Figure 1, it appears that an image source with last index n plays a role, for point for instance, if the ray from that source 212 SQR P goes through wall to reach . In these conditions, source is of no use for point Q as the ray from does not go through wall to reach Q. nP 212 S212 S 2 ī Having thus in mind the algorithm for determining the sources, the motivation of the present work arises from both diagrams in Figure 2. The sector is now defined by the semiinfinite straight lines 1 * and . The configuration on the left leads to four sources (three images and the real source). However, the validity of the field obtained at point is not guaranteed. On the contrary, to the right, the only image source available definitively instills a doubt regarding the field obtained, as it is not expected that wall 2 ī 1 Q 2 * plays no role at all (source reveals the presence of wall only). Nevertheless, it is possible to enlighten the degree of precision of the field obtained by rays by comparing it to the exact solution given by the integral representation. To begin with, the particularly simple situation of a single reflecting plane is observed to gain access to the definition of the first order image source, the wall pressure (in a discretized form) and an iterative access to it. Notations will be defined in the course of development. In Figure 3, a point source radiates an acoustic pressure. In particular at point Q on the perfectly reflecting wall , the elementary solution of the Helmholtz operator is shown to be 1 S1 * 0 S ī (1) 00 S) f p(Q) = G (Q,S ) + G (Q,S ) ff where is the elementary solution of the Helmholtz operator in an open domain (the source flow amplitude is chosen so that the right hand side of the Helmholtz wave equation is 0 G(Q, 8 unity); here, in 2D, it has the form - 0 i -H(kQ-S) 40 with the Hankel function of the 2 - 0 Hnd kind of 0th order. In fact, for any point R in the domain, the integral representation (also called Green’s third formula) leads to (2) M 0n ī p(R) = G (R,S ) + G (R,M) p(M) dM ff w ³ where is the result of the operation taking into account the excitation on the right hand side of the wave equation. 0 G(R,S) f0 ȍ G(R,S)į(S - S ) dS f ³ When point R in the domain tends toward point Q on , the principal part of the double layer potential leads to ī 0 1 p(Q) = G (Q,S ) + p(Q) 2 f (3) which can also be written 0 p(Q) = 2G (Q,S ) Q ī f (4) and will now be a shorthand representation for when the observation points Q are on the boundary *. The subscript 0 is linked to the source (the index of G has been removed when indication of the source occurs). The image source can be made apparent by noting 0 p(ī) = 2G (ī)0 p(Q) = 2G (Q,S ) f 0 S f1 S N 0 1 0 coming from S coming from S 1 p(ī) = G (ī) + p(ī) 2  (5) In fact, the term 1p(ī) 2 in (5) is equal to by identification in (3) and (4). It indeed represents the free-field pressure on the geometrical locus 0 G(Q,S) f * due to but also the pressure on 0 S 9 In the whole domain, with noting the vector containing pressure values on points inside the domain and with noting the matrix originating from the discretization of E(ȍ)p ȍī A  M nī ī G(R,M) + ikȕ G (R,M) p(M) dM ff w ³ (29) the components of which represent the influence on the facets on the reception point in the domain, equation (25) becomes -1 Eȍī īī F0 (ȍ) = [ - ] (ī)()pAIApg: (30) This can be interpreted as follows: is the pressure coming from source radiated on the geometrical locus defined by the wall F(ī)p0 S *, loaded by both its internal impedance and the radiating impedance . This pressure is therefore more precisely written 0 S0 S Z ȍ rad Z (31) 0ȍ geom F0Srad (ī) = (Z Z , ī)pg The operator applied to this quantity does consider the wall impedance (in fact the combination of the wall and propagation medium impedances) and one writes -1 īī [- ]IA 0ȍ 1S rad 0ȍ -1 Eīī F0Srad (Z Z , ī(Z )) (ī) = [ - ] (ī) = (Z Z , ī(Z )) * *   g pIApg  (32) where it has been emphasized that * is now no longer only a geometrical locus but also an actual wall with some kind of internal impedance if this wall is to be seen as a source. Furthermore, as for equation (5), it appears that the pressure emitted by on the wall is also the pressure emitted by image . 