Nonlinear Gravity Waves in Deep Water

In consideration of the problem on the boundary condition of nonlinear free water wave,co-ordinate transform is used to handle the free boundary.Supposing the solution form be the traveling wave,the ordinary differential equations of the one-order autonomous system with two variables are caused,thenexpanding the nonlinear terms at the equilibrium point with the Taylor expansion,we obtained the solutionto traveling wave.The linear approximate equation near the equilibrium point is the small amplitude wave.Anew nonlinear periodic traveling wave and nonlinear dispersion relation are shown when expanding to the sec-ond-order terms.A conclusion that the expansion of dispersion relation does not contain any odd-powerterms of wave steepness and because of the nonlinear effort,an oscillate structure is produced in the verticaldirection is drawn.