A New Expansion for the Velocity Potential of a Piston Source

Abstract

The Rayleigh surface integral, giving the velocity potential for a plane piston source surrounded by an infinite rigid flange, reduces to a line integral when the coordinates are suitably chosen. As shown by Schoch, for points within the geometrical cylinder whose base is formed by the piston surface, the line integral is expressible as a plane wave term plus a “perturbation” integral. For external points, a different integral results. In the present work, these two complementary expressions are evaluated for a circular piston, as series of half-integral order Hankel functions in kz and polynomials in x/a; k is the propagation constant, a the piston radius, z the axial and x the radial coordinate of a field point. The resulting rigorous equation (valid for points not on the piston surface) converges for any value of ka, provided z>a. For large values of kz, where asymptotic formulas apply, the expression assumes a particularly simple form. Sample calculations have been made for ka = 10, z = 10a and ka = 50, z = 50a. Also, an approximate expansion has been derived which may be more useful than the rigorous result in paraxial regions.