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This paper revisits the previously developed NH-2LR (reduced two-layer non-hydrostatic) model. The governing equations and numerical schemes are written in terms of normalized variables, with two dimensionless parameters representing dispersion and non-linearity. By utilizing analytical solutions and laboratory experiments, this study aims to validate the numerical NH-2LR model and investigate the effects of dispersion and non-linearity on the resulting waves. The first validation employs the analytical solution of the linear and fully dispersive model of a landslide moving with constant velocity on a flat bottom. The second validation involves a landslide hump sliding over a constant beach slope. A closer look at the run-up height reveals that this case is non-dispersive. Furthermore, we found that the dispersion effect was evident from the beginning of the wave formation process. Finally, we compare our numerical results to experiments on submarine landslides on sloping beaches. We found that dispersion is essential in the early generation and propagation of waves in off-shore regions. Moreover, non-linearity significantly influences the maximum run-up of landslide-generated waves.

The viability of the Whitham equation as a nonlocal model for capillary-gravity waves at the surface of an inviscid incompressible fluid is under study. A nonlocal Hamiltonian system of model equations is derived using the Hamiltonian structure of the free-surface water-wave problem and the Dirichlet–Neumann operator. The system features gravitational and capillary effects, and when restricted to one-way propagation, the system reduces to the capillary Whitham equation. It is shown numerically that in various scaling regimes the Whitham equation gives a more accurate approximation of the free-surface problem for the Euler system than other models like the KdV and Kawahara equation. In the case of relatively strong capillarity considered here, the KdV and Kawahara equations outperform the Whitham equation with surface tension only for very long waves with negative polarity.

Three weakly nonlinear but fully dispersive Whitham–Boussinesq systems for uneven bathymetry are studied. The derivation and discretization of one system is presented. The numerical solutions of all three are compared with wave gauge measurements from a series of laboratory experiments conducted by Dingemans (Comparison of computations with Boussinesq-like models and laboratory measurements. Delft Hydraulics memo H168412, 1994). The results show that although the models are mathematically similar, their accuracy varies dramatically.

Recently, the Whitham and capillary Whitham equations were shown to accurately model the evolution of surface waves on shallow water. In order to gain a deeper understanding of these equations, we compute periodic, traveling-wave solutions for both and study their stability. We present plots of a representative sampling of solutions for a range of wavelengths, wave speeds, wave heights, and surface tension values. Finally, we discuss the role these parameters play in the stability of these solutions.