On the analytical solutions for water waves generated by a prescribed landslide

This paper presents a suite of analytical solutions, for both the free-surface elevation and the flow velocity, for landslide-generated water waves. The one-dimensional (horizontal, 1DH) analytical solutions for water waves generated by a solid landslide moving at a constant speed in constant water depth were obtained for the linear and weakly dispersive wave model as well as the linear and fully dispersive wave model. The area enclosed by the landslide was shown to have stronger lasting effects on the generated water waves than the exact landslide shape. In addition, the resonance solution based on the fully dispersive wave model was examined, and the growth rate was derived. For the 1DH linear shallow water equations (LSWEs) on a constant slope, a closed-form analytical solution, which could serve as a useful benchmark for numerical models, was found for a special landslide forcing function. For the two-dimensional (horizontal, 2DH) LSWEs on a plane beach, we rederived the solutions using the quiescent water initial conditions. The difference between the initial conditions used in the new solutions and those used in previous studies was found to have a permanent effect on the generated waves. We further noted that convergence rate of the 2DH LSWE analytical solutions varies greatly, and advised that case-by-case convergence tests be conducted whenever the modal analytical solutions are numerically evaluated using a finite number of modes.