Particle trajectory and mass transport of finite-amplitude waves in water of uniform depth

A set of governing equations in Lagrangian form is derived for propagating gravity waves in water of uniform depth. The Lindstedt-Poincaré perturbation method is used to obtain approximations up to fifth order. Recognizing the Lagrangian frequency to be a position function for all particles is a key to find these higher-order approximations. The present solution has zero pressure at the free surface and satisfies exactly the dynamic boundary condition. Under the present approximations, the Lagrangian frequency is composed of two parts. The first part is constant for all particles and equivalent to the term in the fifth-order Stokes' wave theory [J.D. Fenton, A fifth-order Stokes theory for steady waves, J. Waterway, Port, Coastal Ocean Eng. 111 (1985) 216-234]. The second part is a function of the depth. All the particles move as open (nonclosed) loops and have mean drift displacements that decrease exponentially with the water depth. Thus, a new fourth-order mass transport velocity is found.