Boussinesq and Serre type models with improved linear dispersion characteristics: Applications

Abstract

The classical Boussinesq equations only incorporate weak dispersion and weak nonlinearity, and are valid only for long waves in shallow waters. The classical Serre equations (or Green and Naghdi) are fully-nonlinear and weakly dispersive. Thus, as for the classical Boussinesq models, Serre's equations are valid only for shallow water conditions. To allow applications in a greater range of depth to wavelength ratio, a new set of extended Serre equations with additional terms of dispersive origin is proposed in this work. The equations are solved using an efficient finite-difference method, which consistency and stability are tested by comparison with a closed-form solitary wave solution of these equations. It is shown that computed results agree closely with the analytical ones and test data. An equivalent form of the Boussinesq equations, also with improved linear dispersion characteristics, is solved using a numerical procedure similar to that used to solve the extended Serre equations. Numerical solutions of both approaches, in the case of waves generated by disturbances on the water surface, are compared to each other and with available data, testing different functions commonly used in modelling the generation and propagation of ship waves.