The radiating near-field asymptotics of a normal time-harmonic circular ultrasonic transducer in an elastic half-space

Abstract

The modern diffraction theory is applied to analyze the radiating near zone of a normal time-harmonic circular transducer directly coupled to a homogeneous and isotropic solid. The two-tier asymptotic approach is used first to find the far-field asymptotics of a point source and then the radiating near-field asymptotics of the circular transducer. All the known ray-theoretical solutions for body waves, such as the plane P wave and the toroidal edge waves, both P and S, are obtained. The non-ray-theoretical solutions, such as the edge waves present in the axial boundary layer and the total field inside the penumbral boundary layer, are also described. The asymptotic formulas produced all have immediate physical interpretation, give explicit dependence on model parameters and involve in geometrical region elementary functions and inside boundary layers, well-known special functions. It is argued that asymptotic results may be used to write computer codes which simulate the radiating near field of the circular transducer orders of magnitude faster than the exact numerical schemes, but more accurately than other known approximations.