NONLINEAR EVOLUTION OF A UNIDIRECTIONAL SHOALING WAVE-FIELD

Nonlinear energy transfer in the wave spectrum is very important in th e shoaling region. Existing theories are limited to weakly dispersive situations (i.e. shallow water or narrow spectrum). A nonlinear evolut ion equation for shoaling gravity waves is derived, describing the pro cess all the way from deep to shallow water. The slope of the bottom i s taken to be smaller, or of the order of the wave steepness (epsilon) . The waves are assumed unidirectional for simplicity. The shoaling do main extends up to, and excluding, the first line of breaking of the w aves. Reflection by the shore is neglected. Dispersion is fully accoun ted for. The model equation includes terms due to quadratic interactio ns, which are effective over characteristic time and spatial scales of order (T/epsilon) and (lambda/epsilon), respectively, where lambda an d T are wavelength and period at the spectral peak. In the limit of sh allow water, the quadratic interaction model tends to the Boussinesq m odel. By discretizing the wave spectrum, mixed initial and boundary va lue problems may be computed. The assumption of the existence of a ste ady state, transforms the problem into a boundary value one. For this case, solutions for a single triad of waves describing the subharmonic generation and for a full discretized spectrum were computed. The res ults are compared and found to be in good agreement with laboratory an d field measurements. The model can be extended to directionally sprea d spectra and two dimensional bathymetry.