Mathematical modelling of generation and forward propagation of dispersive waves

This dissertation concerns the mathematical theory of forward propagation and generation of dispersive waves. We derive the AB2-equation which describes forward traveling waves in two horizontal dimension. It is the generalization of the Kadomtsev-Petviashvilli (KP) equation. The derivation is based on the variational principle of water waves. Similar to its predecessor, the AB-equation, the AB2-equation is dispersive, accurate in second order and can be adjusted for any water depth. Using pseudo-spectral method, the numerical implementation of the AB2-equation can be done easily since exact dispersion is described by a nonpolynomial pseudo-differential operator that can easily be dealt with in spectral space. For wave generation, we derive various models that describe excitation if the wave elevation (or fluid potential) at a certain position is given. The wave generation discussed in this dissertation is done by an embedded source term added to the equation(s) of water wave motion. In this way, we transform the problem of homogeneous boundary value problem into an inhomogeneous problem. We derive the source functions for any kind of waves to be generated and for any dispersive equation including the general case of (linear) dispersive Boussinesq equations. For a dispersive wave equation, the source is not unique; many choices can be taken as long as they satisfy a certain source - influx signal relation. This is different from the actual condition in a hydrodynamic laboratory where there is a one to one correspondence between influx signal and the generated waves. We designed a set of experiments for oblique wave interaction. The aim of the experiment is to test the applicability and the performance of the AB2-equation and the influxing technique. These experiments were executed in a water tank of MARIN hydrodynamic laboratory. The experiments are performed by generating two oblique waves from two sides of the basin and let the waves collide. We compare the measurements from the experiments and the AB2 simulation results. The AB2 simulations and the MARIN measurements are in satisfactory agreement, showing the bichromatic beat wave pattern, even for large nonlinear effects.