A combination of the acoustic radiosity and the image source method

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1 Downloaded from orbit.dtu.dk on: Sep 20, 2018 A ombination of the aousti radiosity and the image soure method Koutsouris, Georgios I. ; Brunskog, Jonas; Jeong, Cheol-Ho; Jaobsen, Finn Published in: Internoise 2012 Publiation date: 2012 Doument Version Publisher's PDF, also known as Version of reord Link bak to DTU Orbit Citation (APA): Koutsouris, G. I., Brunskog, J., Jeong, C-H., & Jaobsen, F. (2012). A ombination of the aousti radiosity and the image soure method. In Internoise 2012 General rights Copyright and moral rights for the publiations made aessible in the publi portal are retained by the authors and/or other opyright owners and it is a ondition of aessing publiations that users reognise and abide by the legal requirements assoiated with these rights. Users may download and print one opy of any publiation from the publi portal for the purpose of private study or researh. You may not further distribute the material or use it for any profit-making ativity or ommerial gain You may freely distribute the URL identifying the publiation in the publi portal If you believe that this doument breahes opyright please ontat us providing details, and we will remove aess to the work immediately and investigate your laim.

2 A ombination of the aousti radiosity and the image soure method Georgios I. Koutsouris 1 Jonas Brunskog 2 Cheol-Ho Jeong 3 Finn Jaobsen 4 Aousti Tehnology, DTU Eletro, Tehnial University of Denmark, Kongens Lyngby DK-2800, Denmark A ombined model for room aousti preditions is developed, aiming to treat both diffuse and speular refletions in a unified way. Two established methods are inorporated: aoustial radiosity, aounting for the diffuse part, and the image soure method, aounting for the speular part. The model is based on onservation of aoustial energy. Losses are taken into aount by the energy absorption oeffiient, and the diffuse refletions are ontrolled via the sattering oeffiient, whih defines the portion of energy that has been diffusely refleted. The way the model is formulated allows for a dynami ontrol of the image soure prodution, so that no fixed maximum order is required. 1 MODELS BASED ON GEOMETRICAL ACOUSTICS Geometrial aoustis models are widely used in the study of sound fields in rooms, beause they provide low ost preditions for ompliated ases, suh as theaters, onert halls, et. In these models various wave phenomena degenerate to simple geometrial tasks. Furthermore, in most of them only the propagation of aoustial energy is onsidered, any phase information being negleted. For years, the most ommon room aousti models, ray/beam traing 1, 2 and image soure model (ISM), 3 were based on purely speular refletions. On the other hand, a less used method, aoustial radiosity (AR), 4, 5 is based on purely diffuse refletions, represented by Lambert s sattering law. 4 In reality, neither purely diffuse nor purely speular refletions our in rooms. Indeed the refletion on a real wall seems to be a mixture of these two extremes, and several authors have presented hybrid models in this diretion. In 1993, Lewers 6 proposed a ombination of beam traing and AR. In 1996, Dalenbäk 7 proposed a one traing method that aounts for the diffuse refletion by the notion of seondary soures. In this paper a ombination of AR and ISM is proposed for 1 gk@odeon.dk 2 jbr@elektro.dtu.dk 3 hj@elektro.dtu.dk 4 fja@elektro.dtu.dk

3 general polyhedral room ases. The new model is developed for energy impulse response preditions. Initially, the priniples of the two methods are presented and afterwards ISM is merged into the AR equations, resulting in the formulation of the proposed model. In the omplete version ISM and AR are oupled in both diretions, where ISM is onverted to AR and vise versa. A simplifiation is proposed so that only the onversion of ISM to AR is taken into aount. 