Transient Representation of the Somerfeld–Weyl Integral with Application to the Point Source Response from a Planar Acoustic Interface

Abstract

Point source responses from a planar acoustic and/or elastic layer boundary (as well as from a stack of planar parallel layers) are generally obtained by using as a starting point the Somerfeld-Weyl integral, which can be viewed as decomposing a time-harmonic spherical source into time-harmonic homogeneous and inhomogeneous plane waves. This paper gives a powerful extension of this integral by providing a direct decomposition of an arbitrary transient spherical source into homogeneous and inhomogeneous transient plane waves. To demonstrate with an example the usefulness of this new point source integral representation, a transient solution is formulated for the reflected/transmitted response from a planar acoustic reflector. The result is obtained in the form of a relatively simple integral and essentially corresponds to the solution obtained by Bortfeld (1962). It, however, is arrived at in a physically more transparent way by strictly superimposing the reflected/transmitted transient waves leaving the interface in response to the incident transient homogeneous and in-homogeneous plane waves coming from the centre of the point source.