On complex waves

A widely used approximate technique for treating radiation problems involving a source-excited interface is derived from a saddle-point evaluation of the inverse transform-integral representation of the solution. The total solution is approximated to by the first saddle-point term plus residue terms in their respective wedges of contribution, i.e. space wave plus complex waves. The paper considers the well known approximation of space wave plus complex waves in the light of the modified saddle-point technique. The condition for validity of this approximation is shown to be that the numerical distance, a quantity involving both the distance from pole to saddle point and the distance from origin to observation point, should be greater in magnitude than ten. The discussion is numerically illustrated using the example of unidirectional travelling-leaky-wave and surface-wave aperture distributions. For these examples, graphs are compared of the approximate solutions and of solutions including the correction terms. Diagrams which show where in space the complex wave is a valid and significant part of the total solution are also constructed. The leaky-wave case is particularly interesting, since the correction terms are required in a rather large region of space.