We derive the Whitham equations from the water waves equations in the shallow water regime using two different methods, thus obtaining a direct and rigorous link between these two models. The first one is based on the construction of approximate Riemann invariants for a Whitham–Boussinesq system and is adapted to unidirectional waves. The second one is based on a generalisation of Birkhoff’s normal form algorithm for almost smooth Hamiltonians and is adapted to bidirectional propagation. In both cases we clarify the improved accuracy on the fully dispersive Whitham model with respect to the long wave Korteweg–de Vries approximation.
In this paper, we analyze the relevance of the use of the shallow water model and the Boussinesq model to simulate tsunamis generated by a landslide. In a first part, we determine if the two models are able to reproduce waves generated by a landslide. Each model has drawbacks but it seems that it is possible to use them together to improve the simulations. In a second part we try to recover the landslide displacement from the generated wave. This problem is formulated as a minimization problem and we limit the number of parameters to determine assuming that the bottom can be well described by an empirical law.
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