Mathematical/Numerical Modelling of Water Waves

The motion of water waves plays an important role in coastal-, ocean- and maritime engineering, and for most geographical areas, waves are the major source of environmental actions on beaches or on man-made fixed or floating structures. Natural wave trains are irregular in shape and they interact due to non-linear processes.

Classical wave theories fail to describe the combined effect of these processes and they can be divided in two categories: 1) A non-linear description of monochromatic waves of a specific frequency or wave length; B) A linear description of irregular waves based on superposition of individual frequency components with random phases. For the determination of natural wave trains both approaches have their obvious limitations.

At MEK we have derived a new formulation for water waves that includes the description of the dynamics and kinematics of irregular and non-linear waves propagating over an uneven bottom. The theory belongs to the so-called Boussinesq family but in contrast to conventional Boussinesq formulations, which are restricted to weakly non-linear shallow water phenomena, the new formulation is applicable for fully non-linear waves in deep as well as in shallow water. Numerical models solving these equations in two-horizontal dimensions have been developed. In the following we highlight some of the phenomena that have been studied with the new model.


Wave loads on structures

As a first step towards the deterministic identification of wave loads, we have modelled the velocity and pressure variation in the highest possible wave in shallow water and in deep water. Results compare favourably to solitary wave and stream-function wave theories, respectively. Furthermore, velocity and pressure distributions have been determined in irregular wave trains shoaling up to the point of wave breaking, and results have been shown to be significantly better than conventional methods such as Wheeler-stretching. The new results have been used to establish design criteria for offshore windmill foundations in shallow water. Recently, the model has also been extended to allow bottom-mounted structures in two horizontal dimensions. In this connection a highly non-linear run-up on a vertical plate has been computed and found to be in excellent agreement with experimental data

(Figure 1). 


Waves over a rapidly varying bathymetry

When surface waves propagate over a rapidly varying sea bed as e.g. sand dunes, ripples or trenches, they are exposed to resonant interaction which may lead to significant reflection (Figure 2). An example is the so-called Bragg-scatter from a sinusoidal sea bed involving an interaction between the surface (water) wave and the bottom (sand) wave, which is analogous to the mechanism of non-linear wave-wave interaction for surface waves. We have investigated and simulated the three established classes involving the interaction between: a) two surface waves and one bottom wave (Class I); b) two surface waves and two bottom waves (Class II); c) three surface waves and one bottom wave (Class III).


(Figure 2).


Instabilities of water waves

Non-linear waves travelling in deep water are exposed to different types of instabilities that may completely change their shape. A classic example is the two-dimensional

Benjamin-Feir instability, which dominates for a wave steepness less than 0.1. During this process, energy is transferred from a carrier wave to two sideband frequencies and as a result we get a spatial focusing with a significant amplification of the wave height (see Figure 2). Another example is the three-dimensional McLean instability, which dominates for a wave steepness greater than 0.1. In this case we get systematic crescent or horseshoe patterns, which can frequently be observed on the sea surface or in tank experiments (see Figure 4).  

The research is supported by the Danish Technical Research Council (STVF) through an 8-year Frame Program "Computational Hydrodynamics", and by the Danish National Research Foundation through a 5-year Research Professorship. We thank the Danish Center for Scientific Computing for providing the necessary supercomputing resources.