Wave–wave interactions on a sloping bottom

•Analytic solution for wave–wave interaction on a sloping bottom•The solution includes the wave shoaling, bottom effect and wave–wave interactions.•The nonlinear dispersion relation has been discussed.•The resonance caused by wave–wave interaction has been analyzed.

This study derived an analytic solution for two progressive waves propagating on a sloping bottom. The solution of flow field in explicit form is obtained as a three-parameter function of rendering the two wave ordering parameter ε1, ε2 and the bottom slope perturbed to the third order. The results can be simplified to a single propagating wave on sloping beach and wave–wave interaction in uniform depth. The wave number and angular frequency are related to water depth, bottom slope, and wave steepness and are also affected by another propagating wave, the increasing wave steepness or wave length of which will enhance the influence of the dispersion relation. The solutions include the wave–wave interaction, wave shoaling, nonlinearity, and the effect of bottom slope. Those are observed by means of the discussion of the free-surface displacement. The resonance caused by wave–wave interaction will occur in a specified water depth and the ratio of two wave numbers, leading to the singularity of free-surface displacement.