Water Wave Solutions Using Lagrangian Function from Hamilton’s Principle
Mst. Shamima Sultana *
Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh.
Md. Abu Bkar Pk
Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh.
*Author to whom correspondence should be addressed.
Abstract
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 \(\mu\) and the nonlinearity \(\varepsilon\) satisfied \(\varepsilon\) = O(\(\mu^2\)). Here the Lagrangian function is generalised up to O(\(\mu^8\)). The elevation of the free surface should be expanded to \(\mu^4\) order to get the Lagrangian function is in \(\mu^8\) 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.
Keywords: Water wave equation, Lagrangian function, Laplace equation, Hamilton’s principle