2 THEORETICAL BACKGROUND 2.1 Aousti Radiosity The initial formulation of the method was performed by Kuttruff in the early 1970s 4 and Miles 8 applied it for a retangular room. The governing equation needs to be solved numerially, whih involves disretization of the boundary into elements. The alulation of the response using AR is usually implemented in two parts. In the first part the energy history is omputed for eah element. This energy is a ombination of the diret sound from the soure and the interations with the other elements due to diffuse refletions. Afterwards, the response at the reeiver is alulated by olleting the ontributions from the elements. Following the omputer graphis approah, the first part is alled rendering and the seond part gathering. The rate of energy that leaves a unit area of surfae is defined as radiation density B(t). It has the same unit as intensity but it is a salar quantity, desribed in detail by Nosal et al. 5 The disretized form of the main AR equation reads: N B i (t) = ρ i j=1 B j (t R i,j ) F i,je α mr i,j + B i,q (t), (1) where the radiation density of the element i is obtained by the ontributions of all other elements on the boundary in addition to the diret ontribution B i,q (t) from the soure Q. The refletion oeffiient ρ i aounts for the portion of energy that is radiated by element i. The absorbed energy is represented by the absorption oeffiient α i = 1 ρ i. In Eqn. (1) the exponential fator represents the air absorption, where α m is the air absorption exponent. R i,j is the distane between the entroids of elements i and j and is the sound speed. The term in the brakets represents the time delay involved between i and j. F i,j is the so alled form fator, 9 whih defines the fration of energy that leaves element i and is reeived by element j, aording to Lambert s law: F i,j = 1 os θ os θ ds ds. (2) S i Si Sj Here θ and θ define the angles formed between the line onneting the entroids of the elements i, j and the orresponding normal vetors. Normally Eqn. (2) has to be evaluated numerially. The diret ontribution from the soure to element i is B i,q (t) = πr 2 i,j ρ i 4πS i W Q (t R i,q ) H i,qe αmr i,q, (3) where W Q is the power of the soure, R i,q is the distane between the element and the soure, and H i,q is the integral over the solid angle subtended by the element at the soure: H i,q = Si os θ Q R 2 i,q ds. (4) In this paper all the integrals over solid angles are omputed by a fast and aurate method, the spherial triangle, proposed by Nosal et al. 5 So far, it has been implied that when an element ats

4 as a soure it is named j, while when it ats as a reeiver it is named i. This onvention is followed in the whole artile. One every B i (t) has been alulated, the energy density at the reeiver an be found by E(t) = 1 π N j=1 B j (t R P,j ) H P,je α mr P,j os θ P + E Q (t), H P,j = Si RP,j 2 ds (5) is the integral over the solid angle subtended by the element j at the reeiver P. E Q (t) is the diret ontribution from the soure, given by 2.2 Image Soure Model E Q (t) = 1 4πR 2 P,Q W Q (t R P,Q ) e α mr P,Q. (6) In ISM the energy is assumed to be refleted purely speularly. The refletion is modeled by mirroring the soure behind the orresponding wall. 3 This leads to an image soure q, whose ontribution to the reeiver is idential to the refletion from the wall. The first mirroring of the original soure Q gives a first order refletion. Further mirroring of the image soure behind another wall gives a higher order refletion. It is onvenient to all every new image soure as daughter soure and its predeessor as mother soure. Every time a soure is mirrored bak from a wall, its power is attenuated by the refletion oeffiient of the wall, ρ. In an image soure sequene, the power of the last image soure has been attenuated by the produt of all refletion oeffiients of the preedent refletions. This produt is alled the soure fator. The general ase of an arbitrary polyhedral room ISM proves to be very time onsuming for typial impulse response lengths. The number of image soures follows an exponential growth with inreasing order. Moreover, only a small part from the ensemble of image soures orresponds to feasible refletion paths and ontributes to the impulse response. 10 Therefore, the most time onsuming task in the general ISM is to filter out all the unwanted image soures. 