0 S 1 S Finally, the right side of equation (30), except the direct contribution, is interpreted as the pressure coming from the image source on a geometrical locus of the domain and will be written (33) 0ȍ -1 geom ȍī īī F1Srad rad [- ] (ī) = (Z Z , Z Z , ȍ) * *  AIA p g 16 This rich notation reveals that the image source emits a pressure towards a point inside the domain , taking into account the fact that it is the image of source (together with its internal impedance and load ) relatively to wall 1 S :0 S 0 S Zȍ rad Z* (with internal impedance Z* and load ). rad Z* The notion of a wall seen as a source with its own pressure and having an internal impedance that must be combined with the load impedance in order to radiate into the domain – or toward the geometrical locus of another wall – is the key to interpreting the terms of the series development of the solution obtained by the integral equations method. Going back to the situations in Figure 2, the continuous form of the coupled problem on both walls is now the extended form of (16) (34)    M 1 M 2 M 1 110 n1 11 ī n1 21 ī 220 n2 12 ī p(Q ) = G (Q ,S ) + p(M) G (Q ,M) + ik ȕ G(Q,M)dM + p(M) G (Q ,M) + ik ȕ G(Q,M)dM p(Q ) = G (Q ,S ) + p(M) G (Q ,M) + ik ȕ G(Q,M)dM fff ff fff w w w ³ ³ ³  M 2 n2 22 ī + p(M) G (Q ,M) + ik ȕ G(Q,M)dM ff  ° ° ° ° ° ® ° ° °w ° ° ¯³ or in discrete form (this time, unlike (21), without simplification) -1 -1 11101 1112 -1 -1 22202 2221 = [ - ] (ī) + [ - ] = [ - ] (ī) + [ - ]  ° ®2 1   ° ¯ pIA g IA Ap pIA g IA Ap (35) out of which one obtains for example the extended form of (22) -1 -1 -1 111122221 -1 -1 -1 11 0 1 11 12 22 0 2 = - [- ] [- ] [- ] (ī) + [- ] [- ] (ī) ªº   ¬¼ ªº   ¬¼ pIIA AIA A IA g IA A IA g (36) 17 Using the notations the terms responsible for the pressure transfer from to -1 -1 21 22 21 12 11 12 = [ - ] and = [ - ]DIA A DIA A 1 *2 * and inversely appear; they generalize matrices 2B and 2C in (21). The order of the indices comes from the matrix equations and must be read from right to left to reveal the direction of transfer from one wall to the other. The analysis of , for example, shows that , which originates from 21 D 21 A (37)  M 1 n2 12 ī G(Q,M) + ikȕ G (Q ,M) p(M) dM ff w ³  applied to makes wall (with pressure and internal impedance ) radiate towards the locus defined by wall . Moreover, according to equation (27), the operator modifies the pressure radiated at so as to consider the absorption described by . Figure 5 illustrates the action of the operators and . At this stage, the pressure at equation (36) is now 1 p(ī)1 ī1 p(ī)1 Z 2 ī -1 22 [- ]IA 2 ī2 ȕ 21 D12 D > @ -1 -1 -1 11221 110112220 = - [- ] (ī) + [ - ] (ī) ªº     ¬¼ p I D D IA g D IA g 2 (38) and, using Eq. (12) as well as the remarks in Eq. (21), it can be written > @  -1 1 12 21 01 111 12 02 22 2 = - (ī) + (ī,ȕ) + (ī) + (ī,ȕ)  ªº ¬¼ pIDD g g D g g   (39) where the expression stipulates that radiates on , taking the wall admittance into account, and similarly for . By developing the inverse term in (35) we obtain 11 1 (ī,ȕ)g1 S1 ī1 ȕ 222 (ī,ȕ)g    1 01 111 12 02 22 2 12 21 0 1 1 1 1 12 21 12 0 2 2 2 2 =(ī) + (ī,ȕ) + (ī) + (ī,ȕ) + (ī) + (ī,ȕ) + (ī) + (ī,ȕ) + ...    pg g D g g DD g g DDD g g (40) which is a generalization of (23). 18 The terms in equation (40) should be analysed as follows. The symbol is the pressure radiated at geometrical locus coming from source (with its own pressure and impedance). The radiation uses a combination of source and radiation impedances and one notes more precisely as in (31) 01 (ī)g 1 ī0 S (41) 0ȍ geom 01 0S rad 1 (ī) = (Z Z , ī)gg and accordingly (42) 0 geom 02 0S rad 2 (ī) = (Z Z , ī) : gg Similarly to the case with reflecting walls, it is supposed that the term is at the origin of term , but now for absorbing walls a more precise interpretation is sought. With the notations introduced before, this term is 12 0 2 .