Abstract
We relate together different models of non linear acoustic in thermo-elastic media as the Kuznetsov equation, the Westervelt equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) and estimate the time during which the solutions of these models keep closed in the \begin{document}$ L^2 $\end{document} norm. The KZK and NPE equations are considered as paraxial approximations of the Kuznetsov equation. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. Aiming to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Kuznetsov equation in a half-space with periodic in time initial and boundary data) are obtained.
Highlights
Where the pressure p is given by the state law p = p(ρ, S)
For the acoustical framework the wave motion is supposed to be potential and the viscosity coefficients are supposed to be small in terms of a dimensionless small parameter ε > 0, which characterizes the size of the perturbations near the constant state (ρ0, 0, S0, T0)
The interest to study how closed are the solutions of the general model of the nonlinear wave motion, described by the Kuznetsov equation, and of simplified models with more particular area of application is naturally motivated by the questions about the accuracy of the approximations and of a comparative analysis of the solutions of these models
Summary
In this article we derive the KZK and the NPE equations from the Kuznetsov equation just performing the corresponding paraxial change of variables and show that the Westervelt equation can be viewed as an approximation of the Kuznetsov equation by a nonlinear perturbation. The interest to study how closed are the solutions of the general model of the nonlinear wave motion, described by the Kuznetsov equation, and of simplified models with more particular area of application (such as the KZK equation and the NPE equation which are valid only with additional assumptions on the wave propagation describing by the paraxial changes of variables) is naturally motivated by the questions about the accuracy of the approximations and of a comparative analysis of the solutions of these models. We present the structure of the paper and its mains results in the subsection
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