Water wave solutions obtained by variational method

摘要

Variational problem for irrotational, incompressible inviscid fluid in finite water depth is considered. Based on the variational principle, a special solution of the problem is presented under the assumption that the dispersion μ and the nonlinearity ε satisfied ε=O(μ2) as the Lagrange function is expanded up to O(μ8).It is shown that the elevation of the free surface should be expanded toμ4 order to ensure the Lagrange function is inμ8 order. Comparison the nonlinear free surface profiles obtained from the solution with the corresponding ones obtained from linear solutions showed that the wave crest of the nonlinear wave is steepened but the trough is flattened compared to the linear wave as expected.