SCATTERED SOUND POWER FROM A RIGID STRUCTURE USING NON-NEGATIVE INTENSITY

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1 SCATTERED SOUND POWER FROM A RIGID STRUCTURE USING NON-NEGATIVE INTENSITY Daipei Liu, Herwig Peters, Nicole Kessissoglou School of Mechanical and Manufacturing Engineering, UNSW Australia, Sydney, NSW 2052, Australia daipei.liu@unsw.edu.au Steffen Marburg LRT4-Institute of Mechanics, Universität der Bundeswehr München, D Neubiberg, Germany In this work, the non-negative intensity is used to predict the scattered sound power from a rigid structure excited by incident plane waves. The scattered sound power is also predicted from the scattered acoustic intensity integrated over the surface area of the rigid structure, whereby the scattered acoustic intensity is obtained from the scattered sound pressure and scattered particle velocity at the structural surface. The scattered sound power from a rigid sphere under plane wave excitation using the two numerical intensity-based techniques is compared. Furthermore, the scattered acoustic intensity in the far field is evaluated from the scattered sound pressure and scattered velocity at three different shapes for a far-field boundary enclosing the rigid sphere, corresponding to a sphere, a half sphere and a cube. The scattered sound power based on integration of the far-field scattered acoustic intensity and calculated from the surface of the scatterer are then compared. 1. Introduction When a sound wave encounters an obstacle, part of the wave is deflected from its original course. The difference between the actual wave and the undisturbed wave, which would be present if the obstacle was not present, is usually defined as the scattered wave [1]. Lord Rayleigh [2] mathematically investigated the scattering of sound waves from a small, rigid and immovable sphere and a circular cylinder. However, due to the complexity of the analytical solutions, he only considered the limiting case where the scatterers were small compared with the wavelength, which is now referred to as Rayleigh scattering. King [3] calculated the scattered pressure from a small rigid sphere excited by a plane wave in a frictionless fluid. He used spherical wave functions to describe the incident pressure and obtained the scattered pressure in terms of the density and velocity potential of the incident wave. The scattering of acoustic waves by rigid, immoveable circular cylinders and spheres which are not acoustically compact has been extensively studied by Morse [1]. Numerical techniques such as the finite element method (FEM) and the boundary element method (BEM) are more appropriate to predict the vibro-acoustic responses of complex structures. The BEM is often used to solve exterior acoustic problems since it only requires the generation of a mesh on the structural boundary. Seybert et al. [4] introduced a numerical method to implement the Helmholtz ICSV22, Florence, Italy, July

2 q Incident waves k ff Scattered waves Figure 1: Acoustic scattering problem showing a rigid boundary of a rigid sphere and a partial boundary ff in the far field integral formula for acoustic radiation and scattering problems from structures of arbitrary shapes. Quadratic boundary elements were used to mesh different test cases involving spherical, cylindrical and cubical geometries. Numerical results showed good agreement with analytical solutions. Several papers then introduced the boundary integral approach to implement the Kirchhoff formula in acoustic radiation and scattering problems [5, 6, 7]. Marburg et al. [8] investigated a new numerical technique to predict the radiated sound power by computing the surface contributions of a vibrating structure. The surface contribution was calculated based on the acoustic radiation modes, acoustic radiation efficiencies and either acoustic pressure or normal structural velocity. Similar to the surface contribution method to identify the radiated sound power from a vibrating structure, the non-negative intensity was recently presented by Williams [9]. Non-negative intensity is always positive and as such avoids cancellation effects. Furthermore, non-negative intensity provides higher spatial resolution of the radiation hot spots than supersonic acoustic intensity, also developed by Williams [10, 11]. In this work, non-negative intensity is used to predict the scattered sound power from a rigid sphere excited by incident plane waves. The scattered sound power is also obtained by integrating the scattered acoustic intensity over the surface area of the scatterer. The patterns of both the incident and scattered fields on the structural surface are compared at both lower and higher frequencies. The scattered pressure and scattered velocity at three different shapes representing a far-field boundary enclosing the rigid structure, corresponding to a sphere, a half sphere and a cube, are obtained to calculate the scattered acoustic intensity in the far field. The scattered sound power based on integration of the far-field scattered acoustic intensity and calculated from the surface of the scatterer are compared. 2. Numerical BEM Based Methods 2.1 Incident Wave Source A uniform plane wave incident on a rigid sphere is shown in Fig. 1. For this problem, the inhomogeneous Helmholtz equation becomes [12] (1) p (x) + k 2 p (x) = q where q is the source, p is the acoustic pressure and k is the acoustic wavenumber. The solution of Eq. (1) is the sum of the complementary solution p c and the particular solution p p. The complementary 2 ICSV22, Florence, Italy, July 2015

