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.