Abstract
An extended Boussinesq model is developed using a mathematical artifice that decomposes the velocity on a series of functions on the water depth. The aim of this paper is to present the theoretical bases of this finite elements model and the different series of functions available in this model with their respective physical range of validity. We then examine its linear frequency behaviour, its ability to reproduce the vertical variations of velocity and also its linear shoaling behaviour. A special attention is paid to the Legendre polynomials series. One of the advantages of this model is to implicitly improve its behaviour without any addition of higher order derivative terms and without any calibration parameters to fit theoretical curves. This improvement is made only by adding extra functions on the chosen series. Another advantage of this model is to implicitly include slope effect and evanescent modes. It can therefore deal with propagation of waves over relatively steep slopes.
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