3 solution is referred as the scattered pressure p s and the particular solution is referred as the incident pressure field p i. The entire sound pressure p is given by (2) p (x) = p c (x) + p p (x) = p s (x) + p i (x) Similarly, the fluid particle velocity v f can be represented as (3) v f (x) = v c (x) + v p (x) = v s (x) + v i (x) where v c is the complementary solution, v p is the particular solution, v s is the scattered velocity and v i is the incident velocity. The uniform incident wave field is given by (4) p i (x) = p 0 e i(kx+ϕ 0) where p 0 is the magnitude of p i (x), ϕ 0 is the reference phase angle and the vector k contains the wavenumber components that determine the direction of the incident waves. The magnitude of k is the acoustic wavenumber k. The corresponding particle velocity is given by (5) v i (x) = p 0 ϱ 0 ω k nei(kx+ϕ 0) = p 0 k ϱ 0 ω n ei(kx+ϕ 0) where ϱ 0 is the average fluid density, ω is the angular frequency and n is the outward normal vector on the boundary. Discretization of Eq. (1) using the collocation boundary element method yields [12] (6) (H D) p = G (v f v i ) + Hp i = f + f i where G and H are boundary element matrices and D is the boundary admittance matrix. f and f i are the structural forces and the forces due to incident waves, respectively. Substitution of Eq. (2) into Eq. (6) yields (7) (H D) p s = G (v f v i ) + Dp i For a rigid boundary, the matrix D vanishes and the fluid particle velocity on the boundary of the sphere is zero, that is (8) v f = v s + v i = 0 Hence the scattered pressure can be expressed in terms of the incident fluid velocity as follows (9) p s = H 1 Gv i The entire sound pressure p can also be expressed in terms of the incident pressure p i by (10) p = H 1 p i (10) Substituting Eqs. (9) and (10) into Eq. (2), the following relationship between v i and p i can be obtained (11) Gv i = (H I) p i where I is the identity matrix. 2.2 Scattered Sound Power based on the Structural Surface The total scattered sound power is defined as P s = 1 I s n d = 1 { (12) 2 2 R } p s (v s ) d ICSV22, Florence, Italy, July

4 where I s = 1R {p 2 sv s } is the scattered acoustic intensity on and () denotes the complex conjugate. After discretization, Eq. (12) can be expressed as [8] P s = 1 { (13) 2 R (p s ) Θ (v s ) } where () is the matrix transpose and Θ is the boundary mass matrix which is defined as [13] (14) Θ = φφ d (x) where φ are boundary element method basis functions. Substituting Eqs. (8) and (9) into Eq. (13), the total scattered sound power can be expressed in terms of the incident fluid velocity as follows P s = 1 { 2 R (v i ) G H Θ (v i ) } = 1 { (15) 2 R (v i ) Z (v i ) } where Z = G H Θ is the acoustic impedance matrix [14, 15]. The total scattered sound power can also be expressed using non-negative intensity by the following boundary surface integral P s = Is NN (x) d (x) = 1 (16) β s (x) (β s (x)) d (x) 2 Using Eq. (14) to discretize the boundary surface integral in Eq. (16), β s (x) can be expressed as (17) β s (x) = φ (x) β s where β s is a vector without physical significance and is computed using the acoustic radiation modes Ψ, eigenvalues Λ, the boundary mass matrix Θ and incident velocity v i as follows [8] (18) β s = Ψ ΛΨ Θv i (18) The discretized non-negative intensity on the structural surface (I NN s ) in matrix form can be expressed as (19) I NN s = 1 2 β s (β s ) 2.3 Scattered Sound Power based on a Far-Field Boundary Eq. (1) can be rewritten in terms of a Kirchhoff-Helmholtz integral equation by G(x, y) (20) c(y)p(y) + p(x) d(x) = G(x, y) a v(x) d(x) + p i (y) n(x) where c(y) is equal to 1 for y in the acoustic domain and equal to 0.5 for y on the boundary [12]. a = iωϱ 0 and G is the free space Green s function which is given by [12] (21) G(x, y) = 1 e ikr(x,y) 4π r(x, y) ; x, y R3 The entire sound pressure p ff at a point on the far-field boundary ff is calculated using the column matrices g and h as follows [16] (22) p ff = g v f h p + p i,ff Using Eq. (2), the scattered sound pressure p s,ff at a point on the far-field boundary is given by (23) p s,ff = g v f h p 4 ICSV22, Florence, Italy, July 2015

