A generalized Westervelt equation for nonlinear medical ultrasound

A model equation is derived for nonlinear medical ultrasound. Unlike the existing models, which use spatial coordinates, material coordinates are used and hence a model for a heterogeneous medium is able to be derived. The equation is a generalization of the Westervelt equation, and includes the nonlinearity, relaxation, and heterogeneity of soft tissue. The validity of the generalized Westervelt equation as a model equation for a Piola-Kirchoff acoustic pressure and as an equation for the acoustic pressure is discussed. In the second case it turns out that the model follows from two geometric approximations which are valid when the radius of curvature of the phase fronts is much larger than the particle displacements. The model is exact for plane waves and includes arbitrary nonlinearity in the stress-strain relation.