Sound Radiation from a Moving Rigid Sphere

Abstract

Most treatments of moving spatially extended sources in some way model the source as a point. Thus, in effect they approximate the solution by the first term in a multipole expansion. The latter is similar to a Taylor expansion in the sense that it is valid when evaluated at a retarded time at which the velocity has time derivatives of all orders [T. J. Eisler, J. Acoust. Soc. Am. 52, 210 (1972)]. In contrast, the present treatment is similar to an eigenfunction expansion which can be used for less regular functions, for example, a velo-city-versus-time curve which has a corner. We find that such singularities cause a burst of sound radiation. Our method, which is applicable to source distributions which move as a rigid sphere, is based on an expansion theorem for spherical wave functions [R. A. Sack, J. Math. Phys. 5, 252 (1964)]. This is used to expand the integrand of the retarded solution of the wave equation, and the resulting expansion can be summed in certain special cases. The solution is found to depend on the history of the source over an interval of finite duration which includes the retarded time of the center. The method has been applied to a moving rigid sphere using the theory of Ffowcs-Williams and Hawkings [Phil. Trans. Roy. Soc. London A264, 321 (1969)].