LAGRANGIAN AND HAMILTONIAN ASPECTS OF WAVE MIXING IN NONUNIFORM MEDIA- WAVES ON STRINGS AND WAVES IN GAS-DYNAMICS

Hamiltonian and Lagrangian perturbation theory is used to describe lin ear wave propagation in inhomogeneous media. In particular, the proble ms of wave propagation on an inhomogeneous string, and the propagation of sound waves and entropy waves in gas dynamics in one Cartesian spa ce dimension are investigated. For the case of wave propagation on an inhomogeneous heavy string, coupled evolution equations are obtained d escribing 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 w ave interact with each other via gradients in the background flow The wave coupling: coefficients in the gas-dynamical case depend on the gr adients of the Riemann invariants R+/- and entropy S of the background flow. Coupled evolution equations describing the interaction of the d ifferent wave modes are obtained by exploiting the Hamiltonian and Poi sson-bracket structure of the governing equations. Both Lagrangian and Clebsch-variable formulations are used. The similarity of the equatio ns to equations obtained by Heinemann and Olbert describing the propag ation of bidirectional Alfven waves in the solar wind is pointed out.