A mathematical model for weakly nonlinear water wave propagation

Nonlinear wave equations are derived in this paper based on combining the Stokes second-order wave theory with the Boussinesq theory. In shallow water the new equations reduce to the improved Boussinesq equations and in deep water the second-order Stokes waves can be simulated directly without water depth restriction. There is no contradiction between the Boussinesq theory and the second-order Stokes theory for the new equations. Numerical results of the wave equations are compared with analytical solutions of nonlinear Cnoidal waves and second-order Stokes waves. Good agreement has been achieved. The new equations can be easily and directly applied for random wave modelling in finite water depth.