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Abstract: An asymptotic solution that describes a small amplitude gravity-capillary wave propagating on the surface of a gentle sloping beach is derived in the Lagrangian coordinates. The analytical solution in Lagrangian form satisfies the zero pressure at the free surface. In the Lagrangian approximation, the parametric expression of water particles can be obtained directly and explicitly as a function of the wave steepness, the bottom slope and surface tension. The analytical solution for wave asymmetry parameter up to the breaker line for an arbitrary bottom slope can also be derived. The Lagrangian solution enables the description of the features of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the process of successive deformation of a wave profile which leads to wave breaking. Furthermore, by comparing the theoretical values of wave asymmetry with experimental results, it is found that theoretical results of the present solution are in good agreement with the experimental data. It is also found that surface tension lower the breaking wave height, lengthen the wave length and increase the breaking water depth. [Copyright &y& Elsevier]

Abstract: Within the Lagrangian reference frame we present a third-order solution for water particles in a two-dimensional wave and shear current interaction flow. For the case of linear shear current, parametric solutions of particle trajectory are obtained in the Lagrangian coordinates. In particular, the Lagrangian wave frequency and the Lagrangian mean level of particle motion can be obtained which are different from those in an Eulerian description. For the case of negative vorticity, when compared with the pure wave case, particle trajectory traveling in the positive -direction is found to be larger, and faster, while the opposite is true when traveling in the negative direction. Moreover, the influence of a negative vorticity is found to increase the relative horizontal distance traveled by a water particle, while the converse is true in the case of a positive vorticity. [Copyright &y& Elsevier]

Abstract: A new second-order asymptotic solution that describes short-crested waves is derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid satisfies the irrotational condition and there being zero pressure at the free surface, in contrast with the Eulerian solution, in which there is residual pressure at the free surface. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level. The mass transport velocity and Lagrangian mean level associated with particle displacement can also be obtained directly. In particular, the mean level of the particle motion in a Lagrangian form differs that of the Eulerian form. The new formulation reduces to second-order standing or progressive wave solutions in Lagrangian coordinates at the limiting angles of approach. Expressions for kinematic quantities are also presented. [Copyright &y& Elsevier]

Abstract: A third-order asymptotic solution in a Lagrangian description is presented for the finite-amplitude standing waves in water of uniform depth. Regarding the frequency of particle motion to be a function of the wave steepness and using a successive Taylor series expansion to the path and the period of particle motion, the explicit parametric equation of water particles and the Lagrangian wave frequency up to third-order could be obtained. In particular, the Lagrangian mean level which differs from that in the Eulerian approach is also found as a part of the solutions. The variations in the wave profile and the water particle orbits for the nonlinear standing waves are also investigated. Comparison on the third-order wave profiles given by the Eulerian and Lagrangian solution with the experiments reveals that the latter is more accurate than the former in describing the shape of the wave profile. [Copyright &y& Elsevier]

This paper presents a second-order asymptotic solution in the Lagrangian description for nonlinear partial standing wave in the finite water depth. The asymptotic solution that is uniformly valid satisfies the irrotationality condition and zero pressure at the free surface. In the Lagrangian approximation, the explicit nonlinear parametric equations for the particle trajectories are obtained. In particular, the Lagrangian mean level of a particle motion for the partial standing wave is found as a part of the solution which is different from that in an Eulerian system. This solution enables the description of wave profile and particle trajectory, which can be progressive, standing or partial standing waves. The dynamic properties of nonlinear partial standing waves, including mass transport velocity, radiation stress, wave setup and pressure due to reflection are also investigated. [ABSTRACT FROM AUTHOR]

A third-order analytical solution for the gravity–capillary standing wave is derived in Lagrangian coordinates through the Lindstedt–Poincare perturbation method. By numerical computation, the dynamical properties of nonlinear standing waves with surface tension in finite water depth, including particle trajectory and surface profile are investigated. We find that the presence of surface tension leads to a change of the crest form. Moreover, we also find that the particle trajectories near the surface oscillate back and forth along the arcs which will change from concave to convex as the inverse Bond number increases. There is no mass transport of the particles in a wave period. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an experimental and theoretical investigation of nonlinear water wave propagation over a sloping bed. Firstly, a series of monochromatic wave laboratory experiments were performed in order to measure the particle trajectories, evolution of wave profile, and wave phase velocity as wave propagates on a sloping bottom. The particle trajectories are quantified by means of images from a high speed camera, whereas the evolution of wave profile and variation of wave phase velocity are measured by a wave gauge array. Subsequently, the free-surface elevation, phase velocity, particle trajectories, and breaking wave height are estimated using a Lagrangian nonlinear wave transformation model. Model predictions show a reasonable agreement with experimental data. [Copyright &y& Elsevier]

In this paper, we use the perturbation method to develop a new mathematical derivation to describe nonlinear partial standing wave over uniformly sloping bottoms. In the Lagrangian coordinate system, the particle trajectories are obtained as a function of the nonlinear order parameter ɛ and the bottom slope α to the second order of the perturbation. The setups and mean return flow for an arbitrary bottom in the Lagrangian framework are also found as part of the solutions. The direct influences of reflection coefficient, wave steepness and bottom slope on the surface profiles of partially reflected waves are also derived. This nonlinear analytical solution is verified by reduction to the classical Stokes solution of progressive waves in both the deep water and constant depth limit. This solution allows to describe the successive deformation of partial reflected wave profiles and water particle trajectories prior to wave breaking to be described. The dynamic properties including mass transport, and Lagrangian mean level and radiation stress for nonlinear partially reflected waves on various sloping bottoms are investigated. [ABSTRACT FROM AUTHOR]