WEAKLY NONLINEAR KELVIN-HELMHOLTZ WAVES.

The Lagrangian L for gravitiy waves of finite amplitude in an N-layer, stratified shear flow is constructed as a functional of the generalized coordinates of the N plus 1 interfaces. The explicit expansion of L is constructed through fourth-order. Progressive interfacial waves and Kelvin-Helmholtz instability in a two-layer fluid are examined, and the earlier results of P. G. Drazin, A. H. Nayfeh & W. S. Saric and M. A. Weissman are extended to finite depth. It is found that the pitchfork bifurcation associated with the critical point for Kelvin-Helmholtz instability, which is supercritical for inifinitely deep layers, may be subcritical (inverted) for finite depths. The evolution equations that govern Kelvin-Helmholtz waves in the parametric neighbourhood of this critical point are shown to be equivalent to those for a particle in a two-parameter, central force field. The wave motion forced by flow over a sinusoidal bottom is examined and the corresponding resonance curves and Hopf bifurcations determined.