Lagrangian approach to interfacial water waves with free surface

This is a theoretical study on the interfacial water waves with a free surface in a two-layer system, where the lower fluid is taken to be heavier than the upper one. Lagrangian matching conditions are introduced for the physical fields separated by the interface. And a perturbation analysis is carried out to the third order to find the particle trajectory in the Lagrangian description. Observing the derived solution, a symbolic computation is introduced for obtaining the fifth order solution. The Lagrangian drifts, wave frequency, and set-up are also given as part of the solutions. Discontinuities across the interface are found for all of these physical quantities. A generalized set-up effect is found that the Lagrangian mean levels come near to both of the free surface and internal interface. Through some numerical calculations, it is shown that the larger density difference or relative wave height results in the larger drift velocity. The direction of particle motion in the upper layer is anti-clockwise in contrary to that in the lower layer. Better convergence for the Lagrangian solution than the Eulerian one is numerically demonstrated for the barotropic mode that the wave motion is dominated on the free surface.