5 Substituting Eqs. (8) and (10) into Eq. (22), the scattered sound pressure p s,ff can be expressed in terms of the incident pressure as follows (24) p s,ff = h H 1 p i The scattered sound pressure p s,ff at all points on ff can be expressed in matrix form (25) p s,ff = HH 1 p i where H is the matrix containing the transpose of each vector h j. Similarly, by taking the normal derivative of Eq. (20), the particle velocity v ff at a point on the far-field boundary ff can be calculated, from which the scattered velocity v s,ff can then be derived. The total scattered sound power of boundary ff can then be expressed as (26) P s,ff = 1 2 R {(p s,ff ) Θ ff (v s,ff ) } (26) where Θ ff is the far-field boundary mass matrix. Hence, the scattered acoustic intensity on the farfield boundary ff can be expressed as (27) I s,ff = 1 2 R {p s,ffv s,ff } 3. Numerical Results A rigid sphere in air of radius a = 0.5 m is discretized with 384 linear discontinuous boundary elements. Density ϱ 0 = 1.3 kg/m 3 and speed of sound c = 340 m/s are assumed for air. The centre of the sphere is located at point (0, 0, 0) of the Cartesian coordinate system. A uniform plane wave with ϕ 0 = 0 and p 0 = 1 Pa in the direction ( x, y, z) = (0, 0, 1) is incident on the rigid sphere. The three different shapes for the far-field boundary enclosing the rigid spherical scatterer correspond to a 200 m diameter sphere, a 200 m long cube and a half sphere of 200 m diameter. The method has been implemented using the boundary element code AKUSTA [17]. Figure 2 presents the magnitude of the incident pressure, incident velocity, scattered pressure, scattered velocity, scattered intensity and non-negative intensity on the surface of the sphere at frequencies of 10 Hz, 100 Hz and 1000 Hz. The first column shows the same magnitude pattern of the incident pressure at all frequencies. The incident velocity, scattered pressure and scattered velocity exhibit very similar scattered sound patterns at a low frequency of 10 Hz, however the pattern of the scattered intensity I s and non-negative intensity I NN s are notably different. The scattered intensity indicates that the bottom half of the sphere scatters the maximum sound; this result is consistent with the well-known Rayleigh scattering [1]. For very long wavelengths (ka << 1), diffraction around the rigid body results in scattered waves at the bottom half of the sphere. In contrast, non-negative intensity shows a scattered pattern similar to that of the scattered pressure, whereby the top and bottom of the sphere exhibit scattered waves. As the frequency increases, the sphere is no longer acoustically compact and the diffracted waves no longer dominate the rigid body scatterer. At both 100 Hz and 1000 Hz, the scattered intensity and non-negative intensity are similar showing that two opposite sides of the rigid sphere scatter the maximum energy. Figure 3 presents the patterns of the scattered intensity (I s,ff ) on the three different shapes of the far-field boundary at frequencies of 10 Hz, 100 Hz and 1000 Hz. At a given frequency, the far-field scattered acoustic intensity exhibits very similar intensity distributions for each boundary shape. As the frequency increases, the scattered waves cancel each other due to destructive acoustic interference at the top of the body. The maximum energy occurs at the bottom of the boundary due to the diffraction of acoustic waves to the bottom of the body. ICSV22, Florence, Italy, July

6 p i v i p s v s I s I NN s (a) Scattered sound at 10 Hz, ka = (b) Scattered sound at 100 Hz, ka = (c) Scattered sound at 1000 Hz, ka = max 0 max 0 max 0 max min max 0 max Figure 2: Incident and scattered patterns of the rigid sphere The scattered sound power calculated using scattered intensity I s on the surface of the rigid sphere given by Eq. (13), non-negative intensity I NN s also on the surface of the body given by Eq. (16), and the scattered intensity I s,ff on a far-field boundary given by Eq. (26) for three different boundary shapes, are compared in Figure 4 for a frequency range up to 500 Hz. Results show that the scattered sound power gradually increases with increasing frequency. The scattered sound power calculated from the two intensities on the rigid body surface (I s and I NN s ) show good agreement with the scattered sound power calculated from the far-field scattered intensity (I s,ff ) based on the two whole far-field boundary shapes corresponding to a sphere and a cube. The scattered sound power calculated from the far-field scattered intensity based on two partial far-field boundary shapes, corresponding to the top half and the bottom half of a sphere, are also compared. The summation of the scattered sound power from the top and bottom half sphere boundary shapes is equivalent to that from the entire sphere boundary shape. Figure 4 also shows that below 200 Hz, the scattered sound power calculated using the top half of the sphere for a far-field boundary shape is greater than the scattered sound power using the bottom half. Above 200 Hz, the scattered sound power from the bottom half steadily increases while the scattered sound power from the top half remains reasonably constant. This change in contribution to the total scattered power is due to the fact that when the size of the scatterer is close to half of the acoustic wavelength, it is no longer acoustically compact. Both diffracted and reflected waves exist on the rigid body scatterer and contribute to the scattered sound power from the bottom half of the far-field boundary. 4. Summary Scattered sound power from a rigid structure excited by incident plane waves has been evaluated in terms of non-negative intensity. The scattered sound power is also calculated by integrating the scattered intensity both on the surface of the scatterer and on three different shapes representing a far-field boundary enclosing the scatterer. The scattered sound power based on the two intensity 6 ICSV22, Florence, Italy, July 2015

