The reflexion of a pulse within a sphere

Abstract

A pulse sets out from an interior point of a uniform compressible liquid sphere. The formal solution for the velocity at a surface point at any subsequent time is derived by operational methods. This solution, having the form of an infinite series of Legendre polynomials, with coefficients which are the ratios of Bessel functions, converges too slowly to be useful. We use a modification of a method of van der Pol & Bremmer, in which the first part of the series is transformed by Watson’s device into a series of multiple contour integrals, which are evaluated at saddlepoints. The expressions thus derived are shown to correspond term by term to the pulses whose times of arrival can be calculated by geometrical ray theory. The shape of each arriving pulse, not obtainable by ray theory, is given by the present method. It is shown that stationary time pulses which are extremals are related to stationary time pulses which are not extremals as Fourier integrals to the allied Fourier integrals. In particular, it is shown that the two singly-reflected pulses which travel within the shorter arc of a great circle (see figure 14) are so related. If the initial disturbance is given by a Heaviside unit function, the minimum time pulse arrives as a unit function, but the stationary time pulse which is not extremal arrives gradually and rises to a logarithmic infinity. This suggests an explanation of the difference observed on seismograms between the sharp arrivals ofpPand the blunt arrivals ofPP.