Lie Symmetry analysis of partial differetial equations associated with the Westervelt Model in non-linear acoustics

Abstract

Sound-wave propagation in compressible media has significant non-linearities. It is for this reason, that modelling such phenomena necessitates a partial differential equation that considers these non-linear effects. That is where the Westervelt model comes into action, it is a non-linear partial differential equation used to model the propagation of high finite amplitude sound waves in non-linear acoustics, i.e., sonar systems, medical ultrasound imaging and non-destructive testing. The propagation of such waves takes place in non-linear media that exhibit thermal and viscous characteristics, e.g. human tissue. Two equations that represent the Westervelt model are considered in this work, the first one is the usual equation that has the dissipative term as the third-order temporal derivative. The second equation is where the linear wave relation has been inserted for the dissipative term. Symmetry analysis is performed on each of the models individually. This involves generating an over-determined system of linear homogeneous partial differential equations, which is solved to get the Lie point symmetries. Then, with the aid of the adjoint and commutator tables, an optimal system of sub-algebras is found and used in the similarity reductions to get the invari- ant sub-models. One-parameter Lie point groups are constructed, followed by exact invariant solutions for the sub-models, with the modified simple equation method applied to find some solitary wave solutions. Lastly, simulations in terms of 2D and 3D graphs representing the in- variant solutions are presented.