Reflection and Transmission of Sound by Thin Curved Shells

Abstract

In this paper the reflection and transmission of sound by thin curved shells, as well as several related problems, are treated mathematically. First, one derives an inhomogeneous integral equation for the sound field in an infinite medium containing a thin curved shell of different material. The solution of the integral equation is then reduced, approximately, to the evaluation of a surface integral not too different from that obtained in the usual Kirchhoff diffraction theory. The integral is evaluated approximately and gives expressions for the pressure waves reflected from and transmitted through a thin curved shell. The reflection and transmission coefficients of the shell are obtained from these expressions. It is found that the reflection coefficient can be expressed as the product of a geometrical factor, a phase-cancellation factor, and a reflectivity factor. When the reflection problem for a rigid obstacle is solved with the aid of the assumptions of the Kirchhoff diffraction theory, the reflection coefficient thus obtained is the product of the geometrical and phase factors of the previous solution. The geometrical factor alone is obtained as the reflection coefficient when the reflection problem is solved exactly by geometrical acoustics. The agreement among the various solutions may be considered as a partial justification for the Kirchhoff theory as well as for geometrical acoustics. The Kirchhoff method is also applied to the problem of refraction at a curved surface. From the solutions of the reflection and refraction problems, the laws of reflection and refraction for curved surfaces are obtained. In addition, the mirror and lens laws, the conditions for point images, and the change of phase at a focus are obtained.