Phase velocity effects in tertiary wave interactions

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 Δ c 2 of the wave with amplitude a 2 , wave-number k 2 and frequency α 2 resulting from the interaction with the wave ( a 1 , k 1 , σ 1 ) is given by Δ c 2 = a 2 1 k 1 σ 1 , provided k 1 k 2 . If k 1 > k 2 , then Δ c 2 is given by the same expression multiplied by k 2 / k 1 . 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.