A computationally efficient elastic wave model for media with power-law absorption

The absorption of ultrasound waves in biological tissue has been experimentally shown to follow a frequency power law. This type of behaviour can be modelled using fractional derivative operators. However, previous elastic wave equations are based on fractional derivatives that are non-local in time. This makes them difficult to solve using standard numerical techniques in a memory efficient manner. Here, a fractional Kelvin-Voigt model is derived based on the fractional Laplacian. This is obtained by splitting the particle velocity into compressional and shear components using a dyadic wavenumber tensor. This allows the temporal derivatives to be replaced with spatial derivatives using the lossless dispersion relation with the appropriate compressional or shear wave speed. If the spatial gradients are computed using the Fourier collocation spectral method, this results in a computationally efficient elastic wave model that can account for arbitrary power law absorption of both compressional and shear waves.