Abstract
The main aim of this work is to introduce the analytical approximate solutions of the water wave problem for a fluid layer of finite depth in the presence of gravity. To achieve this aim, we begun with the derivation of the Korteweg-de Vries equations for solitons by using the method of multiple scale expansion. The proposed problem describes the behavior of the system for free surface between air and water in a nonlinear approach. To solve this problem, we use the well-known analytical method, namely, variational iteration method (VIM). The proposed method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. The proposed method provides a sequence of functions which may converge to the exact solution of the proposed problem. Finally, we observe that the elevation of the water waves is in form of traveling solitary waves.
Highlights
We are concerned with a two-dimensional, irrotational flow of an incompressible ideal fluid with a free surface under the gravitational field
Schneider and Wayne gave the justification without assuming the one directional motion of the wave and extended it to the capillary-gravity waves. They showed that the interactions between two waves were negligible so that the solution of the full water wave problem was approximated by a sum of the solutions, which were appropriately scaled, of the decoupled Korteweg-de Vries equation (KdV) for the long time interval
We present model equations for surface water waves by using a new method of multiple scale technique
Summary
We are concerned with a two-dimensional, irrotational flow of an incompressible ideal fluid with a free surface under the gravitational field. Where X stands for a space coordinate, T denotes time, and Z represents the surface elevation of a liquid in a shallow duct This equation can be derived perturbatively from the Euler equation for the motion of an incompressible and inviscid fluid [4]. Schneider and Wayne gave the justification without assuming the one directional motion of the wave and extended it to the capillary-gravity waves They showed that the interactions between two waves were negligible so that the solution of the full water wave problem was approximated by a sum of the solutions, which were appropriately scaled, of the decoupled KdVs for the long time interval. The main aim in this work is to effectively derive the Korteweg-de Vries equations and employ VIM to establish approximate solutions of waves propagating along the interface between air-water.
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