The classic "hydraulic analogy" between supercritical shallow water flows and gas dynamics was leveraged in the early to mid 20th century as a means to model the flow patterns around wings with a relatively simple water table. But this analogy breaks down when water wave dispersion is strong relative to dissipative effects as in oscillatory, undular bores. In this regime, the more appropriate analogy is to dispersive hydrodynamic media such as superfluids and nonlinear optical diffractive or dispersive patterns, where dispersive shock waves (DSWs), also called quantum and optical shock waves, have been observed. In this talk, the problem of steady pattern formation in supercritical, shallow water flows will be examined theoretically and experimentally. Two new, non-classical types of oscillatory DSWs will be identified within the context of the fifth order Kawahara equation resulting from a balance between capillarity and gravity at water depths on the order of 0.5 cm. Experiments with a shallow water table demonstrate their physical existence and properties. The results presented have implications for dispersive hydrodynamic media with non-convex linear dispersion curvature such as ultracold atomic superfluids and nonlinear fiber optics.