Bi-directional wave propagation

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 /ρgh 2 0 , 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.

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter a ¼ a=h 0 and wavelength parameter b ¼ ðh 0 =lÞ 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. These equations are also characterized by the surface tension parameter, namely the Bond number s ¼ C=qgh 2 0 , where C is the surface tension coefficient, q is the density of water, and g is the acceleration due to gravity. The traveling solitary wave solutions are explicitly constructed for a class of lower order Boussinesq system. From the Boussinesq equation of higher order, the appropriate equations to model solitary waves are derived under appropriate scaling in two specific cases: (i) b (ð1=3 À sÞ 6 1=3 and (ii) ð1=3 À sÞ ¼ OðbÞ. The case (i) leads to the classical Boussinesq equation whose fourth-order dispersive term vanishes for s ¼ 1=3. This emphasizes the significance of the case (ii) that leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation.: S 0 0 2 0-7 2 2 5 (0 2) 0 0 1 8 0-5

The Euler's equations describing the dynamics of capillary-gravity water waves in two-dimensions are considered in the limits of small-amplitude and long-wavelength under appropriate boundary conditions. Using a double-series perturbation analysis, a general Boussi-nesq type of equation is derived involving the small-amplitude and long-wavelength parameters. A recently introduced sixth-order Boussinesq equation by Daripa and Hua [Appl. Math. Comput. 101 (1999), 159– 207] is recovered from this equation in the 1/3 Bond number limit (from below) when the above parameters bear a certain relationship as they approach zero.