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Mathematical Problems in Engineering
Volume 2011, Article ID 147327, 21 pages
Research Article

Approximate Method for Studying the Waves Propagating along the Interface between Air-Water

1Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt
2Department of Mathematics, Faculty of Science, Umm Al-Qura University, 21955, Saudi Arabia

Received 23 November 2010; Accepted 6 January 2011

Academic Editor: Ezzat G. Bakhoum

Copyright © 2011 M. M. Khader and R. F. Al-Bar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is devoted to consider the approximate solutions of the nonlinear water wave problem for a fluid layer of finite depth in the presence of gravity. The method of multiple-scale expansion is employed to obtain the Korteweg-de Vries (KdV) equations for solitons, which describes the behavior of the system for free surface between air and water in a nonlinear approach. The solutions of the water wave problem split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as the solutions of KdV equations. The solution of KdV equations is obtained analytically by using a reliable modification of Laplace decomposition method (LDM), namely, the modified Laplace decomposition method (MLDM) is presented. This procedure is a powerful tool for solving large amount of nonlinear problems. The proposed method provides the solution as a series which may converge to the exact solution of the problem. Also, the convergence analysis of the proposed method is given. Finally, we observe that the elevation of the water waves is in form of traveling solitary waves. The horizontal and vertical of the velocity components have nonlinear characters.