On the exact analytical solutions to equations of nonlinear acoustics

Abstract

Some different equations derived as second-order approximations to complete system of equations of nonlinear acoustics of Newtonian media (such as Lighthill-Westerwelt equation, Kuznetsov one, etc.) are usually solved numerically or at least approximately. A general exact analytical method of solution of these problems based on a short chain of changes of variables is presented in the work. It is shown that neither traveling-wave solutions nor classical soliton-like solutions obey these equations. There are three types of possible forms of acoustical pressure depending on parameters of initial equation: so-called continuous shock (or diffusive soliton), a monotonously decaying solution as well as a sectionally continuous periodic one. Obtained results are in good qualitative agreement with previously published numerical calculations of different authors.