Gm. Webb et al., LAGRANGIAN AND HAMILTONIAN ASPECTS OF WAVE MIXING IN NONUNIFORM MEDIA- WAVES ON STRINGS AND WAVES IN GAS-DYNAMICS, Journal of Plasma Physics, 60, 1998, pp. 341-382
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.