Interaction of Linear Modulated Waves and Unsteady Dispersive Hydrodynamic States with Application to Shallow Water Waves

A new type of wave-mean flow interaction is identified and studied in which asmall-amplitude, linear, dispersive modulated wave propagates through anevolving, nonlinear, large-scale fluid state such as an expansion (rarefaction)wave or a dispersive shock wave (undular bore). The Korteweg-de Vries (KdV)equation is considered as a prototypical example of dynamic wavepacket-meanflow interaction. Modulation equations are derived for the coupling betweenlinear wave modulations and a nonlinear mean flow. These equations admit aparticular class of solutions that describe the transmission or trapping of alinear wave packet by an unsteady hydrodynamic state. Two adiabatic invariantsof motion are identified that determine the transmission, trapping conditionsand show that wavepackets incident upon smooth expansion waves or compressive,rapidly oscillating dispersive shock waves exhibit so-called hydrodynamicreciprocity recently described in Phys. Rev. Lett. 120, 144101 (2018) in thecontext of hydrodynamic soliton tunnelling. The modulation theory results arein excellent agreement with direct numerical simulations of full KdV dynamics.The integrability of the KdV equation is not invoked so these results can beextended to other nonlinear dispersive fluid mechanic models.