Cylindrical wave expansions of the fundamental singular potentials of an elastodynamic half‐plane

The singular solutions of Lamb's problem are of great importance to the theory of scattering of elastic waves from defects lying in or close to a free surface, and specifically the scattering of Rayleigh waves from a perpendicular edge crack. Such problems occur in the fields of NDT and seismology. The singular solutions correspond to line sources of P and SV waves within a homogeneous, isotropic, linearly elastic half‐plane. They have until now been expressed as Fourier integrals, but, by considering the continuation of these functions into the whole plane, we derive convergent expansions for them in series of cylindrical waves. The coefficients are functions of the depth of the source, but the expansions are uniformly convergent as this tends to zero. The expansions provide an efficient means of evaluating the expressions arising in the integral equations or infinite system of equations of a numerical approach to the scattering problem. They also yield explicit power series about the singular points, which are required for any low‐frequency treatment. [Work supported by S.E.R.C. (UK) and DOE (USA).]