3 COMBINED MODEL The refleted energy is now further separated to a diffuse part, desribed by the sattering oeffiient s and a speular part, (1 s). As a result, the diffuse refletion oeffiient is s(1 α) while the speular is (1 s)(1 α). Conservation of energy implies that 3.1 Image Soures α + s(1 α) + (1 s)(1 α) = 1. (7) The idea now is to represent all the speular refetions from the original soure by its image soures up to order λ. If l denotes the intermediate refletions up to λ, then eah refleting wall an be represented by w l, taking values from 1 to the total number of walls N w in the room. With this nomenlature, the soure fator of an image soure q k is r qk = λ l=0 (1 s wl )ρ wl. (8) The speial produt (1 s w0 )ρ w0 = 1 symbolizes the diret ontribution, so that the original soure, Q has a soure fator of 1. The subindex k = 1, 2,... denotes the numbering of the image soure. When k = 0, define q 0 Q. The purely speular energy density reahing any reeiver P in the

5 room is omputed as the sum of the ontributions from all effetive image soures and the diret ontribution: E qk (t) = 1 r qk 4π RP,q 2 W Q (q k, t R P,q k ) e a mr P,qk. (9) k k=0 The boundary of the room is subdivided into elements, in order to represent the diffuse refletions. Eah of the elements ats as a new reeiver of speular refleted energy from the image soures. The speular omponent of the radiation density for eah element, together with the diret ontribution from the image soures, is B i,qk (t) = s iρ i 4πS i k=0 r qk W Q (q k, t R i,q k ) H i,qk e a mr i,qk, (10) where S i is the area of the element, R i,qk is the distane between the image soure and the element, and H i,qk = Si os θ qk R 2 i,q k ds (11) is the integral over the solid angle subtended by the element at the soure. 3.2 Image Elements The diffuse energy refleted by an element on the boundary is free to be refleted afterwards either diffusely again or speularly. The first ase is already represented by AR and the other elements on the boundary. The seond ase an be modeled by the notion of image elements, similar to the image soures of the original soure. The priniple is illustrated in Fig. 1, where the mirroring of the soure element behind the wall gives the full path of the first order refletion at the reeiving element. For every element j a sequene of image elements is generated. Similar to the image soures, the index m is used to number these image elements. Therefore, a speifi image element assoiated with the element j is denoted by the subindex jm. The radiation density of eah image element is the same as that of the soure element, attenuated by the orresponding soure fator, r jm = λ l=0 (1 s wjl )ρ wjl, (12) similar to Eqn. (8), where λ indiates the refletion order of jm and jl is the indexing of the mirroring walls involved in the generation of jm. As before, the produt (1 s wj0 )ρ wj0 = 1 orresponds to the diret ontribution from the soure element j. The energy transfer from an image element jm to a reeiving element i an be implemented by generalizing the lassial form fator of Eqn. (2) for any pair of elements, image or real: F i,jm = 1 os θ os θ ds ds. (13) S i Si Sjm πr 2 i,jm This new form fator is alled extended form fator. Apparently, S jm = S j. For optimizing purposes it an be assumed that the energy does not vary over i so that Eqn. (13) is simplified to F i,jm Sjm os θ os θ πr 2 i,jm ds. (14) This simplifiation an be justified by the fat that with inreasing order the image elements are plaed further and further away from i, so that the five times rule from omputer graphis an be applied. 9

6 3.3 Formulation of the Proposed Model Equation (1) an now be extended to inlude the speular omponents by the image soures and the image elements. The radiation density of the reeiving element i beomes N B i (t) = s i ρ i j=1 m=0 r jm B j (t R i,jm ) F i,jm e amr i,jm + B i,qk (t). (15) The first double summation represents the ontribution from all soure elements and their images. The first term of the inner summation (for m = 0) is idential to the first term in Eqn. (1), orresponding to the diret ontribution from j to i. In the extended AR equation the sattering oeffiient has been used, indiating that only a fration s i of the refleted energy is diffusely