(ī)Dg 21 (ī)g (43) -1 12 0 2 11 12 0 2 .(ī)=[ - ] . . (ī)Dg IA A g The operator applied to (the precise form of which is written as (42)) arises from the continuous term 12 A02 (ī)g   M 0 ȍ 2 geom n1 21 0Srad2 ī(*) G(Q,M) + ikȕ G(Q,M) G(Z Z ,ī(M)) dM ff w ³ (44) ( is the Green function corresponding to vector 0 G0 g ) where some conjectures had to be accepted in order to go further in the interpretation. In equation (44), the term could have the role of the pressure coming from seen as a source radiating toward geometrical locus . To this end, it should have a source impedance and a radiation impedance. This source would then be revealed by the existence of the image source . The term (*) in equation (44) could have this role of combining both source and radiation impedances. In these conditions, 0ȍ geom 0S rad 2 G(Z Z ,ī(M))2 * geom 1 ī 2 S 19  0ȍ0ȍī 2 geom geom 12 0 S rad 2 2 S rad 2 rad 1 (Z Z , ī)ZZ, ZZ,ī { Ag g (45) and, following the interpretation of (37),  0ȍ0ȍī 2 -1 geom 12 0 2 11 12 0 S rad 2 2 S rad 2 rad 1 1 (ī) = [ - ] (Z Z , ī) (Z Z ),(Z Z ), ī(Z ){Dg IA A g g (46) thus confirming the first supposition. Encouraged by this understanding of the term 12 0 2 (ī)Dg , a similar interpretation of term 21 12 0 2 (ī)DDg is sought. It is expected that expression (46) allows for (47)  0ȍī ī 21 21 12 0 2 21 S rad 2 rad 1 rad 2 2 (ī) Z Z , Z Z , Z Z , ī(Z ) {   DDg g Indeed, multiplied by equation (46) can be understood as the pressure coming from 21 A1 * radiating towards with the needed impedances. Particularly, the “internal” impedance of 2 *1 * and the pressure on are united in the existence of source , so the right-hand term of equation (46) can now be understood as the source pressure 1 *21 S (48)  0ȍī 2 21 S rad 2 rad 1 1 ZZ, ZZ,ī(Z )g  Upon multiplication by , (48) becomes 12 A  0ȍī ī 21 geom 21 S rad 2 rad 1 rad 2 Z Z , Z Z , Z Z , īg (49) and finally via -1 22 [- ]IA (50)  0ȍī ī 21 21 S rad 2 rad 1 rad 2 2 Z Z , Z Z , Z Z , ī(Z )g Each impedance grouping is linked to a particular propagation path, so there are as many reflections as groupings. The interpretation of all other terms follows the same procedure. But even if these conclusions give meaning to the image sources and to the number of reflections that are associated with them, they still remain to be formally demonstrated. For it is at first sight 20 surprising that an incident pressure wave on a wall would lead to a source pressure as soon as the wall impedance is considered and that this very impedance would be considered a second time when this source radiates (again, part (*) of equation (44) shows a combination of this wall impedance and the radiation impedance). C. Integral representation inside the domain and series development The investigation proposed in this paper of the rational origins of the notion of image sources associated with the acoustical ray method rests entirely on the series development of the exact solution of the wall pressure. This development could not have been directly applied to the exact solution within the domain. Whereas, now that the wall pressure can be developed as a series, an extension toward an expression of the pressure inside the domain is possible. Only the case of perfectly reflecting walls is considered here. Again, some preliminary remarks are needed. In the elementary configuration of Figure 6a, pressure at point R is expressed by M n ī contribution from S', or G (R,S') p(R) = G (R,S) + p(M) G (M,R)dM f ff w ³  (51) with (according to (4)) p (M) = G (M,S) + G (M,S') = 2G (M,S') ff f (52) thus leading to the following matrix equation (with our notation conventions) (53) ( ) () ( ) 2'() ff : * :  *pg Ep g E g For the two-walled configuration under study here (Figure 6b), we write similarly () : 0112 p2 g Ep Ep (54) During the analysis of the series development to obtain the source contribution, it appears that the terms are counted by pairs. Indeed, the formulation is also 21 (55) 11 1 1 11 1 1 1 1 2 (ī)2 (ī) 2 (ī) 11 1 2 (ī) = (ȍ) + .