An infinite-order, Boussinesq-type differential equation for wave shoaling over variable bathymetry is derived. Defining three scaling parameters -nonlinearity, the dispersion parameter, and the bottom slope -the system is truncated to a finite order. Using Padé approximants the order in the dispersion parameter is effectively doubled. A derivation is made systematic by separately solving the Laplace equation in the undisturbed fluid domain and then addressing the nonlinear free-surface conditions. We show that the nonlinear interactions are faithfully captured. The shoaling and dispersion components are time independent.

We derive a deterministic directional shoaling model and a stochastic directional shoaling model for a gravity surface wave field, valid for a beach with parallel depth contours accounting for refraction and nonlinear quadratic (three wave) interactions. A new phenomenon of non-resonant spectral evolution arises due to the medium inhomogeneity. The kernels of the kinetic equation depend on the bathymetry through an integral operator. Preliminary tests carried out on laboratory data for a unidirectional case indicate that the stochastic model also works rather well beyond the region where the waves may be regarded as nearly Gaussian. The limit of its applicability is decided by the dispersivity of the medium (relative to the nonlinearity). Good agreement with both laboratory data and the underlying deterministic model is found up to a value of about 1.5 for the spectral peak Ursell number. Beyond that only the deterministic model matches the measurements.

Evolution of a nonlinear wave field along a laboratory tank is studied experimentally and numerically. The numerical study is based on the Zakharov nonlinear equation, which is modified to describe slow spatial evolution of unidirectional waves as they move along the tank. Groups with various initial shapes, amplitudes and spectral contents are studied. It is demonstrated that the applied theoretical model, which does not impose any constraints on the spectral width, is capable of describing accurately, both qualitatively and quantitatively, the slow spatial variation of the group envelopes. The theoretical model also describes accurately the variation along the tank of the spectral shapes, including free wave components and the bound waves.

No abstract