Higher-order Boussinesq equations for two-way propagation of shallow water waves

Abstract

Standard perturbation methods are applied to Euler's equations of motion governing the capillary-gravity shallow water waves to derive a general higher-order Boussinesq equation involving the small-amplitude parameter, α = a / h 0 , and long-wavelength parameter, β = ( h 0 / l ) 2 , where a and l are the actual amplitude and wavelength of the surface wave, and h 0 is the height of the undisturbed water surface from the flat bottom topography. This equation is also characterized by the surface tension parameter, namely the Bond number τ = Γ / ρ g h 0 2 , where Γ is the surface tension coefficient, ρ is the density of water, and g is the acceleration due to gravity. The general Boussinesq equation involving the above three parameters is used to recover the classical model equations of Boussinesq type under appropriate scaling in two specific cases: (1) | 1 3 − τ | ≫ β , and (2) | 1 3 − τ | = O ( β ) . Case 1 leads to the classical (ill-posed and well-posed) fourth-order Boussinesq equations whose dispersive terms vanish at τ = 1 3 . Case 2 leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [P. Daripa, W. Hua, A numerical method for solving an illposed Boussinesq equation arising in water waves and nonlinear lattices, Appl. Math. Comput. 101 (1999) 159–207] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation. The relationship between the sixth-order Boussinesq equation and fifth-order KdV equation is also established in the limiting cases of the two small parameters α and β.