A numerical comparison of the Westervelt equation with viscous attenuation and a causal propagation operator

The Westervelt wave equation can be used to describe non-linear propagation of finite amplitude sound. If one assumes that the medium can be treated as a thermoviscous fluid, a loss mechanism can be incorporated, but such a loss mechanism is not adequate if the medium is dispersive. In order to accurately describe pulse propagation in a dispersive medium the Westervelt equation must incorporate attenuation and dispersion correctly. Szabo has shown that the effects of frequency dependent attenuation and dispersion can be included by the use of a causal time-domain propagation factor (TDPF) which is obtained from a corresponding time domain convolution operator. In previous work the TDPF has been successfully employed in the linear wave equation for both isotropic and non-isotropic media, and the authors recently carried out a comparison of numerical solutions, in one dimension, to the Westervelt equation using the TDPF with those obtained using a traditional loss mechanism for a themoviscous fluid. These computations showed that the TDPF correctly incorporated the full dispersive characteristics of the media, and that the results may differ significantly from those obtained using the traditional loss term. In this work the problem of propagation of ultrasonic acoustic energy through human tissue in two dimensions is solved numerically using the Westervelt equation with the TDPF, and comparisons are made with computations treating the human tissue as a thermoviscous fluid. The equations are solved using the method of finite differences.