Critical ratio between the amplitudes of two overtaking solitary water waves

Abstract

In this paper, we study the critical ratio between the amplitudes of two overtaking solitary waves on a layer of water with uniform depth. At the center of encounter, the wave profile is fore-and-aft symmetric, but it could have a single peak or double peaks. The critical ratio separates these two regimes. At the critical point, the wave peak is flat with zero slope and curvature. We solve the full water wave problem numerically by using a fully nonlinear and highly dispersive Boussinesq model. The model is numerically justified to be a good approximation of the Euler equations for solitary waves with very large amplitude. For small amplitude water waves, our calculated critical ratio reduces to the well-known result of 3 predicted by the Korteweg-de Vries equation, a weakly nonlinear and weakly dispersive model. For large amplitude water waves, the nonlinear effect is significant; we find that the critical ratio deviates significantly from 3. For water waves with very high amplitude, e.g., 0.6 relative wave height, the critical ratio could be as large as 4. Our result suggests that higher-order nonlinear and dispersive effects are important when modeling the strong interaction between large amplitude water waves.