Some contributions to the theory of edge waves

Abstract

The classical edge-wave problem is addressed by scaling the governing equations, for small slope ($\varepsilon$) at the shore, according to the exact edge-wave solution (for uniform slope) which is based on the Gerstner solution of the water-wave problem. The bottom is allowed to be any suitable profile which varies on the scale of this small parameter; a multiple-scale method is then employed to construct the solution. The leading-order equations – which are a version of the shallow-water equations – are fully nonlinear, but an appropriate exact travelling-wave solution exists; the next term in the asymptotic expansion, valid for $\varepsilon\,{\to}\,0$, is also found and, from this, uniformity conditions are deduced. The results are used to describe the run-up pattern produced by edge waves at the shoreline, based on any mode other than the first; this pattern corresponds closely with what is observed, and also with the exact solution for uniform slope everywhere. The surface wave, from the shoreline, seawards, is described for various depth profiles (such as a constant depth at infinity or with a sand bar close inshore). The problem for the first mode, which corresponds to a non-uniformity in the expansion, is briefly discussed; in this case, it is not possible to find an exact closed-form solution.

The corresponding analysis in the case when a longshore current (varying on the same scale as the depth) is flowing, in addition to a general depth profile, is also presented, and the notion of an effective depth profile is confirmed. Finally, a brief mention is made of model equations for edge waves (which have single-mode exact solutions); these may provide the basis for further investigations into the interaction of modes.