On the dimensionality of elastic wave scattering within heterogeneous media

Elastic waves scatter when the wavelength becomes comparable to random spatial fluctuations in the elastic properties of the propagation medium. It is postulated that within the long-wavelength Rayleigh regime, the scattering induced attenuation obeys a D = 1,2,3 dimensional dependence on wavenumber, k^D+1, whilst within the shorter-wavelength stochastic regime, it becomes independent of the dimensions and thus varies as k². These predictions are verified numerically with a recently developed finite element method in three dimensions (3D), two dimensions (2D), and one dimension (1D), for the example of ultrasonic waves propagating within polycrystalline materials. These findings are thought to be practically useful given the increasing uptake of numerical methods to study highly scattering environments which exhibit multiple scattering, but often remain limited to 2D given computational constraints. It is hoped that these results lay the groundwork for eventually producing computationally efficient 2D simulations that are representative of 3D.