Bifurcation and instability in gravity waves

Abstract

Calculations of the two-dimensional normal-mode perturbations of gravity waves on deep water (Longuet-Higgins, Proc. R. Soc. Lond . A 360, 471-488 (1978 a ); Longuet-Higgins, Proc. R. Soc. Lond . A 360, 489-505 (1978 b )) 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 < 2 m 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 = ½ 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.