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Higher–order Boussinesq–type equations for surface gravity waves: derivation and analysis

P. A. Madsen, H. A. Schäffer


Boussinesq–type equations of higher order in dispersion as well as in nonlinearity are derived for waves (and wave–current interaction) over an uneven bottom. Formulations are given in terms of various velocity variables such as the depth–averaged velocity and the particle velocity at the still water level, and at an arbitrary vertical location. The equations are enhanced and analysed with emphasis on linear dispersion, shoaling and nonlinear properties for large wave numbers.

As a starting point the velocity potential is expanded as a power series in the vertical coordinate measured from the still water level (SWL). Substituting this expansion into the Laplace equation leads to a velocity field expressed in terms of spatial derivatives of the vertical velocity ŵ and the horizontal velocity vector û at the SWL. The series expressions are given to infinite order in the dispersion parameter, μ. Satisfying the kinematic bottom boundary condition defines an implicit relation between ŵ and û, which is recast as an explicit recursive expression for ŵ in terms of û under the assumption that μ ≪ 1. Boussinesq equations are then derived from the dynamic and kinematic boundary conditions at the free surface. In this process the infinite series solutions are truncated at O6), while all orders of the nonlinearity parameter, ϵ are included to that order in dispersion. This leads to a set of higher–order Boussinesq equations in terms of the surface elevation η and the horizontal velocity vector û at the SWL.

The equations are recast in terms of the depth–averaged velocity, U leaving out O2μ4, which corresponds to assuming ϵ =O(μ). This formulation turns out to include singularities in linear dispersion as well as in nonlinearity. Next, the technique introduced by Madsen and others in 1991and Schäffer & Madsen in 1995 is invoked, and this results in aset of enhanced equations formulated in U and including O4,ϵμ4)terms. These equations contain no singularities and the embedded linear and nonlinear properties are shown to be significantly improved. To quantify the accuracy, Stokes's third–order theory is used as reference and Fourier analyses of the new equations are carried out to third order (in nonlinearity) for regular waves on a constant depth and to first order for shoaling characteristics. Furthermore, analyses are carried out to second order for bichromatic waves and to first order for waves in ambient currents. These analyses are not restricted to small values of the linear dispersion parameter, μ. In conclusion, the new equations are shown to have linear dispersion characteristics corresponding to a Pade [4,4] expansion in kh (wave number times depth) of the squared celerity according to Stokes's linear theory. This corresponds to a quite high accuracy in linear dispersion up to approximately kh = 6. The high quality of dispersion is also achieved for the Doppler shift in connection with wave–current interaction and it allows for a study of wave blocking due to opposing currents. Also, the linear shoaling characteristics are shown to be excellent, and the accuracy of nonlinear transfer of energy to sub– and super–harmonics is found to be superior to previous formulations.

The equations are then recast in terms of the particle velocity, ũ, at an arbitrary vertical location including Oμ45μ4)terms. This formulation includes, as special subsets, Boussinesq equations in terms of the bottom velocity or the surface velocity. Furthermore, the arbitrary location of the velocity variable can be used to optimize the embedded linear and nonlinear characteristics. A Fourier analysis is again carried out to third order (in ϵ) for regular waves. It turns out that Padé [4,4] linear dispersion characteristics can not be achieved for any choice of the location of the velocity variable. However, for an optimized location we achieve fairly good linear characteristics and very good nonlinear characteristics.

Finally, the formulation in terms of ũ is modified by introducing the technique of dispersion enhancement while retaining only O45μ4) terms. Now the resulting set of equations do show Padé [4,4] dispersion characteristics in the case of pure waves as well as in connection with ambient currents, and again the nonlinear properties (such as second– and third–order transfer functions and amplitude dispersion) are shown to be superior to those of existing formulations of Boussinesq–type equations.

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