Propagation of wave packets along large-scale background waves

We study propagation of high-frequency wave packets along a large-scale background wave which evolves according to dispersionless hydrodynamic equations for two variables (fluid density and flow velocity). Influence of the wave packet on evolution of the background wave is neglected, so the large-scale evolution can be found independently of the wave packet's motion. At the same time, propagation of the packet depends in essential way on the background wave and it can be considered in framework of geometric optics approximation with the use of Hamilton equations for the carrier wave number and the mean coordinate of the packet. We derive equations for the carrier wave number as a function of the parameters which describe the background wave. When they are solved, the path of the packet can be found by simple integration of the Hamilton equation. The theory is illustrated by its application to the problem of propagation of wave packets along expanding large-scale wave which evolution is described by the shallow water equations. In particular, they correspond to the dispersionless limit of the defocusing nonlinear Schr\"{o}dinger equation, and then the expanding wave can be considered as an expanding cloud of the Bose-Einstein condensate. Reflection of wave packets from upstream flows and their propagation along stationary flows are also discussed. The analytical solutions found for these particular cases agree very well with exact numerical solution of the nonlinear Schr\"{o}dinger equation.