Accurate modelling of uni-directional surface waves

This paper shows the use of consistent variational modelling to obtain and verify an accurate model for uni-directional surface water waves. Starting from Luke’s variational principle for inviscid irrotational water flow, Zakharov’s Hamiltonian formulation is derived to obtain a description in surface variables only. Keeping the exact dispersion properties of infinitesimal waves, the kinetic energy is approximated. Invoking a uni-directionalization constraint leads to the recently proposed AB-equation, a KdV-type of equation that is also valid on infinitely deep water, that is exact in dispersion for infinitesimal waves, and that is second order accurate in the wave height. The accuracy of the model is illustrated for two different cases. One concerns the formulation of steady periodic waves as relative equilibria; the resulting wave profiles and the speed are good approximations of Stokes waves, even for the Highest Stokes Wave on deep water. A second case shows simulations of severely distorting downstream running bi-chromatic wave groups; comparison with laboratory measurements show good agreement of propagation speeds and of wave and envelope distortions.