Comparison Between Scattering Coefficients Determined By Specimen Rotation And By Directivity Correlation Tetsuya Sakuma, Yoshiyuki Kosaka Institute of Environmental Studies, University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-33 Japan sakuma@k.u-tokyo.ac.jp ABSTRACT For measuring the random-incidence scattering coefficient of wall surfaces, a diffuse-field method and a free-field method have been developed, in both of which a specimen is rotated on a turntable. Besides, a way to determine the scattering coefficient at a fixed orientation has been proposed, where the coefficient can be derived from the two reflection directivities for the specimen and for the flat plate. If using the latter way, a random-incidence coefficient can be obtained by integrating the coefficient at every incidence angle. However, it is not theoretically equivalent to that measured by the former way regarding the integration process. There is a possibility that some types of surfaces have large fluctuation in the distribution of directional scattering coefficients, which causes disagreement in the random-incidence values determined by the two ways. Numerical simulation demonstrates the difference between the values in the cases with sinusoidal surfaces. KEYWORDS: Scattering coefficient, Wall surface, Numerical analysis INTRODUCTION In recent years, acoustical measurement techniques to evaluate scattering performance of wall surfaces are developed [1, 2], and an ISO standard for measuring the random-incidence scattering coefficient in the diffuse field is about to be published [3]. The usefulness of the coefficient is promising in the application to computational sound field simulation. However, at the present, some unclear points remain regarding the behavior of the coefficient related to the principle of the measurement. In particular, the ISO method employs an unique procedure, that is rotation of a circle specimen on a turntable, to extract the specular component of the reflected energy. Its basic theory was already developed [2], but there seems still room for detailed discussion on the
effect of the procedure. On the other hand, according to Mommertz s definition [4], the scattering coefficient at a specific incidence angle is determined by the correlation between the two reflection directivities for the specimen and for the flat reference plate in the free filed. In principle, the randomincidence scattering coefficient can be obtained by statistically averaging the directional coefficients over all incidence angles. This way is difficult to be applied to measurement, but can be realized by computational analysis. Theoretically, the above two ways give different definitions of the random-incidence scattering coefficient regarding the averaging process. In this paper, the two ways in the free field are numerically simulated with the boundary element method, and the difference in the calculated coefficients is investigated in detail. Calculation of Reflection Directivity NUMERICAL METHOD As illustrated in Figure 1, it is assumed that a plane wave impinges on a specimen with negligible thickness in the free field. With sampling N directions in the upper hemisphere at constant intervals of polar and azimuthal angles, applying the boundary element method in the normal derivative form on every incidence condition gives the following matrix equation. where A ij = e j 2 G(r i,r q ) n i n q A p 1,p 2,,p l,,p N = d 1,d 2,,d l,,d N, (1) ds q, d il = n i exp jk l r i, p jl is the sound pressure difference on both surfaces of j-th element, k l is the wave number vector for l-th condition with the incidence angle (θ l, l ). Therefore, after calculating A 1, p l for every incidence condition is obtained simultaneously. Considering the discrete reflection angle, the reflected sound pressure distribution for every incidence condition is calculated by the following equation. p 1,p 2,,p l,,p N = H p 1,p 2,,p l,,p N, (2) G(r m,r q ) where H mj = ds n q, p ml is the sound pressure at m-th receiving point with the e q j reflection angle (θ m, m ) for l-th incidence condition, and supposing that every receiving point is sufficiently far from the origin. For the detailed set-up of computation, refer to [5]. Calculation of Scattering Coefficients Directivity Correlation Method. For determining the scattering coefficient by Mommertz s definition [4], the two reflection directivities for the specimen and for the flat reference plate are required to be calculated. The directional scattering coefficient is represented by s(θ l, l )=1 N p ml N * p ml p ml sin θ m 2, (3) 2 N 2 sin θ m p ml sin θ m