refleted from i in ontrast to Eqn. (1). The seond term is the speular ontribution from Q, given by Eqn. (10). The energy density at the reeiver an be alulated by extending Eqn. (5) in the same way as for Eqn. (15): E(t) = 1 π N j=1 m=0 r jm B j (t R P,jm ) H P,jm e a mr P,jm + E qk (t). (16) Again, the first double summation orresponds to the ontribution from all elements (soure and image), that is from all speular refletions of the diffuse energy that reah the reeiver. The seond term is the speular ontribution from Q, given by Eqn. (9). At this point it seems impratial to inlude all the orders of refletion for the usual impulse response lengths. The omputation time would raise abruptly and the information gained would be useless beause the energy at the late part of the response would be already diffuse. Thus, it seems wise to use both ISM and AR for the early part of the response but only AR for the late part. The number of image soures an be finite, K, orresponding to the first few refletion orders. The energy ontribution from the last image soure K is refleted totally in a diffuse manner at element i, sine no further speular refletions are assumed. Figure 2 illustrates two unique speular refletion paths and their onversion to totally diffuse ones. The upper one represents a sequene of image soures while the lower one represents a sequene of image elements. During the final refletion the energy is assumed only diffusely refleted. Eqn. (15) is now modified as where N B i (t) j=1 [s i ρ i M 1 m=0 +ρ i r jm B j (t R i,jm B i,qk (t) s K 1 iρ i 4πS i k=0 r jm B j (t R i,jm ) F i,jm e a mr i,jm ) F i,jm e a M R i,jm ] + B i,qk (t), r qk W Q (q k, t R i,q k ) H i,qk e a mr i,qk + ρ i r qk W Q (q K, t R i,q K 4πS i (17) ) H i,qk e a mr i,qk. (18) Seondly, the double summation in Eqns. (15) and (16) is the most omputationally ostly term, when implementing ombined model. A question arises now whether the role of image elements is really important or not. It may be argued that one the energy is diffusely refleted any subsequent speular refletions an be judged as diffuse anyway. In that ase, the two primitive models, ISM and AR, an be onsidered as oupled only towards one diretion, from ISM to AR. As a result, Eqns. (15) and (16) are simplified to N B i (t) ρ i j=1 B j (t R i,j ) F i,je amr i,j + B i,qk (t), (19)

7 E(t) 1 π N j=1 B j (t R P,j ) H P,je amr P,j + E qk (t). (20) So now the energy from the soure element j is refleted only diffusely at element i and the total refletion oeffiient ρ i is used. The validity of this simplifiation will be examined in Se IMPLEMENTATION OF THE MODEL For the implementation of the model an impulsive original soure is assumed, i.e., W Q = 1 for t = 0 and W Q = 0 for t 0. Therefore, Eqn. (18) is normalized with W Q. Moreover, the air absorption an be fully negleted until the final step of the alulations. One the energy density at the reeiver has been omputed without air absorption let us all it E 0 (t) the orreted value is given by: E am (t) = E 0 (t)e amt. (21) The time is disretized by a sampling frequeny f s. Eah time step is now denoted by an integer n, so that the atual time at n is t n = n dt. Now, t = 0 is represented by n = 1. Evidently, the higher the sampling frequeny the more aurate the response is, but the omputation time beomes longer. A ompromise between the duration of the response, the auray desired and the omputation time allowed should lead to an optimal seletion of f s. Following Nosal et al., 5 all the delays between the elements, the soure and the reeiver an be expressed by a number of time steps, utilizing the eiling funtion ( ). For example, the delay between i and j is expressed as T i,j = R i,j /( dt). Similarly, the length of the disretized impulse response is T = t max / dt, where t max is the atual duration. 