Low-frequency elastic-wave scattering by an inclusion: limits of applications

The present investigation considers various approximations for the problem of low-frequency elastic waves scattered by a single, small inclusion of constant elastic parameters. For the Rayleigh approximation containing both near- and far-field terms, the scattered amplitudes are investigated as a function of distance from the scatterer. Near-field terms are found to be dominant for distances up to two wavelengths, after which far-field solutions correctly describe the scattered field. At a distance of two wavelengths the relative error between the total and the far-field solution is about 15 per cent and decreases with increasing distance. Deriving solutions for the linear and quadratic Rayleigh-Born approximation, the relative error between the non-linear Rayleigh approximation and the linear and quadratic Rayleigh-Born approximation as a function of the scattering angle and the parameter perturbation is investigated. The relative error reveals a strong dependence on the scattering angle, while the addition of the quadratic term significantly improves the approximation for all scattering angles and parameter perturbations. An approximation for the error caused by linearization of the problem, based entirely on the perturbations of the parameters from the background medium, and its validity range are given. We also investigate the limit of the wave parameter for Rayleigh scattering and find higher values than previously assumed. By choosing relative errors of 5 per cent, 10 per cent and 20 per cent between the exact solution and the Rayleigh approximation, we find the upper limits for the parameter kpR to be 0.55, 0.7 and 9.9, respectively.