Boussinesq-type equations for nonlinear evolution of wave trains

Nonlinear evolution of wave trains involves amplitude dispersion and four-wave resonant interaction and hence is difficult to describe using a simple wave equation such as the cubic Schrödinger equation or conventional Boussinesq equations. The present study develops a set of improved higher-order Boussinesq equations with a wide accuracy range of third-order nonlinear characteristics, including amplitude dispersion, and with superior performance for simulations of the nonlinear evolution of wave trains. The equations are obtained by enhancing the higher-order Boussinesq-type equations developed by Zou [Z.L. Zou, A new form of high-order Boussinesq equations, Ocean Eng. 27 (2000) 557–575] through introducing two nonlinear terms into the expression for the computation velocity. The new terms can improve the nonlinear property at higher order by adjusting their free parameters to match the theoretical solutions for amplitude dispersion and the third-order transfer function. Super- and sub-harmonics of bichromatic waves are also improved. The new equations are applied to simulate the nonlinear evolution of wave groups along a 1D wave tank with flat bottom, and nonlinear refraction and diffraction of regular wave trains over a cylindrical ramp, good effectiveness is found.