Capillary-Gravity Waves Research Papers - Academia.edu (original) (raw)

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