From deformation theory to a generalized Westervelt equation

Abstract

The Westervelt equation describes the propagation of pressure waves in continuous nonlinear and, eventually, diffusive media. The classical framework of this equation corresponds to fluid dynamics theory. This work seeks to connect this equation with the theory of deformations, considering the propagation of mechanical waves in nonlinear and loss-energy media. A deep understanding of pressure wave propagation beyond fluid dynamics it is required to be applied to medical diagnosis and therapeutic treatment. A deduction of a nonlinear partial differential equation for pressure waves is performed from first principles of deformation theory. The nonlinear propagation of pressure waves in tissue produces high-frequency components that are absorbed differently by the tissue, thus, distinguishing each of these modes is essential. An extension of the Westervelt equation beyond fluids media is required. In order to include the behaviour of any order harmonics, a generalization of this equation is also developed.