Wave transformation models with exact second-order transfer

Fully dispersive deterministic evolution equations for irregular water waves are derived. The equations are formulated in the complex amplitudes of an irregular, directional wave spectrum and are valid for waves propagating in directions up to ±90° from the main direction of propagation under the assumptions of weak nonlinearity, slowly varying depth and negligible reflected waves.

A weak deviation from straight and parallel bottom contours is allowed for. No assumptions on the vertical structure of the velocity field is made and as a result, the equations possess exact second-order bichromatic transfer functions when comparing to the reference solution of a Stokes-type analysis.

Introduction of the so-called ‘resonance assumption’ leads to the evolution equations of among others Agnon, Sheremet, Gonsalves and Stiassnie [Coastal Engrg. 20 (1993) 29–58]. For unidirectional waves, the bichromatic transfer functions of the ‘resonant’ models are found to have only small deviations in general from the reference solution.

We demonstrate that the ‘resonant’ models can be solved efficiently using Fast Fourier Transforms, while this is not possible for the ‘exact’ models. Simulation results for unidirectional wave propagation over a submerged bar show that the new models provide a good improvement from linear theory with respect to wave shape.

This is due to the quadratic terms, enabling a nonlinear description of shoaling and de-shoaling, including the release of higher harmonics after the bar. For these simulations, the similarity between the ‘exact’ and ‘resonant’ models is confirmed. A test case of shorter waves, however, shows that the amplitude dispersion can be quite over-predicted in the models.

This behaviour is investigated and confirmed through a third-order Stokes-type perturbation analysis.