The Stability of Steep Gravity Waves

The linear stability of steep gravity waves is investigated for superharmonic disturbances. As to this problem, Longuet-Higgins presented a conjecture that the gravity wave would become unstable to superharmonic disturbances at the steepness where the phase speed of the wave attains a maximum. Contrary to this conjecture, it is found that the instability first occurs at the steepness which corresponds to the maximum not of the phase speed but of the total energy and the impulse of the wave. This discrepancy implies the possibility of the superharmonic bifurcation of steady gravity waves in the region of large steepness.