4.1 The Main Algorithm The model is implemented in an iterative algorithm, where eah iteration is equivalent to the order of refletion. A entral feature is the radiation response of eah element i, desribing the whole time vetor of the radiation density assoiated with it. This is symbolized by B[i, 1 T ]. Similar to the radiation response, the reeiver s response is denoted by E[1 T ]. These vetors essentially represent the refletograms, with respet to the impulsive soure. During eah iteration the whole vetors are updated, so that they beome more dense and longer, oupying progressively a larger part of the desired total duration. This an been seen as a onvergene towards a final solution, where the reeiver s response E[1 T ] has been stabilized and it does not hange with further iterations. Initially, the simplified version of the model is onsidered, as given by Eqns. (19) and (20). The orresponding algorithm is alled main algorithm. In Se. 4.3 the omplete problem is treated as given by Eqns. (15) and (16). During the primary iteration the diret ontribution from the original soure to the elements and the reeiver is alulated. In addition, the first order image soures are generated for any semi-diffuse surfae. At the end of the primary iteration eah B[i, 1 T ] and E[1 T ] ontain only one bin orresponding to the impulse from the soure. In the next iteration every element ollets the radiation response from all other elements, updating its old radiation response. The orresponding assignment for every pair i and j reads B[i, T i,j + 1 T ] = B[i, T i,j + 1 T ] + β i B [j, T T i,j ] F i,j, (22) where β i = ρ i, beause the energy from j is assumed to be refleted totally diffusely at i. The symbol = implies that a new value is assigned to the old value of the variable. One the soure element has distributed all of its present energy, B[j, 1 T ] is reset to zero. In the same way to Eqn. (22) the reeiver gathers energy from eah element aording to E[T P,j + 1 T ] = E[T P,j + 1 T ] + 1 π B [ j, 1 T T P,j]. (23)

8 Equations (22) and (23) essentially represent the diffuse model for the first order refletion. As for the speular model, all first order image soures are proessed one after the other. For eah soure the reeiver is heked whether it is visible. If so, the ontribution is added: r qk E q [T P,qk ] = E q [T P,qk ] + 1 4π RP,q 2. (24) k Now, all walls are proessed in a loop that serves two purposes: 1) The energy ontribution from the image soure is omputed for all visible elements on eah wall. For simpliity, only the enter of the element is taken into aount for the visibility hek. 2) The valid daughter image soures are generated. Aording to the assumption of Eqn. (18), the inident energy from the image soure is refleted by a portion s i ρ i from element i, as long as the soure produes a daughter soure behind the wall of i. If it is not the ase, the orresponding speular refletion path is terminated and the portion is ρ i. Compatly, the ontribution from the image soure reads B[i, T i,qk ] = B[i, T i,qk ] + β i 1 4πS i r qk H i,qk, (25) where β i = ρ i, if the soure is the last one and β i = s i ρ i, otherwise. At the end of the first order iteration, the diffuse and speular ontributions have updated the radiation and the reeiver s responses. Every suessive iteration an be exeuted with the same proedure, with the image and seondary soures ontributing the orresponding order of speular and diffuse refletion, respetively. 4.2 Dynami Termination of the Image Soure Sequene The upper limit of the summation in Eqn. (18) an be predefined by fixing the maximum order of image soures, like in the pure ISM. However, surfaes with high sattering oeffiient are expeted to reflet only a small portion of energy in a speular manner. In suh ases, the orresponding image soures are very weak. Furthermore, as more and more speular rays beome diffuse during eah iteration, the energy in the speular model is expeted to derease, raising the energy in the diffuse model. Consequently, it would be wise to onvert the ombined model dynamially to pure AR, whenever the diffuse model dominates the speular, aelerating the omputational effort. For that reason the diffuse and speular radiation densities of a wall an be defined, denoted by B dw and B sw, respetively. The first one represents the energy of the wall in the diffuse model without adding the ontribution from the image soures. The seond one ontains only the speular ontribution. Every time a daughter soure is to be produed behind a wall w, the speular radiation density arriving at the wall