Propagation of wave packets along intensive simple waves

Abstract

We consider propagation of high-frequency wave packets along a smooth evolving background flow whose evolution is described by a simple-wave type of solutions of hydrodynamic equations. In geometrical optics approximation, the motion of the wave packet obeys the Hamilton equations with the dispersion law playing the role of the Hamiltonian. This Hamiltonian depends also on the amplitude of the background flow obeying the Hopf-like equation for the simple wave. The combined system of Hamilton and Hopf equations can be reduced to a single ordinary differential equation whose solution determines the value of the background amplitude at the location of the wave packet. This approach extends the results obtained in the paper by Congy et al. [J. Fluid Mech. 875, 1145 (2019)] for the rarefaction background flow to arbitrary simple-wave type background flows. The theory is illustrated by its application to waves obeying the KdV equation.