Lagrangian and Hamiltonian aspects of wave mixing in non-uniform media: waves on strings and waves in gas dynamics

Abstract

Hamiltonian and Lagrangian perturbation theory is used to describe linear wave propagation in inhomogeneous media. In particular, the problems of wave propagation on an inhomogeneous string, and the propagation of sound waves and entropy waves in gas dynamics in one Cartesian space dimension are investigated. For the case of wave propagation on an inhomogeneous heavy string, coupled evolution equations are obtained describing the interaction of the backward and forward waves via wave reflection off gradients in the string density. Similarly, in the case of gas dynamics the backward and forward sound waves and the entropy wave interact with each other via gradients in the background flow. The wave coupling coefficients in the gas-dynamical case depend on the gradients of the Riemann invariants R± and entropy S of the background flow. Coupled evolution equations describing the interaction of the different wave modes are obtained by exploiting the Hamiltonian and Poisson-bracket structure of the governing equations. Both Lagrangian and Clebsch-variable formulations are used. The similarity of the equations to equations obtained by Heinemann and Olbert describing the propagation of bidirectional Alfvén waves in the solar wind is pointed out.