A weakly nonlinear wave equation for damped acoustic waves with thermodynamic non-equilibrium effects

TL;DR

This paper derives a weakly nonlinear wave equation for damped acoustic waves incorporating thermodynamic non-equilibrium effects, using a novel discontinuous Lagrangian approach based on Hamilton's principle, extending Kuznetsov's equation.

Contribution

It introduces a new derivation method for nonlinear acoustic wave equations using a discontinuous Lagrangian framework, accounting for thermodynamic non-equilibrium effects.

Findings

Derived a generalized wave equation including thermodynamic effects.

Showed the applicability of Hamilton's principle to viscous flow.

Extended Kuznetsov's equation with additional thermodynamic terms.

Abstract

The problem of propagating nonlinear acoustic waves is considered; the solution to which, both with and without damping, having been obtained to-date starting from the Navier-Stokes-Duhem equations together with the continuity and thermal conduction equation. The novel approach reported here adopts instead, a discontinuous Lagrangian approach, i.e. from Hamilton's principle together with a discontinuous Lagrangian for the case of a general viscous flow. It is shown that ensemble averaging of the equation of motion resulting from the Euler-Lagrange equations, under the assumption of irrotational flow, leads to a weakly nonlinear wave equation for the velocity potential: in effect a generalisation of Kuznetsov's well known equation with an additional term due to thermodynamic non-equilibrium effects.