reflection θ z θ' incidence y z.2m y z y O ' x sample x x.6m (a) 1D sinusoidal (Type 1) (b) 2D sinusoidal (Type 2) Figure 1. Geometry of the numerical model. Figure 2. Test specimens with periodical surface. where p and p denote the complex sound pressure for the specimen and for the flat plate, respectively, and the intervals θ and are constant. According to Paris formula [6], integrating the directional coefficients gives the random-incidence coefficient. Specimen Rotation Method. As substitute for the diffuse-field method with specimen rotation [3], a free-field method has been also proposed, where the polar angle of the incident wave is fixed during the rotation [1]. In this method, the specular reflection factor is obtained by coherent averaging of the complex reflection factors under variation of the specimen orientation. However, in the following calculation, another way with coherent averaging of the complex sound pressure at each receiving point is adopted to extract the specular component of the reflected energy. The averaging is also done under variation of the specimen orientation, and the directional scattering coefficient is calculated by s(θ l )=1 N 1 p M ml ( j ) j =1, (4) N M 1 p M ml ( j ) 2 j =1 M 2 where M directions are sampled for j the rotation angle of the specimen at constant intervals. Finally, the random-incidence coefficient is obtained by Paris formula. Behavior of Reflection Directivity RESULTS AND DISCUSSION Figure 2 illustrates two types of test specimens with 1D sinusoidal periods and with 2D ones, which have a diameter of 3 m, and perfectly reflective surface. Figure 3 shows examples of the reflected energy distributions calculated for the two specimens and the flat reference plate. Compared with the distribution for the flat plate, the specular component is clearly weakened and some unique components caused by periodical shape appear in those for the speciments. Next, the distributions with coherent averaging under one rotation are demonstrated in Figure 4. It is natural that the distribution for the flat plate does not change at all, and furthermore, a similar pattern is exposed in those for the specimens as for the flat plate. In Equation (4), the total energy of this remaining pattern is considered as the specular component.
45 9 θ 45 9 θ 45 9 θ -1-2 -3-4 -5-6 Relative SPL (db) (a) Reference (b) Type 1 (c) Type 2 Figure 3. Distributions of the reflected energy for the flat reference plate and for the two types of specimens, on the incidence condition θ = 28.1 and = 18.1 at 2 khz. 45 9 θ 45 9 θ 45 9 θ -1-2 -3-4 -5-6 Relative SPL (db) (a) Reference (b) Type 1 (c) Type 2 Figure 4. Distribution of the reflected energy with coherent averaging under the polar incidence angle θ = 28.1 at 2 khz. Behavior of Scattering Coefficient Directional Scattering Coefficient. Figures 5 and 6 show the distributions of the directional scattering coefficients calculated with the directivity correlation method for the specimens Types 1 and 2, respectively. Generally, the coefficients become higher as the frequency goes up and as the polar incidence angle approaches 9 degrees, and large deviation appears in the distribution at 1 khz. In the case with Type 1, it is clearly seen that the incidence directions for lower values correspond to that of the ribs on the surface. The scattering coefficients of polar incidence angle dependence can be calculated both with the directivity correlation method and with the specimen rotation method. Figures 7 and 8 show the results with the two methods for Types 1 and 2, respectively. In the case with Type 2, good agreement is seen in both results regardless of the frequency, however, considerable differences arises at 1 khz in the case with Type 1, which cause the disagreement in the randomincidence values. Random-Incidence Scattering Coefficient. Figure 9 shows the frequency characteristics of the random-incidence scattering coefficients calculated with the above two methods. In addition, regarding the specimen Type 1, the results obtained by the theoretical analysis assuming the
' ' ' 45 9 θ' 45 9 θ' 45 9 θ' 1..8.6.4.2 Scattering coefficients (a) 5 Hz (b) 1 khz (c) 2 khz Figure 5. Distribution of the directional scattering coefficients for the specimen Type 1, calculated with the directivity correlation method. ' ' ' 45 9 θ' 45 9 θ' 45 9 θ' 1..8.6.4.2 Scattering coefficients (a) 5 Hz (b) 1 khz (c) 2 khz Figure 6. Distribution of the directional scattering coefficients for the specimen Type 2, calculated with the directivity correlation method. infinite surface, and those by the measurement with the diffuse-field method [7] are shown for reference. Compared between the values obtained by the two methods, good agreement is seen in the case with Type 2 at all frequencies, however, differences of about.2 arise at middle frequencies in the case with Type 1. Regarding the curve for the infinite surface, that by the directivity correlation method is relatively close, especially at high frequencies. This reason is considered that the effect of the edge diffraction of the finite specimen becomes