An approximation to the far field and directivity of elastic wave transducers

The far field of surface mounted elastic wave transducers, as used in Non-Destructive Testing (NDT), is analyzed in order to understand the mechanisms that determine the directivity of these devices. To model the physical configuration, a moving distribution of normal traction, acting at the free surface of an elastic half-space, is used. The exact field expression consists of a spatial superposition of the elementary waves emitted by the hypothetical point sources that make up the finite source. The current aim is to derive approximate far field expressions that are physically instructive and numerically fast. This is achieved through a paraxial approximation of the elementary waves and a Taylor approximation of their arrival times. The approximate expressions thus obtained, consist of a time convolution of a paraxial wave and a transfer function. The latter represents the influence of the source geometry. Besides elucidating the mechanisms of directivity, the analysis renders a rule of thumb for synthesizing a source with a desired directivity pattern. Numerical experiments show that the approximations reduce the computational effort by a factor of 1000, and that the accuracy of the approximate far field and directivity proves excellent at distances beyond a few times the characteristic source size.