Leaky waves on weakly curved scatterers. II. Convolution formulation for two-dimensional high-frequency scattering

A simple high-frequency approximation is developed for leaky wave contributions to two-dimensional scattering by curved elastic surfaces. Following Bertoni and Tamir [Appl. Phys. 2, 157–172 (1973)] the method relies on general features of the Laurent expansion of the plane surface reflection coefficient R(kx) about the leaky wave pole of interest at the complex surface wave number kx=kl+iα. The formulation uses the real part kl and radiation damping rate α for leaky waves on the curved elastic surface of interest rather than an analysis of the response of any specific class of structures such as thin shells. The high frequency limit of the complex coupling coefficient Gl [see, e.g., P. L. Marston, J. Acoust. Soc. Am. 83, 25–37 (1988)] is recovered for right circular cylinders and the physical origin of the π/4 phase shift is discussed. An O(kh)−1 phase correction important for empty thin shells of thickness h is obtained in agreement with results from other approaches. The importance of the Fresnel width of the coupling region is illustrated by consideration of a cylinder with an ideal coating having an abrupt edge. The leaky wave contribution becomes proportional to a Fresnel integral having a complex argument. The integral manifests the degree to which launching of a leaky wave can be considered to be a local process. The product of α and the Fresnel width is an important parameter. The detachment of the ray to the far field is taken to be separated from its launching by more than the Fresnel width. Leaky wave contributions to scattering by surfaces of variable curvature are approximated and applications for ultrasonic beams are noted.