Bifurcation in gravity waves

Abstract

A new method is proposed for the calculation of gravity waves on deep water. This is based on some recently discovered quadratic identities between the Fourier coefficients an in Stokes's expansion. The identities are shown to be derivable from a cubic potential function, which in turn is related to the Lagrangian of the motion. A criterion for the bifurcation of uniform waves into a series of steady waves of non-uniform amplitude is expressed by the vanishing of a particular determinant with elements which are linear combinations of the coefficients an. The critical value of the wave steepness for the symmetric bifurcations discovered by Chen & Saffman (1980) are verified. It is shown that a truncated scheme consisting of only the coefficients a0, a1 and a2 already exhibits Class 2 bifurcation, and similarly for Class 3. Asymmetric bifurcations are also discussed. A recent suggestion by Tanaka (1983) that gravity waves exhibit a Class 1 bifurcation at the point of maximum energy is shown to be incorrect.