Bifurcation in gravity waves

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 a n 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 a n . 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 a 0 , a 1 and a 2 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.