Leaky wave excitation on concave surfaces and hyperbolic Fresnel zones

Abstract

Leaky waves are waves guided by elastic structures having phase velocities exceeding the speed of sound in the surrounding fluid. When analyzing the excitation of leaky waves on smoothly curved elastic structures it is often helpful to determine the Fermat paths. Those paths satisfy Fermat’s principle for the combined acoustic propagation in the fluid and guided propagation on the structure. It is also helpful to determine the shape of the Fresnel zones associated with deviations from Fermat paths. That information is directly useful for calculating guided leaky wave amplitudes [P. L. Marston, ‘‘Spatial approximation of leaky wave surface amplitudes for three-dimensional high-frequency scattering: Fresnel patches and applications to edge-excited and regular helical waves on cylinders,’’ J. Acoust. Soc. Am. 102, 1628–1638 (1997)]. The Fresnel zones associated with the external excitation of helical leaky waves on submerged circular pipes or cylinders were previously shown to be elliptical. The resulting leaky wave amplitudes are in agreement with amplitudes derivable by other methods and with experiments. In the present work, the analysis is generalized to excitation of leaky waves by ultrasound incident from the concave side of cylindrical shells or pipes. Even though the resulting Fresnel zones are approximately hyperbolic in shape, ray approximations for the amplitude are applicable when expressed using the Hessian of the phase deviation. [Supported by the Office of Naval Research.]