The near-field singularity predicted by the spiral Green’s function in acoustics and electrodynamics

Abstract

The retarded solution of the wave equation for a point source in circular motion, whose speed exceeds the wave speed, is singular on a spiralling tube-like surface that is at rest in the rest frame of the point source. When solving the wave equation for a corresponding extended source, therefore, we are faced with integrals over the volume of the source which are improper and need to be handled either with the aid of the theory of generalized functions or by Hadamard’s method of finite parts. In this paper, after isolating the finite part of the gradient of the retarded potential due to a rotating extended source, we calculate the asymptotic values of the coefficients in its Fourier representation and show that the radial component of this gradient does not remain finite at those points within the source which move with the wave speed, and so lie on the boundary of the domain of hyperbolicity of the equation of the mixed type to which the wave equation in this case reduces. This latter singularity arises because the problem in question, though well posed physically, is in fact mathematically ill posed.