A new approximate formulation of periodic gravity waves

Periodic gravity waves travelling in irrotational deep water flows are examined. A new method of implicit approximation to the travelling waves along the free water surface rather than the explicit Stokes wave expansions along the calm water surface is formulated. Therefore, the approximate waves satisfy exactly the dynamic free surface boundary condition along the free water surface, while the corresponding Stokes waves satisfy the dynamic free surface boundary condition approximately along the calm water surface based on the Taylor expansion. The distinction between the proposed wave and the corresponding Stokes wave can be ignored at small wave steepness but becomes clear with the increment of the wave steepness due to the nonphysical form of the Stokes wave at large wave steepness. Approximation to the Stokes highest wave is demonstrated.