from the mother soure q k at time step T w,qk is ompared with the diffuse radiation density at the same time step. If B sw > B dw [w, T w,qk ], the daughter soure is produed behind the wall, beause the energy in the speular model is still important, ompared to the energy in the diffuse one. On the other hand, if B sw < B dw [w, T w,qk ] the image soure sequene is terminated. The termination ondition an be subjeted to a predefined margin. Someone ould demand, for example, that it is enough for the speular energy to be higher than 50% of the diffuse energy, in order for the image soure sequene to be ontinued. The lower the margin is, the later the image soure sequene is terminated. 4.3 The Full Algorithm Now the omplete oupling between AR and ISM is inluded, by letting the energy emitted from the elements suffer several speular refletions while it is distributed on the boundary. Aording to Se. 3, these speular refletions are represented by image elements. Now β i = s i ρ i in Eqn. (22). In order to simplify the algorithm, the termination order for the image elements is fixed. The refletion

9 orders of the image elements an work independently of the orders/iterations of the main algorithm. This allows to set a relative low number of refletions for the image elements, in order to keep the omputational ost as low as possible. 4.4 Impulse Response Due to the Image Elements During eah iteration a new ensemble of image elements is formed for every soure element j, representing the lower path in Fig. 2. However, the way all the image elements of j ontribute at element i is a matter of geometry and remains the same for all iterations. Thus, there is no need to generate them from srath. It is enough to store the radiation response of eah element i due to the image elements of j, when it emits a unit impulse (B[j, 0] = 1): M 1 B m [i, j, T i,jm ] = B m [i, j, T i,jm ] + s i ρ i r jm F i,jm + ρ i r jm F i,jm. (26) m=1 Disrete onvolution of B m [i, j, n] with the real B [ j, n], gives the real radiation density as a funtion of time: B[i, n] = B[i, n] + B m [i, j, n] B [ j, n] = B[i, n] + T τ=n B m [i, j, τ] B [ j, τ n + 1]. (27) Equation (26) leads to a three dimensional (3D) matrix: for every soure element j, every reeiving element i and every time step n. 3D matries are very impratial and diffiult to be stored. Transformation into two dimensional (2D) is quite benefiial. For that reason a very low sampling frequeny ould be adopted only for the B m [i, j, 1 T ]. Hene, the responses due to the image elements will be a bit oarse but this does not seem to harm the result. The important thing is that many refletions an be grouped into one bin, as long as they arrive at the same time interval. Then it seems useful to store only the bins and their arrival times, not the whole time vetors B m [i, j, 1 T ], whih ontain usually a large number of zeros among the bins. Hene two 2D matries should be reated; one for the values of B m [i, j, 1 T ] and one for the orresponding arrival time steps. Their olumns represent the reeivers i, while the lines are divided into groups of soure elements j. In eah group, every line orresponds to one bin of the response. Following the same proess, the reeiver s response due to the image elements of j an be omputed when the element ats as a unit impulse soure: E m [ j, T P,jm ] = E m [ j, T P,jm ] + 1 π M r jm H P,jm. (28) m=1 The result an be stored diretly in a 2D matrix with j lines and n olumns. Then the real energy density, E[n], due to the image elements of j, an be omputed by disrete onvolution with B [j, n]: E[n] = E[n] + E m [ j, n] B [ j, n] = E[n] + 5 RESULTS AND ANALYSIS T τ=n 5.1 A Polyhedral Room with Varying Sattering E m [ j, τ] B [ j, τ n + 1]. (29) A small polyhedral room of 7 walls Fig. 3(a) was studied. The boundary was subdivided into 276 elements and a sampling frequeny of 16 khz was used. The sampling frequeny was hosen relatively low in order to help for better visibility of the refletograms. The absorption and sattering oeffiients of the walls are given in Table 1. In the first ase the sattering oeffiient

