We employ a fully nonlinear numerical model to generate and propagate long-crested gravity waves in a tank containing an incompressible inviscid homogeneous fluid, initially at rest, with a horizontal free surface of finite extent and of finite depth. A non-orthogonal curvilinear coordinate system is constructed which follows the free surface and is “fitted” to the bottom topography of the tank and therefore tracks the entire fluid domain at all times. A waveform relaxation algorithm provides an efficient iterative method to solve the resulting discrete Laplace equation, and the full nonlinear kinematic and dynamic free surface boundary conditions are employed to propagate the solution. In addition, a bi-chromatic deterministic theoretical wave-maker, employing a Dirichlet type boundary condition, and a suitably tuned numerical beach are utilized in the numerical model.
Using our fully nonlinear model, we calculate the energy and energy flux for both small steepness bi-chromatic waves and for larger steepness bi-chromatic waves and compare these results with the results from linear theory. In particular, this paper evaluates the validity of the superposition principle inherent in linear estimations of energy and energy flux calculations for bi-chromatic and, in general, multi-chromatic waves. Special attention is given to the phenomenon of the nonlinear interaction of the “higher order components”, especially near resonance situations, and its effect on energy and energy flux.