Propagation of wave packets along intensive simple waves

TL;DR

This paper develops a theoretical framework for analyzing high-frequency wave packet propagation along simple-wave background flows in hydrodynamics, extending previous results and applying to KdV waves.

Contribution

It introduces a unified approach combining Hamilton and Hopf equations to describe wave packet dynamics on arbitrary simple-wave backgrounds.

Findings

Extended previous models to general simple-wave flows.

Derived a single ODE for background amplitude at wave packet location.

Applied the theory to KdV equation 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 Ref.~\cite{ceh-19} 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.