Bifurcation and Instability in Gravity Waves

Calculations of the two-dimensional normal-mode perturbations of gravity waves on deep water (Longuet-Higgins, Proc. R. Soc. Lond. A 360, 471-488 (1978a); Longuet-Higgins, Proc. R. Soc. Lond. A 360, 489-505 (1978b)) are here extended to values of the wave steepness ak as high as 0.43. This is achieved (a) by using a new method to evaluate the coefficients in Stokes's series to high order, and (b) by rearrangement of the matrix equations so as to reduce the order by half. The behaviour of the normal-mode frequencies σ_n in the range 0.35 < ak < 0.43 is clarified. Subharmonics of the form n = (l/m, 2-l/m) where l and m are integers and l < 2m are shown to combine in pairs to form type II instabilities with relatively high rates of growth. For these modes, the critical values of ak at which σ vanishes correspond precisely to bifurcation points. In the special case l/m = 1/2 the two modal curves coincide. The family of frequency curves is bounded by the lowest superharmonic (n = 2). It is verified that this mode becomes unstable when ak = 0.4292, corresponding to the lowest maximum of the energy density E. The boundaries of the parameter regions for instabilities of both type I and type II are determined.