Flow down a wide inclined channel such as a canal or dam spillway is typically modeled by the inclined Saint Venant equations for shallow water flow, a system of hyperbolic balance laws w_t+f(w)_x + g(w)_y= r(w) in the form of a relaxation system. Related in structure to combustion models, and in various asymptotic limits to KdV and Kuramoto-Sivashinsky equations, these provide fascinating examples of pattern formation in quasilinear hyperbolic flow. In particular, they give rise to dramatic 1d hydraulic shocks or bores and to ``rogue'' periodic patterns known as roll waves, which can reach heights several times that of a smooth laminar flow with equivalent fluid throughput, both relevant to design and safety issues in hydraulic engineering.

The 1d existence and stability theory for these waves has a long and fascinating history, now essentially complete. Here, we initiate a new chapter with a first study of transverse, or 2d stability of these waves, with the twin goals to (i) refi ne the 1d stability picture and thus the description of sustainable rogue waves, and (ii) to look for steady transverse bifurcation to genuinely multi -d waves, in particular the familiar crosshatch or ``herringbone'' flow readily found in conditions when roll waves are observed. We indeed find a much narrower zone of stability when transverse effects are included; and, as hoped, we observe transition to herringbone flow. However, curiously, the transition is for hydraulic shocks rather than roll waves and of a previously unknown ``2d essential bifurcation'' type. There do occur also transverse bifurcations of roll waves, but these are of a quite different ``Floquet-Hopf'' type that we have dubbed ``flapping waves,'' and so far as we know have not previously been observed.