The dispersion of short wavelets in the presence of a dominant long wave

Abstract

The characteristics are studied of short surface waves superimposed upon, and interacting with, a long, finite-amplitude dominant wave of frequency N. An asymptotic analysis allows the numerical investigation of Longuet-Higgins (1978) to be extended to higher superharmonic perturbations, and it is found that, although they are distorted by the underlying finite-amplitude wave, gravity wavelets continue to propagate freely provided the dominant wave does not break. Capillary waves can, however, be blocked by short, steep, non-breaking gravity waves, so that in a wind-wave tank at short fetch and high wind speed, freely travelling gravity-capillary waves can be erased by the successive dominant wave crests.A train or group of short gravity waves suffers modulations δk in its local wave-number because of the straining of the long wave, and large modulations Cδk in its apparent frequency measured at a fixed point (where C is the long wave phase speed), largely because of the Doppler shifting produced by the dominant wave orbital velocity. The spectral signatures of a wave train are calculated by stationary phase and are found to have maxima at the upper wavenumber or frequency in the range. If an ensemble of short-wave groups is sampled at a given frequency f at a fixed point, the signal is derived from groups with a range of intrinsic frequencies δ, but is dominated by those at the long-wave crest for which f = δ + k.u0, where u0 is the orbital velocity of the dominant wave. The apparent phase speed measured by a pair of such probes is the sum of the propagation speed c of the wavelet and the orbital velocity u0 of the long wave. When f/N is large, the apparent phase speed approaches u0, independent of f. These results are consistent with measurements by Ramamonjiarisoa & Giovanangeli (1978) and others in which the apparent phase speed at high frequencies is found to be independent of the frequency — the measurements do not therefore imply a lack of dispersion of short gravity waves on the ocean surface.