( (ī) + (ī) + (ī) + (ī) + (ī) + (ī) +...) +.((ī) + (ī) + (ī) + ( 121 12121 1 21 0 1 0 1 12 121 1212 12121 gg g 1 2 21 212 2121 g pg Eg g g g g g Eg g g g     11 22 2 11 2 (ī)2 (ī) 22 2 2 2 2 2 (ī)2 (ī) 2 (ī) ī) + (ī) +... + ...) + .( (ī) + (ī) + (ī) + (ī) + (ī) + (ī) + .. 2121 21212 212 212121 2 21212 gg 2 0 2 21 212 21212 212121 gg g g Eg g g g g g               22 2 2222 2 2 2 (ī)2 (ī)2 (ī) .) + .( (ī) + (ī) + (ī) + (ī) + (ī) + (ī) + ...) 12 1212 121212 2 1 12 121 1212 12121 121212 gg g Eg g g g g g       Transferring the pressures from the walls towards the domain through leads to 1 andE2 E (56) 0 1 2 12 21 121 212 1212 = (ȍ) + (ȍ) + (ȍ) + (ȍ) + (ȍ) + (ȍ) + (ȍ) + (ȍ) +... pg g g g g g g g where assembling terms by pairs always takes into account terms of the same order in the series revealing the pressures on the walls . It must be noted that this particular order of terms is of no significance, and it would have been quite possible to write, for example, (57) 0 2 1 12 21 212 121 = (ȍ) + (ȍ) + (ȍ) + (ȍ) + (ȍ) + (ȍ) + (ȍ) +... pg g g g g g g However, in the present case, the relative order of the terms series stemming from S1 and S2 remains. This question about the order of terms radiating toward the domain will appear in the conclusion. III - NUMERICAL EXPERIMENTS The reasoning correlating the terms of the series development and the image sources may lack rigor and an analysis of this reasoning will sooner or later prove necessary, but as a first step, numerical experiments can yield results faster and provide a factual confirmation of the interpretation presented here. All the experiments presented here were done in the situation depicted in Figure 7, composed of two perfectly reflecting walls 1 * and 2 *, at an angle ș. The present study is 22 concerned with the justification of a possible term-by-term relation between the series development of the integral solution and the series of image sources. Therefore, work has been concentrated on the case of perfectly reflecting walls (except for Situation B presented in Figure 10, see below), deliberately setting aside the case of absorbing walls. Only after this term-byterm correspondence has been assessed will it be possible to compare a series term with wall impedance and an image source contribution with specular absorption. In this second step, the difference between a local specular reaction and the non-local reaction present in the integral equations (diffusion) could then be verified. Both walls, theoretically of infinite length, are in fact 5m long for numerical reasons; the source is located at the coordinates . The values of , and used in the different situations referred to in this section are summed up in Table I. The walls are discretized into 250 facets of a length of 0.02m each (the wavelength is ca. 0.7m). The pressure is computed on both walls at 500 Hz. In the following tests, the solution obtained by the image sources method is compared with the corresponding series development. The reference solution in all cases is computed with the integral method. This solution is assumed to be exact, but with an approximation brought by the discretization and the finite length of the walls. SS (x , y ) șS xS y Seeking a way to observe if there is a correspondence between the terms of the series development and the image sources for the computation of the wall pressure, the first test comes from an intuitive consideration. For an acute angle ș, a great number of reflections can occur between the walls, so a great number of image sources is expected; it is noticeable that the image choice algorithm shows that all image sources are visible for the wall pressure. For ș>ʌ2 (obtuse), a small number of sources should intervene. It could be that the number of image sources is a monotonous function of the angle, so the convergence speed