On reflection of a planar solitary wave at a vertical wall

The reflection of a planar solitary wave at a vertical wall is investigated by solving the Boussinesq equations analytically as well as numerically. The analytical solution is obtained by means of the inner-outer expansions technique, while the numerical solution is based on a finite difference scheme. The maximum wave amplitude at the wall and the time at which this maximum amplitude is reached are presented. It is also found that the incident wave does not reflect immediately at the wall as predicted by the linear wave theory. Rather, the wave suffers a time delay, called the phase lag, during the reflection process. This phase lag is found to be inversely proportional to the square root of the initial wave amplitude. As the reflected wave eventually propagates away from the wall, it has a phase shift in comparison with that obtained by the linear wave theory.