Temporal and spectral analysis of simulated and experimental Boussinesq type waves

Abstract

In this project we are interested in using computers to predict the characteristics of ocean waves where they meet the shore. Waves in the ocean play an important role in a number of areas as follows: Out in the deep sea they impact shipping and fishing. The breaking of waves along the coast results in erosion and beach line changes. The breaking of waves also creates huge forces on shoreline structures and can be very destructive. Thus a knowledge of these waves and their characteristics is useful for beach management and protecting coastal structures and harbours. In order for numerical models to be valid, they must be comparable to real world experimental equivalents. This is particularly true for complex phenomena like Boussinesq beach waves. The Boussinesq equation is highly nonlinear and is further complicated by the boundary conditions that need to be satisfied. In this work we aim to numerically solve the equations for water waves propagating along a narrow and long tank in which a sloping bottom is introduced at one end. The purpose of the sloping bottom is to create/simulate breaking waves. By doing so we aim to determine the domain of validity for the chosen numerical scheme. As a precursor for solving the Boussinesq equation, we first attempt to numerically solve the classical one dimensional wave equation, followed by the numerical solution of the Korteweg-de Vries (KdV) equation. In solving these two partial di!erential equations we set the scene for the more complex Boussinesq equation. In solving each of these equations we set up the discretisation procedure, followed by actual implementation of the discretised equation using MATLAB. In the numerical solution of the one dimensional wave equation and the KdV equation, the accuracy of the numerical techniques is assessed by comparing them to analytical solutions or results published in the open literature. In the case of the Boussinesq equation, the accuracy of the results are assessed by comparing them to a real experiment conducted in a wave tank. In both the numerical and experimental cases, we examine the changes in wave profile, wave speed, and spectral content of the waves as they move from water of constant depth into a region containing a sloping beach where breaking occurred.