A simple and rigorous theory of leaking modes for liquid layer or SH elastic wave propagation problems is presented. By taking the frequency ω and wave-number k simultaneously as complex variables and choosing appropriate paths of integration in the ω-plane, the integration with respect to k is performed exactly using Cauchy residue theory. The remaining integration with respect to ω is then carried out by use of the Fast Fourier transform. The method is simply to apply, accurate, and computationally efficient. There are no spurious arrivals, and provided the number of points in the Fast Fourier transform can be taken sufficiently large, there are no restrictions on distance.
The theoretical results established in the paper show conclusively that complete seismograms, including all possible body waves, can be expressed simply as a superposition of modes. No branch line integrals are required, contrary to the widespread supposition in the literature.
The application of the theory is illustrated by producing complete theoretical seismograms for a model consisting of six liquid layers.