Water Wave Solutions Using Lagrangian Function from Hamilton’s Principle

This paper concerns the development and application of the Lagrangian function which is the difference between kinetic energy and potential energy of the system. Here irrotational, incompressible, inviscid fluid in finite water depth is considered. Our attention is to focus on the problem to solve water wave evolution with Lagrangian function which is obtained from Hamilton’s Principle. Then Lagrangian function is expanded under the assumption that the dispersion μ and the nonlinearity ε satisfied ε = O(μ2). Here the Lagrangian function is generalised up to O(μ8).The elevation of the free surface should be expanded to μ4 order to get the Lagrangian function is in μ8 order. Finally a wave model from Euler- Lagrangian equation of motion has been generalized which follows that the generalized wave velocity decreases with the large value of time and at very large time, a wave crest and trough will be diminished.