An expansion theorem applicable to problems of wave propagation in an elastic half-space containing a cavity

This paper is principally concerned with the two-dimensional time harmonic vibrations of an elastic half-space y ≥ 0, containing a submerged cavity in the form of an infinite circular cylinder. Two sequences of line source potentials are obtained which are singular along the axis of the cylinder, satisfy the free surface conditions on y = 0, and represent outgoing waves at infinity. (The radiation condition.) It is proved that any solution of the governing equations which satisfies the free surface conditions and consists of outgoing waves at infinity is expansible as a sum of these fundamental source potentials, with coefficients to be determined from the boundary conditions on the cylinder only. The requirement of outgoing waves is carefully discussed and it is shown that the conditions taken give rise to, and are satisfied by, potential fields which would be regarded intuitively as representing outgoing waves.