smaller at higher frequencies, and in principle, Mommertz s definition gives the same values as for the infinite one if the edge diffraction vanishes. On the other hand, the curve by the specimen rotation method fairly agrees with that by the measurement, in spite of the different sound field conditions. Therefore, it is considered that the method based on Equation (4) is almost equivalent to the diffuse-field measurement method. Effect of Averaging Processes. As is seen above, the directivity correlation method and the specimen rotation method gives different values on some condition, which is considered to be caused by different averaging processes. For example, this disagreement occurs in the case with Type 1 at 1 khz, where large deviation of directional scattering coefficients is also observed. Considering that R spec,i denotes the specular reflection factor of i-th incidence condition where the polar angle is fixed, the averaging process for the directivity correlation method is represented
Scattering coefficient 1.8.6.4.2 random-incidence 3 6 9 3 6 9 3 6 9 (a) 5 Hz (b) 1 khz (c) 2 khz Figure 7. Scattering coefficients of polar incidence angle dependence for the specimen Type 1, calculated with the directivity correlation method () and with the specimen rotation method (). Scattering coefficient 1.8.6.4.2 random-incidence 3 6 9 3 6 9 3 6 9 (a) 5 Hz (b) 1 khz (c) 2 khz Figure 8. Scattering coefficients of polar incidence angle dependence for the specimen Type 2, calculated with the directivity correlation method () and with the specimen rotation method (). 1 Random-incidence scattering coefficient.8.6.4.2 Measured Infinite 25 5 1 2 Frequency [Hz] 25 5 1 2 Frequency [Hz] Figure 9. Frequency characteristics of the random-incidence scattering coefficinets obtained by: calculation with the directivity correlation method (), with the specimen rotation method (), and with the theory for infinite surface, and measurement with the diffusefield method (N.B. the height of the specimen is equivalent to.63 m after frequency conversion.) [7].
as follows s cor = 1 M M s cor,i i =1 =1 1 M M i =1 R spec,i 2. (5) On the other hand, the averaging process for the specimen rotation method, which is to extract the specular components, is represented by s rot =1 M 1 M 2 M 2 R i 1 1 R i =1 M spec,i. (6) i =1 where R i is the complex reflection factor for the specular and the diffuse components. If R spec,i is constant regardless of the azumithal incidence angle, the above two scattering coefficients give the same values. However, as demonstrated in Figures 5 and 6, R spec,i has variation with changing the azumithal angle. Comparison bewteen Equations 5 and 6 gives the relation s rot s cor, where the difference of the two values becomes greater in the case that the deviation of R spec,i is large, which can explain the case with Type 1. CONCLUSIONS The two ways of determining the scattering coefficinets in the free field, with directivity correlation and with specimen rotation, were numerically simulated, and it was demonstrated that some difference in the calculated random-incidence values is made in the case with 1D periodical surface, but not in the case with 2D one. Theoretical consideration clarified that the dominant reason for this differece is the influence of averaging processes in the two meothds, which are energy-based and amplitude-based averaging in terms of the azumithal angle of the incident wave. Furthermore, the difference becomes remarkable if the directional scattering coefficients have large deviation under each azumithal incidence angle. It suggests that the measurement methods based on specimen rotation also have a possibility to include this influence on the random-incidence scattering coefficient. ACKNOWLEDGEMENTS We would like to thank Prof. M. Vorländer for thoughtful suggestions and Dr. M. Gomes for providing precious measured data. REFERENCES [1] M. Vorländer and E. Mommertz, Definition and measurement of random-incidence scattering coefficients, Appl. Acoust. 6, 187-199 (2). [2] AES-4id-21: Characterization and measurement of surface scattering uniformity (21). [3] ISO/DIS 17497-1: Acoustics - Measurement of the sound scattering properties of surfaces, Part 1: Measurement of the random-incidence scattering coefficient in a reverberation room (2). [4] E. Mommertz, Determination of scattering coefficients from the reflection directivity of architectural surfaces, Appl. Acoust. 6, 21-23 (2). [5] Y. Kosaka and T. Sakuma, Numerical study on the behavior of scattering coefficients of wall surfaces, Proc. 18th Int l Cong. Acoust. (24). [6] H. Kuttruff, Room Acoustics 4th ed., Chap.2 (Spon Press, London, 2). [7] M. Gomes, M. Vorländer and S. Gerges, Aspects of the sample geometry in the measurement of the randomincidence scattering coefficient, Proc. Forum Acusticum Sevilla 22, RBA-6-2-IP (22).