Weakly nonlinear waves in fluids of low viscosity: Lagrangian and Eulerian descriptions

Abstract

Acoustics equations are derived in low-viscosity newtonian fluids, when nonlinear effects are of first order relative to a small dimensionless parameter ∊, which is a measure of the Mach number. Another small dimensionless parameter ζ is used to define low-viscosity precisely. In this context, using conservation of mass and of linear momentum, one derives governing equations for complex motions (simultaneous forward and backward propagation) and simple motions (forward propagation only). Propagation equations are obtained for four physical quantities (particle displacement, particle velocity, mass density, and pressure) in eulerian as well as in lagrangian form. For simple waves, the equations for particle velocity, mass density and pressure are found to be of the Burgers type; that for the displacement is not of the Burgers type. Consistent with the weak nonlinearity, proper boundary conditions for the simple-wave equations are derived in both eulerian and lagrangian forms; these new results are expressed only in terms of the source displacement and the fluid constants.