Phase velocity effects in tertiary wave interactions

Abstract

It is shown that, when two trains of waves in deep water interact, the phase velocity of each is modified by the presence of the other. The change in phase velocity is of second order and is distinct from the increase predicted by Stokes for a single wave train. When the wave trains are moving in the same direction, the increase in velocity Δc2 of the wave with amplitude a2, wave-number k2 and frequency α2 resulting from the interaction with the wave (a1, k1, σ1) is given by Δc2 = a21k1σ1, provided k1 < k2. If k1 > k2, then Δc2 is given by the same expression multiplied by k2/k1. If the directions of propagation are opposed, the phase velocities are decreased by the same amount. These expressions are extended to give the increase (or decrease) in velocity due to a continuous spectrum of waves all travelling in the same (or opposite) direction.