The instabilities of gravity waves of finite amplitude in deep water II. Subharmonics

Calculation of the normal-mode perturbation of steep irrotational gravity waves, begun in part I, is here extended to a study of the subharmonic perturbations, namely those having horizontal scales greater than the basic wavelength $(2\pi /k)$. At small wave amplitudes $a$, it is found that all perturbations tend to become neutrally stable; but as $ak$ increases the perturbations coalesce in pairs to produce unstable modes. These may be identified with the instabilities analysed by Benjamin & Feir (1967) when $ak$ is small. However, as $ak$ increases beyond about 0.346, these modes become stable again. The maximum growth scale of this type of mode in the unstable range is only about 14% per wave period, which value occurs at $ak$ $\approx $ 0.32. At values of $ak$ near 0.41 a new type of instability appears which has initially zero frequency but a much higher growth rate. It is pointed out that this type might be expected to arise at wave amplitudes for which the first Fourier coefficient in the basic wave is at its maximum value, as a function of the wave height. The corresponding wave steepness was found by Schwartz (1974) to be $ak$ = 0.412. A comparison of the calculated rates of growth are in rather good agreement with those observed by Benjamin (1967) in the range 0.07 < $ak$ < 0.17.