Alternative forms of the higher-order Boussinesq equations: Derivations and validations

• Published in: Coastal engineering (Amsterdam) 2008-06, Vol.55 (6), p.506-521
• Main Authors: Zou, Z.L., Fang, K.Z.
• Format: Article
• Language: English
• Subjects: Boussinesq equations; Coastal oceanography, estuaries. Regional oceanography; Earth, ocean, space; Exact sciences and technology; External geophysics; Geophysics. Techniques, methods, instrumentation and models; Nonlinear waves; Physics of the oceans; Wave propagation
• ISSN: 0378-3839; 1872-7379
• DOI: 10.1016/j.coastaleng.2008.02.001

An alternative form of the Boussinesq equations is developed, creating a model which is fully nonlinear up to O( μ 4) ( μ is the ratio of water depth to wavelength) and has dispersion accurate to the Padé [4,4] approximation. No limitation is imposed on the bottom slope; the variable distance between free surface and sea bottom is accounted for by a σ-transformation. Two reduced forms of the model are also presented, which simplify O( μ 4) terms using the assumption ε = O( μ 2/3) ( ε is the ratio of wave height to water depth). These can be seen as extensions of Serre's equations, with dispersions given by the Padé [2,2] and Padé [4,4] approximations. The third-order nonlinear characteristics of these three models are discussed using Fourier analysis, and compared to other high-order formulations of the Boussinesq equations. The models are validated against experimental measurements of wave propagation over a submerged breakwater. Finally, the nonlinear evolution of wave groups along a horizontal flume is simulated and compared to experimental data in order to investigate the effects of the amplitude dispersion and the four-wave resonant interaction.