of the series should 23 increase from acute to obtuse angles. To verify this assertion, a distance between the exact solution (actually the expression “exact” is incorrect since the solution is only numerically approached) and the solution obtained with a number of terms of the series development or obtained with a number of image sources is defined as t N s N 1 1 22 t series t exact s sources s exact īī d(N )= p (N ,x)-p (x) dx and d(N )= p (N ,x)-p (x) dx ³³ (58) Figure 8 shows that the convergence curves of both the ray method and series development solutions are closely related and verify the fact that the convergence is faster for wider angles. For acute angles, the extra terms of the series development (those without an image source equivalent) appear to be of weak or even negligible contribution compared to the first terms. From a more physical point of view, the development series and its interpretation via Huygens’ Principle lead to the same conclusion, since the specular part of the sound field (located in the first terms of the series) is of greater importance than the diffracted part (in the higher order terms). Therefore the correspondence between the terms of the series development and the image sources can be further explored. Let us remark that in the case of an open sector, no resonances with infinite amplitudes at some frequencies are expected, which has indeed been observed. A special situation is the so-called quarter-infinite space (in 2D), where ș=ʌ2 (situation A). It has been previously observed that the image source method is in very good agreement with the reference solution in the case of perfectly rigid walls and also of those with a local impedance [13]. This also means that the visible sources (of which there are 4: the real source plus 3 images) contain the majority if not all of the needed information. Figure 8 shows that the reference solution is reached in four terms both by the series development and by the corresponding image sources. However, a closer observation (induced by the strong 24 convergence, as we shall see) shows a very slight difference between the two convergence curves: the terms of the series development converge a little more slowly than the sum of the image sources contributions. This could be due to the numerical approximation of the exact (reference) solution, and will be further examined. This situation where the first four terms are sufficient to obtain a good solution ought to be revealed in equation (23) if the product of the matrices were null for CB ș=ʌ2 without being zero. In practice, this would be highlighted by a norm of the product. Figure 9 shows the maximum singular value of the product along with the convergence (according to (58)) of the 5 orBC CB th term of the series development on , i.e. . Both values are seen to decrease from acute angles toward 1 *1 22 (* 0 CBg )ʌ2. The convergence stabilizes at zero from there on, signifying that the 5th term is superfluous for obtuse angles. On the other hand, the fact that the maximum singular value of is not null at CB ș=ʌ2 could hint that this norm is perhaps not appropriate to treat the expected vanishing of at CB ș=ʌ2. Despite the fact that the numerical experiments presented in this paper focus on the case , to gain confidence in the well-founded base of the work, a comparison is proposed in Figure 10 between the pressure field on the boundaries of a quart-infinite space (situation B) with an arbitrary impedance (reduced impedance ȕ = 0 9 r Z , which characterises an absorption of about 36% at normal impedance) calculated with the integral method and the pressure calculated with the four images. The very good agreement between both fields lead us to believe that the first four terms of the series still correspond to the four sources, probably resulting therefore in product in equation (39) null without both of the matrices being zero, but this still remains to be demonstrated. 