Modeling wave behavior with linear wave theory

Abstract: This thesis aimed to look at the behaviours of the water beneath waves from a modeling and simulations point of view. We have investigated how to use Linear wave theory (LWT) to model the kinematic movements in water governed by free ocean waves. The model assumes the surface to consist of a superposition of sinusoidal waves. We have used fast Fourier transform (FFT) to move the surface waves from the space domain to the frequency domain from which we used the components to transform back with the inverses described by LWT. We recreated the surface with the expression for surface elevation for both two and three dimensions, and compared to the original surface. Then the same transformed components could be used to calculate the velocity fields beneath the surface. We found that the recreated surface aligns with the original surface in two and three dimensions. For the three dimensional surface we also found that the error is larger on the peaks of the waves and that at the boundary where the surface ends the error is significant due to some implementation error which we disregard. The calculated velocity fields has the expected circular movement over time and it decreases with depth. We found that a surface described with few discrete points significantly overestimates the velocities close to the surface. By increasing the number of points on the surface with simple linear interpolation this issue was resolved. The algorithms used to calculated the inverse transforms had complexity Ο(N) where N is the number of points for which to calculate the velocity, and the FFT has complexity Ο(N log N) where N is the amount of points that the surface consists of. The performance test seems to follow this trend. For applications of the methodology some future work is advised. Firstly the velocities need to be compared to some data to validate the method. Secondly some further time needs to be spent on the three dimensional case to verify that that the velocity fields behave properly and that the cross sections match with the two dimensional case. And finally, we apprehended an issue on which wavenumber to use for each wave component. Since the positive and negative wavenumber is possible and determines the propagation direction of that wave, we need to find a way to make sure that we are using the correct one for each wave component of an unknown surface.