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Abstract.Within the Lagrangian reference framework we present a third-order trajectory solution for water particles in a two-dimensional wave-current interaction flow. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level and within a vertical water column, which were calculated previously by an approximation method using an Eulerian approach. Mass transport associated with a particle displacement can now be obtained directly in Lagrangian form without using the transformation from Eulerian to Lagrangian coordinates. In particular, the Lagrangian wave frequency and the Lagrangian mean level of particle motion can also be obtained, which are different from those in an Eulerian description. A series of laboratory experiments are performed to measure the trajectories of particles. By comparing the present asymptotic solution with laboratory experiments data, it is found that theoretical results show excellent agreement with experimental data. Moreover, the influence of a following current is found to increase the relative horizontal distance traveled by a water particle, while the converse is true in the case of an opposing current.

[1] Wave attenuation over a mud (kaolinite) layer is investigated via laboratory experiments and numerical modeling. The rheological behavior of kaolinite exhibits hybrid properties of a Bingham and pseudoplastic fluid. Moreover, the measured time-dependent velocity profiles in the mud layer reveal that the shear rate under wave loading is highly phase dependent. The measured shear rate and rheological data allow us to back-calculate the time-dependent viscosity of the mud layer under various wave loadings, which is also shown to fluctuate up to 1 order of magnitude during one wave period. However, the resulting time-dependent bottom stress is shown to only fluctuate within 25% of its mean. The back-calculated wave-averaged bottom stress is well correlated with the wave damping rate in the intermediate-wave energy condition. The commonly adopted constant viscosity assumption is then evaluated via linear and nonlinear wave-mud interaction models. When driving the models with measured wave-averaged mud viscosity (forward modeling), the wave damping rate is generally overpredicted under the low wave energy condition. On the other hand, when a constant viscosity is chosen to match the observed wave damping rate (inverse modeling), the predicted velocity profiles in the mud layer are not satisfactory and the corresponding viscosity is lower than the measured value. These discrepancies are less pronounced when waves become more energetic. Differences between the linear and nonlinear model results become significant under low-energy conditions, suggesting an amplification of wave nonlinearity due to non-Newtonian rheology. In general, the constant viscosity assumption for modeling wave-mud interaction is only appropriate for more energetic wave conditions.

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In this paper, analytical particular solutions of the augmented polyharmonic spline (APS) associated with Reissner plate model are explicitly derived in order to apply the dual reciprocity method. In the derivations of the particular solutions, a coupled system of three second-ordered partial differential equations (PDEs), which governs problems of Reissner plates, is initially transformed into a single six-ordered PDE by the Hörmander operator decomposition technique. Then the particular solutions of the coupled system can be found by using the particular solution of the six-ordered PDE derived in the first author's previous study. These formulas are further implemented for solving problems of Reissner plates under arbitrary loadings. In the solution procedure, an arbitrary loading measured at some scattered points is first interpolated by the APS and a corresponding particular solution can then be approximated by using the prescribed formulas. After that the complementary homogeneous problem is formally solved by the method of fundamental solutions (MFS). Numerical experiments are carried out to validate these particular solutions.