Various sets of long-wave equations were rederived to ensure consistency in the formulation and to allow for a deforming bottom boundary (i.e., a non-porous landslide). A shock-capturing numerical model was developed to solve the various long-wave equations in a consistent manner. As a result of the consistency, for a given problem, the numerical results predicted by different wave equations can be directly compared, and the performance of each set of long-wave equations can be directly ranked against each other.

In addition, analytical solutions based on linear wave theory (linear and fully dispersive, as well as the linear shallow water equations) were derived, for water waves generated by a landslide, a moving atmospheric pressure, an earthquake, or due to an initial water surface displacement. Simple analytical expressions that reveal the scaling relations between the forcing function and the generated waves were obtained, which help us better understand the physical processes.

Utilizing the insights provided by the analytical solutions, we proposed a set of criteria, which can be estimated a priori, to predict how well the assumptions behind each set of long-wave equations would hold in a given landslide wave problem. The performance of these criteria were then tested against a large number of numerical simulations based on different long-wave equations. These criteria are of practical value because they provide means to quickly predict whether an approximate water wave model applies in a landslide wave problem.

We are currently working on generalizing the criteria (which are based on idealized scenarios) so that they apply to a wider range of landslide wave scenarios.