Optimized Variational 1D Boussinesq modelling of coastal waves propagating over a slope

The Variational Boussinesq Model (VBM) for waves is based on the essential property that wave phenomena can be exactly described as a Hamiltonian system. In the VBM, the fluid potential in the expression of the kinetic energy is approximated by its value at the free surface plus a linear combination of vertical potential profiles with horizontal spatially dependent functions as coefficients. The vertical potential profiles are chosen a priori and determine completely the dispersive properties of the model. For signalling problems above varying bottom we show how to optimize the wave number of one or more Airy functions as vertical profiles, by minimizing the kinetic energy functional for the given influx signal. The performance of a finite element implementation with piecewise linear basis functions is investigated by comparing simulations with experimental data from MARIN hydrodynamic laboratory for bichromatic and irregular waves running over a sloping bottom. The conclusion is that this code is robust and remarkably accurate and efficient.