12 21 DD 25 256, 873-940 [11]Martin, V. and Guignard, T. (2005). “Justification of the Image Sources in Ray Method,” Proceedings of the 12th Int. Congress on Sound and Vibration, Lisbon, Portugal, Paper 178 [12]Baker, B. B. and Copson, E. T. (1987). The mathematical theory of Huygens’ Principle (3rd edition, Chelsea Publishing Company, New York) [13]Guignard, T., Martin, V. and Courtois, T. (2004). “Interpolated identified reflection coefficients for acoustic ray method,” Proceedings of the 11th Int. Congress on Sound and Vibration, St. Petersburg, Russia, 3707-3714 32 Table I - Considered situations Label T (, ) SS x y in m r ZCorresponds to Figure A2 S (3.0, 3.0) f13 B2 S (0.5, 0.5) 9 10 C58 S (0.5, 2.0) f11a, 14a D58 S (-0.5, 2.0) f11b, 14b E78 S (0.1, 0.3) f14c F6 S (3.0, 3.0 tan( 2) T )f12a, 12b 33 Figure 1 (color online) – Set of image-sources liable to give rise to reflected rays for points in the angular sector; set of rays (i.e., of sources) contributing to the calculation of the acoustic field at point P. Figure 2 – The implementation of the algorithm for determining the useful image-sources for point Q1 results in three images in (a) and only one in (b). Figure 3 – Reflection of an acoustic wave on a totally reflecting plane in the half-infinite space Figure 4 – Matrices 2B and 2C transfer respectively pressure from wall 1 * toward wall 2 * and inversely. Figure 5 – Operators D21 et D12 transfer respectively pressure from wall 1 * toward wall 2 * and inversely (the order of the indices comes from the matrix representation and is to be read from right to left) Figure 6 – (a) elementary configuration with a reflection on the wall; (b) transfer of pressures from the wall toward the inside of the domain. Figure 7 – Geometrical configuration for the numerical tests Figure 8 (color online) – Convergence speed as function of aperture angle T . Figure 9 (color online) – The contribution of  CB decreases when going from acute to obtuse angles. Figure 10 (color online) – Pressure levels on wall 1 * in situation B, with absorbing walls ()9 r Z Figure 11 (color online) – Weak convergence on wall 1 * : (a) Situation C, (b) Situation D Figure 12 (color online) - Strong convergence in situation F: (a) convergence of the series development terms toward the exact solution (for clarity, only the first 10 terms are shown); (b) contribution of the first 6 terms of the series development and corresponding image sources. 34 Figure 13 (color online) - Strong convergence in Situation A Figure 14 (color online) - Effect of an "invisible" source (a) Situation C, (b) Situation D, (c) Situation E 35 S S 1 S 2 S 12 S 121 S 21 S 212 P Q R * 1 * 2 O A B S * 1 * 2 S 1 S 2 S 21 Q 1 * 1 * 2 Q 1 S S 1 * Q S0 S1 *1 *2 2C 2B *1 *2 D12 D21 0 0.005 0.01 0.015 0.02 0.025 4 3 2 1 Distance d(Ns) or d(Nt) Number of sources/terms Image sources Image sources with extra source Series development 0 0.01 0.02 0.03 0.04 0.05 0.06 4 3 2 1 Distance d(Ns) or d(Nt) Number of sources/terms Image sources Image sources with extra source Series development 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Acoustic Pressure [Pa] Observation on wall Γ1 [m] Series development terms Exact solution 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Acoustic Pressure [Pa] Observation on wall Γ1 [m] Series development terms Image sources Exact solution 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Acoustic Pressure [Pa] Observation on wall Γ1 [m] Series development terms Image sources Exact solution 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Acoustic Pressure [Pa] Observation on wall Γ1 [m] Image sources Image sources with extra source Exact solution 0.05 0.1 0.15 0.2 0.25 0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Acoustic Pressure [Pa] Observation on wall Γ1 [m] Image sources Image sources with extra source Exact solution 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Acoustic Pressure [Pa] Observation on wall Γ1 [m] Image sources Image sources with extra source Exact solution