ABSTRACT
Dispersion relation of Peregrine’s system [2] is an accurate approximation to Stokes first order wave theory for very small values of the dispersion parameter μ . Madsen et al [3] improved dispersion characteristics of this system by adding extra dispersive terms to the momentum equations as expressed in terms of depth integrated velocities P h u( )+ and Q h v( )+η . The form of the dispersion relation is determined by matching the
1 INTRODUCTION
The earliest depth-averaged wave model that included weakly dispersive and nonlinear effects was derived by Boussinesq (1871), in which the nonhydrostatic pressure was linearized and included in the momentum equations. The original equations were derived for constant depth only. Later, Mei and LeMeháute [1], Peregrine [2] derived Boussinesq equations for variable depth. While Mei and LeMeháute used the velocity at the bottom as the dependent variable, Peregrine used the depth-averaged velocity and assumed the vertical velocity varying linearly over the depth. Due to wide popularity of the equations derived by Peregrine, these equations are often referred to as the standard Boussinesq equations for variable depth in the coastal engineering community. The standard Boussinesq equations are valid only for relatively small kh and H h/ values where kh and H h/ represents the parameters indicating the relative depth (dispersion) and the wave steepness (nonlinearity), respectively. Madsen et al [3] and Madsen and Sørensen [4] included higher order terms with adjustable coefficients into the standard Boussinesq equations for constant and variable water depth, respectively. Beji and Nadaoka [5] gave an alternative derivation of Madsen et al’s [4] improved Boussinesq equations. Liu & Wu [7] presented a model with specific applications to ship waves generated by a moving pressure distribution in a rectangular and trapezoidal channel by using boundary integral method. Torsvik [9] presented a numerical investigation on waves generated by
dispersion characteristics to linear wave theory. Later, this procedure has been extended to the case of variable depth by Madsen and Sørensen [4]. Alternatively, Beji and Nadaoka [5] introduced a slightly different method to improve the dispersion characteristics by a simple algebraic manipulation of Peregrine’s work for variable depth.
