Oblique dispersive shock waves in steady supercritical shallow water flow

Dispersive shock waves (DSWs) are universal structures arising in hydrodynamic nonlinear wave systems where dissipation is negligible with respect to wave dispersion. DSWs have recently been studied in great detail, due to their ubiquity in physical systems--examples of which range from ultra-cold superfluids at micron scales to atmospheric flows at kilometer scales. Here we consider steady, oblique DSWs generated by deflecting a supercritical shallow water flow past a thin wedge. The boundary value problem associated with the fluid flow can be recast as an initial value problem of a steady Korteweg-de Vries (KdV) equation via a multi-scale asymptotic expansion where one spatial dimension is time-like. For sufficiently shallow flow, surface tension forces are in balance with gravity and the KdV model equation must be adjusted to include a fifth order dispersive term. The high order model equation gives predictions for a bifurcation in the DSW structure as the flow depth passes through approximately 5 mm because of the higher order dispersion. Here, we will detail the structure of DSWs predicted from the approximate model equation using Whitham modulation theory and numerical simulations. The next talk will discuss an experiment where this theory can be tested.