Applications to radiation and scattering of a wave front construction using the involute of circular virtual caustics

Abstract

From the classical theory of plane curves applied to two-dimensional wave propagation in uniform media, it is known that the wave fronts are involutes of caustics or virtual caustics. (The involute of a curve C is constructed by the unwinding of a taut thread tangent to C.) in the present research, the involute construction is applied to wave fronts radiated into water by surface guided elastic waves (SEW) on scatterers having a uniform circular profile. Examples of SEW include circumferential leaky Rayleigh or Lamb waves on solid or hollow circular cylinders, respectively, with an outer radius = a. The outgoing wave appears to radiate from a virtual caustic that is a circle of radius sl = ac/cl, where cl is the phase velocity along the surface of the SEW and c is the sound speed in water. Consequently, the radiated wave fronts are the portion with r > a of the involute of a circle of radius sl. When the SEW are only weakly dispersive, radiated bursts imaged by schlieren photograph [W. G. Neubauer, J. Acoust. Soc. Am. 45, 1134–1144 (1969)] approximate the location of wave fronts. In the present research such images are found to be involutes of circles of radius sl. Applications to scattering from spherical shells and to the localization principle were also investigated. [Work supported by ONR.]