POINT SOURCE DIFFRACTION BY AN ABSORBING STRIP IN A MOVING FLUID

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 4, Number 4, Fall 1996 POINT SOURCE DIFFRACTION BY AN ABSORBING STRIP IN A MOVING FLUID S. ASGHAR, TASAWAR HAYAT AND M. AYUB ABSTRACT. The scattering of an acoustic wave by a strip in the presence of a moving fluid due to a point source is studied. The problem is solved by using integral transforms, the Wiener-Hopf technique and asymptotic methods. The diffracted field from the two edges of the strip and a field due to the interaction of the two edges are obtained. The physical interpretation of the results is then discussed. 1. Introduction. Much interest has been shown in recent years to the problem of noise reduction. An effective method to achieve this is to use absorbing barriers. However, in the case of noise radiated by aeroengines and inside wind tunnels, it is necessary to discuss the acoustic diffraction in the presence of moving fluids. The absorbing half-plane was first considered by Rawlins [a], who discussed the diffraction of a line source of sound by an absorbing half-plane in a moving fluid. Asghar [I]extended this analysis to a finite barrier and calculated the diffracted field by the two edges of the barrier. In this paper we consider the diffraction of a spherical acoustic wave emanating from a point source by a finite absorbing barrier in a fluid moving at subsonic speed. This consideration of the point source is important in the sense that point sources are regarded as better substitutes for real sources. The integral transforms and the Wiener- Hopf technique [7] are employed to obtain the integral representation of the diffracted field. The analytic solution of the integral appearing in the diffracted field is obtained using steepest descent method [3]and the far field is presented. The results for still air can be recovered easily by putting M = Formulation of the problem. We consider the scattering of an acoustic wave from a finite plane occupying a place y = 0, -1 5 x 5 0. The plane is assumed to be of negligible thickness and Accepted for publication on December 2, Copyright Rocky Mountain Mathematics Consortium 327

2 328 S. ASGHAR, T. HAYAT AND M. AYUB satisfying absorbing boundary conditions p - u,z, = 0 [5] on both sides of its surface. Here p is the surface pressure and u, the normal component of the perturbation velocity, Z, is the acoustic impedance of the surface and n a normal pointing from the fluid into the surface. The whole system is now assumed to be in a fluid moving with subsonic velocity U parallel to the x-axis. The perturbation velocity u of the irrotational sound wave can be written as u = grad$. The resulting pressure in the sound field is given by: where po is the density of the undisturbed stream. We shall restrict our attention to the time harmonic variations e-'wt (w is the frequency) and suppose there is a point source at (xo, yo, 20). Thus the convective wave equation satisfied by $ in the presence of the point source is subject to the boundary conditions = 6(a: - ZO)~(Y - YO)~(Z - zo), vj(x, O+, 2) = $(Go-, 2) x < - I, x > 0, where k = w/c, p = poc/z,, c is the velocity of sound, and M = U/c is the Mach number. For subsonic flow IMI < 1 and Rep > 0. In addition, we insist that $ represents an outward traveling wave as (x2 + y2 + z2)ll and satisfy the normal edge condition at the boundary discontinuity (see Noble [7, Chapter Solution of the problem. The Fourier transform and its inverse over the variable z are defined as 00 (4) d(x, v, C) = J $(x, 9, x)e-ikcz dx, -00 k +(x, P, 2) = i;; / 0 (~, Y, ~ ) e ~ ~ c ~ d ~ -

3 POINT SOURCE DIFFRACTION 329 In equation (4) the transform parameter is taken conveniently to be kt, 5 is non-dimensional and k(= wlc) is the wave number. For analytic convenient, it has been assume that w has a small positive imaginary part. The decomposition (4) is common in other field theories as well [6, 41. Transforming equation (1) and the boundary conditions (2) and (3) with respect to z by using equation (4), we obtain Since we are dealing with subsonic flow (IMJ < I), we introduce the following substitutions Using these substitutions, equations (5) to (7) take the form

4 S. ASGHAR, T. HAYAT AND M. AYUB where and y2 = [ I- e2(1 - M2)], In equation (8), Q can be written as the sum of QinC and Qdiff, where Qinc is the solution of inhomogeneous equation (8) and Qdiff satisfies the corresponding homogeneous equation. We observe that the mathematical problem formulated in the transformed plane is the same as in the two-dimensional case [I] except that K2y2 replaces K2 and "6" replaces "a" in the wave equation. Thus the solution of the boundary value problem consisting of equations (8)-(10) can be obtained by employing the procedure used by the author in reference [I.]. In order to avoid repetition, the mathematical details are omitted and the diffracted field in <-plane is directly given by: (11) qdiff (x, y, <) = Qsep (x, y, <) + Qint(x, y, 5). In equation (11) ie-ikm(x-x~) Qsep (x7yl<) = dsk(l - ~2)1/2(RWo)'/2 3 1 (- Ky cos 8) ie-ikm(x-xo) 'int(x'v'o = dnk(l - ~ 2 ) 1 /(R&,)l/ (- Ky cos 19)

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