10 was everywhere 1, meaning that the pure AR algorithm was used in the alulations. In the seond ase the sattering oeffiient varied signifiantly and the ombined model was employed. The refletograms at the reeiver for both ases are given in Fig. 3(b). The line orresponding to AR is very smooth throughout the whole impulse response length. The early refletions have lost all of their speular nature. On the other hand, the refletogram by the ombined model preserves a lot of information for the speular refletions in the early part (highly osillating behavior) and onverges smoothly to the straight AR line only at the late part, as the speular refletions are gradually onverted to diffuse ones. 5.2 Predition of Flutter Eho One interesting appliation of the main algorithm is the predition of flutter ehoes, whih are diffiult to simulate properly by the existing room aousti methods. A retangular room was onsidered with dimensions (L x, L y, L z ) = (8, 5, 3) m and oeffiients from Table 2. The absorption and sattering was quite low for two parallel lateral walls, while it was high for the rest. The boundary was subdivided into 252 elements and the sampling frequeny was 2.7 khz. The soure was plaed at the enter of the room, Q = (4, 2.5, 1.5) m, and the reeiver was plaed between the soure and one of the speular refleting walls, P = (4, 1.25, 1.5) m. The simulation was run with 7 iterations. After the first iteration all the weak image soures had been aborted. Then, only two image soures were produed per iteration, orresponding to the pair of the speular parallel walls. The refletogram at the reeiver is illustrated in Fig. 4(a). There are 14 equally spaed prominent bins, just after the diret ontribution. These bins form 7 pairs, orresponding to the number of iterations used in the simulation. The advantage of the ombined model in the ase of highly symmetrial problems, like flutter eho, is that the omputational load is mitigated by the fat that only the important image soure sequenes are kept, while the rest of the problem is handled by the muh faster AR part. Figure 4(b) gives a global piture of the energy behavior in the room. The energy reeived by eah element at every time step was alulated first. Then the averaged energy from all the elements at every time step was omputed. By bakward integration the energy deay urves were obtained. The individual deay urves from 22 random elements are given by the thin lines. The bold line is the deay urve of the averaged energy. Evidently, the urves orresponding to the elements on the parallel speular walls have shallower slope and they are elevated above the average. The energy deay at these elements reveals the repeatable refletions of the flutter eho and leads to a derease of the slope at the late part of the average energy. 5.3 Computational Performane The refletions in AR are memoryless, so that the exeution time of the method remains stable as the order of refletion is inreased, in ontrast to ISM. As for the exeution time of the ombined model, this highly depends on whih of the two methods is dominant for eah refletion level. During the early refletions ISM arries most of the information, so that the omputational effort per refletion order is inreased. As the speular refletions are onverted to diffuse, AR is dominant and the omputational effort is stabilized. In the following example, the room of Se. 5.1 was used with the absorption data for ase 2 in Table 1. The ombined model was run with the sattering oeffiient varying uniformly from 0 to 1. In total, 18 iterations were performed with a sampling frequeny of 20 khz. The duration of the impulse response was 35 ms. A dynami termination of image soures was used (Se. 4.2). The Cpu time for exeution was reorded for every iteration and the result is illustrated in Fig. 5. The exeution time is proportional to the number of generated image soures. For pure speular refletions it inreases monotonially. Applying s = 0.1 to all walls is enough to introdue a peak"

