A New Expansion for the Velocity Potential of a Piston Source

Abstract

The Rayleigh formula giving the velocity potential for a piston source surrounded by an infinite rigid flange is a surface integral which reduces to a line integral when the coordinates are suitably chosen, as shown by Schoch and Stenzel. For points within the geometrical cylinder whose base is formed by the piston surface, the line integral was expressed by them as a plane wave term plus a “perturbation” integral. For external points a different integral resulted. In the present work these two complementary expressions are evaluated as series of half-order Bessel functions in kz and polynomials in r/a; k is the propagation constant, a the piston radius, z and r the cylindrical coordinates of a field point. The resulting rigorous equation (valid for all points not on the piston surface) appears to converge for any value of ka and distance z from the source. The theoretical sound field on the surface of the cylinder and on the piston surface has also been investigated. Approximate expansions which under some circumstances are more useful than the rigorous result have been derived for paraxial regions, both from the perturbation integral and from an integral solution of the wave equation given by Bateman.