7 I s,ff (sphere) I s,ff (cube) I s,ff (half sphere) (a) Scattered intensity patterns at 10 Hz, ka = (b) Scattered intensity patterns at 100 Hz, ka = (c) Scattered intensity patterns at 1000 Hz, ka = 9.24 min max min max min max Figure 3: Scattered intensity patterns for different shapes of the far-field boundary Figure 4: Scattered sound power from scattered intensity and non-negative intensity methods using different shapes of the far-field boundary ICSV22, Florence, Italy, July

8 techniques evaluated on the rigid body surface shows good agreement with the results calculated using the far-field scattered intensity on a far-field boundary based on two different shapes corresponding to a sphere and a cube. It is also shown that the summation of the scattered sound power calculated from two halves of a far-field boundary shape is equivalent to that from the entire far-field boundary. REFERENCES 1. Morse, P. M. C. Vibration and Sound, Chapter 7 - The radiation and scattering of sound, McGraw-Hill, New York, 2nd edition, pp , (1948). 2. Rayleigh, J. W. S. and Lindsay. R.B., The Theory of Sound, Dover Books on Physics, Dover, vol. 2, (1945). 3. King, L. V. On the acoustic radiation pressure on spheres, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 147 (861), , (1934). 4. Seybert, A. F., Soenarko, B., Rizzo, F. J. and Shippy, D. J. An advanced computational method for radiation and scattering of acoustic waves in three dimensions, Journal of the Acoustical Society of America, 77 (2), , (1985). 5. Miller, R. D., Moyer, E. T., Huang, H. and Überall, H. A comparison between the boundary element method and the wave superposition approach for the analysis of the scattered fields from rigid bodies and elastic shells, Journal of the Acoustical Society of America, 89 (5), , (1991). 6. Farassat, F. and Myers, M. K. Extension of Kirchhoff s formula to radiation from moving surfaces, Journal of Sound and Vibration, 123 (3), , (1988). 7. Myers, M. K. and Hausmann, J. S. Computation of acoustic scattering from a moving rigid surface, Journal of the Acoustical Society of America, 91 (5), , (1992). 8. Marburg, S., Lösche, E., Peters, H. and Kessissoglou, N. Surface contributions to radiated sound power, Journal of the Acoustical Society of America, 133 (6), , (2013). 9. Williams, E. G. Convolution formulations for non-negative intensity, Journal of the Acoustical Society of America, 134 (2), , (2013). 10. Williams, E. G. Supersonic acoustic intensity, Journal of the Acoustical Society of America, 97 (1), , (1995). 11. Williams, E. G. Supersonic acoustic intensity on planar sources, Journal of the Acoustical Society of America, 104 (5), , (1998). 12. Marburg, S. and Nolte, B. Computational Acoustics of Noise Propagation in Fluids - Finite and Boundary Element Methods, Chapter 0 - A unified approach to finite and boundary element discretization in linear time-harmonic acoustics, Springer-Verlag, Berlin, Heidelberg, pp. 1 34, (2008). 13. Peters, H., Kessissoglou, N. and Marburg, S. Enforcing reciprocity in numerical analysis of acoustic radiation modes and sound power evaluation, Journal of Computational Acoustics, 20 (3), , (2012). 14. Naghshineh, K. and Koopmann, G. H. Active control of sound power using acoustic basis functions as surface velocity filters, Journal of the Acoustical Society of America, 93 (5), , (1993). 15. Chen, P. T. and Ginsberg, J. H. Complex power, reciprocity, and radiation modes for submerged bodies, Journal of the Acoustical Society of America, 98 (6), , (1995). 16. Marburg, S. Efficient optimization of a noise transfer function by modification of a shell structure geometry - Part I: Theory, Structural and Multidisciplinary Optimization, 24 (1), 51 59, (2002). 17. Marburg, S. and Schneider, S. Influence of element types on numeric error for acoustic boundary elements, Journal of Computational Acoustics, 11 (3), , (2003). 8 ICSV22, Florence, Italy, July 2015