Angular and modal equipartitioning of elastic waves in scattering media: An illustration based on energy transport

We illustrate the angular and modal equipartitioning of elastic waves in scattering media using two-dimensional elastic radiative transfer equations . To solve these equations, we decompose the P and S specific intensities into direct and scattered components. We handle the direct component analytically, and derive integral equations for the scattered components of the P and S specific intensities. We construct a time-stepping algorithm with which we evolve the scattered components of the specific intensities numerically in time. We handle the advection of P and S energy analytically at the computational grid points and use numerical interpolation to deal with advection terms which do not lie on the grid points. We test this algorithm for a pure P source and a double couple, which radiates both P and S energy. We compare our numerical solutions against known approximations and find good agreement. We use this algorithm to illustrate the local behavior of equipartitioning over wave modes and angular directions. We find that both types of equipartitioning are a function of space and time, depending on the extent of scattering. This local behavior must be taken into account when studying diffusion and equipartitioning of elastic waves.