11 in the omputational time, after whih it delines gradually. For s = 0.2 the exeution time has been dramatially redued and for s > 0.4 it is almost the same as with s = 1 (pure AR). 6 CONCLUSIONS An attempt to ombine AR and ISM has been presented in order to model the mixed kind of speular and diffuse refletions observed on real walls, inside rooms. The ombined model utilizes the most effiient parts of the two methods and makes it feasible to use ISM up to very high order, preserving at the same time the important speular information missed by AR. As a result, interesting phenomena like flutter ehoes an be simulated preisely, with low ost. In general, the model serves for a unified preditor of both the early and the late part of the response. Unlike other room aousti models that inorporate speular and diffuse refletions by a traing proess, the presented one is fully deterministi, providing high detail and reproduibility of the results. 7 REFERENCES [1] A. Krokstad, S. Strøm, and S. Sørsdal, Calulating the aoustial room response by the use of a ray traing tehnique., J. Sound Vib. 8, (1968). [2] C.-H. Jeong, J.-G. Ih, and J. H. Rindel, An approximate treatment of refletion oeffiient in the phased beam traing method for the simulation of enlosed sound fields at medium frequenies, Appl. Aoust. 69, (2008). [3] J. Borish, Extension of the image model to arbitrary polyhedra, J. Aoust. So. Am. 75, (1984). [4] H. Kuttruff, Room Aoustis, 4th ed. (Applied Siene Publishers LTD, London, 2000), Chap. 4, pp [5] E. M. Nosal, M. Hodgson, and I. Ashdown, Improved algorithms and methods for room soundfield predition by aoustial radiosity in arbitrary polyhedral rooms, J. Aoust. So. Am. 116, (2004). [6] T. Lewers, A ombined beam traing and radiant exhange omputer model of room aoustis, Appl. Aoust. 38, (1993). [7] B. I. Dalenbäk, Room aousti predition based on a unified treatment of diffuse and speular refletion., J. Aoust. So. Am. 100, (1996). [8] R. N. Miles, Sound field in a retangular enlosure with diffusely refleting boundaries, J. Sound Vib. 92, (1984). [9] I. Ashdown, Radiosity: A Programmer s Perspetive, 1st ed. (Wiley, New York, 1994), Chap. 2, pp and Chap. 5, pp [10] M. Vorländer, Simulation of the transient and steady state sound propagation in rooms using a new ombined ray-traing / image-soure algorithm, J. Aoust. So. Am. 86, (1989). [11] International Organization for Standardization, Geneva, Switzerland, ISO Aoustis - Measurement of room aousti parameters - Part 1: Performane spaes (2009).

12 Walls S w (m 2 ) α ᾱ = 0.40 ase 1 s s = 1.00 ase 2 s s = 0.24 Table 1: Absorption and sattering oeffiients for the two ases in Se. 5.1 averaged values are shown on the left. Walls S w (m 2 ) α ᾱ = 0.62 s s = 0.70 Table 2: Absorption and sattering oeffiients for investigation of flutter eho in the retangular room averaged values are shown on the left. reeiving element image element soure element Figure 1: The diffusely refleted energy from the soure element an reah the reeiving element either diretly ( ) or via a speular refletion ( ), represented by the image element. α 1 a 2 a k a k+1 1 (1-s 1 )(1-α 1 ) (1-s 1 )(1-α 1 ) (1-s k )(1-α k ) s 1 (1-α 1 ) s 2 (1-α 2 ) s k (1-α k ) (1-α k+1 ) α1 α2 αm αm+1 (1-s1)(1-α1) (1-s2)(1-α2) (1-sm)(1-αm) s1(1-α1) s2(1-α2) sm(1-αm) (1-αm+1) Figure 2: Flow of energy in two refletion paths. The upper path represents a single image soure sequene for the original soure. The lower path represents a single image element sequene for one of the elements ating as a soure. When any of the paths is terminated, the speular oeffiient, (1 s), beomes zero and from now on, the energy is refleted totally diffusely.

13 z (m) y (m) x (m) 0.8 E (db re Ws/m 3 ) Time (ms) (a) (b) Figure 3: (a): A 7-wall polyhedral room meshed with N=276 triangular elements. Soure: Q = (0.3, 0.2, 0.2) m. Reeiver: P = (0.4, 0.7, 0.5) m.. (b): Refletograms at the reeiver in the room. Coeffiients from Table 1. : Case 1. : Case 2. E (db re Ws/m 3 ) (a) Time (s) Energy Deay (db) Number of Mean Free Paths (b) Time (s) Figure 4: Demonstration of flutter eho. Results obtained by the ombined model with oeffiients from Table 2. (a) Refletogram at the reeiver. (b) : Individual deay urves from 22 uniformly hosen elements, : Deay urve of averaged energy. 40 Cpu Time (s) Iteration Number Figure 5: Exeution time per iteration for different uniform sattering oeffiients. : s = 0, : s = 0.1, : s = 0.15, : s = 0.2, : s = 0.3, : s = 0.4, : s = 1.