A common procedure for estimating the apparent width of a ray in volume propagation is to construct the locus of defective paths having a phase shift that deviates from that of the true ray by π radians [Yu A. Kravtsov and Yu I. Orlov, Sov. Phys. Uspekhi 23, 750–760 (1980); P. L. Marston, Phys. Acoust. 21, 1–234 (1992)]. An analogous construction is described for the effective surface region for the high-frequency coupling of sound with a leaky wave on a curved surface along a given ray. Results are shown for right circular cylinders and spheres. For defective paths in the plane of incidence of the ray, the resulting Fresnel angular half-width of the patch is approximately [2π/(ka cos θl)]1/2 radians where a is the radius of the surface, sin θl=c/cl and cl is the leaky wave phase velocity. The analysis is relevant to estimating the degree to which ray coupling processes can be considered to be local processes. It is also relevant to the anticipation of the transition to creeping wave behavior when cl becomes sufficiently small for the Fresnel region to touch the edge of the scatterer. Defective paths outside the plane of incidence are also considered for a right circular cylinder. The Fresnel patch associated with a leaky ray to (or from) a specified point on the cylinder is approximately elliptical and the analysis is relevant to radiation as well as to scattering.
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Philip L. Marston (1994) studied this question.