Akustik I / Acoustics I by Philip M. Morse, K. Uno Ingard (auth.), S. Flügge (eds.)

By Philip M. Morse, K. Uno Ingard (auth.), S. Flügge (eds.)

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1). 1) can be modified, by methods by now familiar, to correspond to this case. We suppose the incident wave vector k i is in the x, z plane at angle to the z axis and that the measurement point per, {}, q;) is a great distance from the origin. The area A, around the origin, has admittance p, whereas the rest of the x, y plane has admittance Po. Using the asymptotic forms of Eq. 1), the 'pressure at Pis P(r) ~ Pi eik(-,cosHxsint} + Pi (~~:: ~;) eik(,cosHxsint} _ e"" ( 2ik cosD 4nr cosD+fJ )ff(p _P ) P (x 0 0' y) e-,"(xosin6costp+Yosin6sintp) dx dy.

In either case, it is usually easiest to adjust the GREEN'S function G (r Iro) so that the boundary condition on Br is that the normal gradient of G is zero there. 6) G(rlr~)vo(r~)dso over B,. where Vo is the normal velocity amplitude at frequency wj2:n:. 6) to calculate the radiated wave. 6) for the time dependent wave, using Eq. 7) for the GREEN'S function for the infinite domain. 7) This is the well-known retarded potential solution of the wave equation. It can also be used when the boundaries are closer than infinity, by suitably modifying G to fit appropriate boundary conditions and by adding the surface integrals of Eq.

If the difference (P - Po) is not large we can consider the factor P in the integral to be approximately equal to the plane-pIus-reflected pressure at the surface (Born approximation), so that the scattered wave P.. for large r, is Pi (eik'/r) X X/(ki' k s ), with -ikcoscpcosD ) i,,· .. 2) -ikcoscpcosD )d A 12) -+ n(coscp+fJ)(cosD+fJ) b(xo' Yo xodyo ( ~JI. If If 0 ) A where the limiting second form is valid when the dimensions of area A are small compared to the wavelength (2n/k). J, p,. ro = k Xo (sin - sin{} cos q;) - k Yo sin{} sin q;.

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