Angular and frequency behaviour of elastodynamic scattering from embedded scatterers.

The elastodynamic scattering behaviour of a finite-sized scatterer in a homogeneous isotropic medium can be encapsulated in a scattering matrix (S-matrix) for each wave mode combination. In a 2-dimension (2D) space, each S-matrix is a continuous complex-valued function of 3 variables: incident wave angle, scattered wave angle and frequency. In this paper, the S-matrices for various 2D scatterer shapes (circular voids, straight cracks, rough cracks and a cluster of circular voids) are investigated to find general properties of their angular and frequency behaviour. For all these shapes, it is shown that the continuous data in the angular dimensions of their S-matrices can be represented to a prescribed level of accuracy by a finite number of complex-valued Fourier coefficients that are physically related to the angular orders of the incident and scattered wavefields. It is shown mathematically that the number of angular orders required to represent the angular dimensions of an S-matrix at a given frequency is a function of overall scatterer size to wavelength ratio, regardless of its geometric complexity. This can be interpreted as a form of the Nyquist sampling theorem and indicates that there is an upper bound on the sampling interval required in the angular domain to completely define an S-matrix. The variation of scattering behaviour with frequency is then examined. The frequency dependence of the S-matrix can be interpreted as the Fourier transform of the time-domain impulse response of the scatterer for each incident and scattering angle combination. Depending on the nature of the scatterer, these are typically decaying reverberation trains with no definite upper bound on their durations. Therefore, in contrast to the angular domain, there is no lower bound on the sampling interval in the frequency domain needed to completely define an S-matrix, although some pragmatic solutions are suggested. These observations may help for the direct problem (computing ultrasonic signals from known scatterers efficiently) and the inverse problem (characterising scatterers from measured ultrasonic signals).