Smooth Convex Surface Research Articles

An Overview of Creeping Wave Formulation and Its Relationship to Surface Waves

Interface phenomena in the propagation of plane or spherical acoustic waves over plane interfaces between two media are well known; the corresponding “lateral” and “Schmidt head waves” were studied theoretically and experimentally. Less well understood are the analogous effects at curved surfaces, although considerable progress was made here in the last few years. For impenetrable obstacles of simple shape (cylinders, spheres), Franz's theory of circumferential waves, based on the Watson transformation, can be directly taken over from the electromagnetic to the acoustic case (Überall, Nussenzveig); the corresponding “creeping waves” can be shown to emerge from the conventional normal-mode solution (Rudgers), and their existence was demonstrated experimentally (Harbold and Steinberg; Neubauer; Wright). For general smooth convex surfaces, Levy's and Keller's theory leads to similar “diffracted surface rays.” This may be generalized to penetrable elastic media (Keller and Karal; Chen; Rulf), and made quantitative for simple shapes (Beckmann and Franz; Mishra; Streifer and Kodis; Doolittle, Überall, and Uginčius; Nussenzveig; Ludwig). Transmitted waves (Brill and Überall) complicate the experimental search for surface waves (Bunney, Goodman, and Marshall; Neubauer and Dragonette). The characteristics of surface waves may be studied independently of their mode of excitation (Viktorov; Wait; Norwood and Miklowitz; Brekhovskikh; Lapin; Junger and Feit).

Asymptotic Theory of Electromagnetic and Acoustic Diffraction by Smooth Convex Surfaces of Variable Curvature

A general method is presented for obtaining successive terms in short wavelength asymptotic expansions of the diffracted field produced by plane acoustic and electromagnetic waves incident on an arbitrary smooth convex surface. By introducing the geodesic coordinate system on arbitrary surfaces of non-constant curvature, both scalar and vector integral equations governing the surface fields are solved directly. The expressions for leading and second-order terms in the asymptotic expansion of the diffracted fields are obtained explicitly and the differences between acoustic and electromagnetic creeping waves are shown.

A high-frequency asymptote of the non-stationary diffraction problem in the illuminated region

Problems of high-frequency diffraction in convex bodies are of great theoretical and practical interest. For approximate calculations of the field in the illuminated region the method of geometric or physical optics is very often used. This is as follows. A system of rays constructed according to the laws of geometric optics is associated with the reflected wave. Then the phase of the reflected wave is taken to be equal to the length of the ray arriving at the given point, and its amplitude determined from the spread of the radial tubes of the given system of rays. The results obtained by this method agree fairly well with experimental data when the incident wave is of sufficiently high frequency. It is therefore usually considered that the approximation of geometric optics gives the first term of the asymptotic expansion of the exact solution in inverse powers of k ( k is the wave number). There have recently been numerous attempts to find a rigorous mathematical proof of this statement. Some results achieved in this direction for stationary diffraction were reported at the Second All-Union Symposium on Wave Diffraction in Gerky in June 1962. No analogous results for the nonstationary diffraction of waves were obtained. In this paper we consider non-stationary high-frequency diffraction on a smooth convex surface going to infinity. We obtain the expansion of the solution in inverse powers of the wave number (or frequency) of the incident wave in the illuminated region and prove a theorem which states that the resulting expansion is the asymptotic expansion of the exact solution.