Investigation into angular and frequency dependence of scattering matrices of elastodynamic scatterers

The scattering behaviour of a finite-sized elastodynamic scatterer in a homogeneous isotropic medium can be encapsulated in a scattering matrix (S-matrix) for each wave mode combination. Each S-matrix is a continuous complex function of 3 variables: incident wave angle, scattered wave angle and frequency. In the paper, the S-matrices for various scatterers (circular holes, straight smooth cracks, rough cracks and 4 circular holes in an area of interest) are investigated. It is shown that, for a given scatterer, the continuous data in the angular dimensions of an S-matrix can be represented to a prescribed level of accuracy by a finite number of complex Fourier coefficients. The finding is that the number of angular orders required to characterise a scatterer is a function of scatterer size and is related to the Nyquist theorem. The variation of scattering behaviour with frequency is examined next and is found to show periodic oscillation with a period which is a function of scatterer size and its geometry. The shortest period of these oscillations indicates the maximum frequency increment required to accurately describe the scattering behaviour in a specific frequency range. Finally, the maximum angular order and frequency increments for the chosen scatterers in a specific frequency range are suggested.