Extended Serre Equations for Applications in Intermediate Water Depths

The Serre or Green and Naghdi equations are fully-nonlinear and weakly dispersive and have a built-in assumption of irrotationality. However, like the standard Boussinesq equations, also Serre’s equations are only valid for long waves in shallow waters. To allow applications in a greater range of h0/l, where h0 and l represent, respectively, depth and wavelength characteristics, a new set of extended Serre equations, with additional terms of dispersive origin, is developed and tested in this work by comparisons with available experimental data. The equations are solved using an efficient finite-difference method, which consistency and stability are analyzed by comparison with a closed-form solitary wave solution of the Serre equations. All cases of waves propagating in intermediate water depths illustrate the good performance of the extended Serre equations with additional terms of dispersive origin. It is shown that the computed results are in conformity with the analytical ones and test data. An equivalent form of the Boussinesq type equations, also with improved linear dispersion characteristics, is solved using a numerical procedure similar to that used to solve the extended Serre equations.