A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth

Abstract

Abstract A formal derivation of the improved Boussinesq equations of Madsen and Sorens (1992) is presented to provide the correct forms of the depth-gradient related terms. Linear shoaling characteristics of the new equations are investigated by the method of Madsen and Sorensen (1992) and by the energy flux concept separately and found to agree perfectly, whereas these approaches give conflicting results for the equations derived by Madsen and Sorensen (1992). Furthermore, Nwogu's (1993) modified Boussinesq model is found to produce a linear shoaling-gradient identical with the present work. Numerical modelling of the derived equations for directional waves is carried out by three-time-level finite-difference approximations. A higher-order radiation condition is implemented for effective absorption of the outgoing waves. Several test cases are included to demonstrate the performance of the model.