Mathieu Progressive Waves

A new family of exact solutions to the wave equation representing relatively undistorted progressive waves is constructed using separation of variables in the elliptic cylindrical coordinates and one of the Bateman transforms. The general form of this Bateman transform in an orthogonal eurvilinear cylindrical coordinate system is discussed and a specific problem of physical feasibility of the obtained solutions, connected with their dependence on the cyclic coordinate, is addressed. The limiting case of zero eccentricity, in which the elliptic cylindrical coordinates turn into their circular cylindrical counterparts, is shown to correspond to the focused wave modes of the Bessel-Gauss type.

The Transient Solution of Plane Progressive Waves

Generated by an ideal sinusoidal motion of the vertical plate, the simplest linear solution in time domain for two-dimensional regular waves is derived. The solution describes the propagation process of the plane progressive wave with a front, and will approach the linear steady- state solution as the oscillation time of the plate approaches infinity. The solution presented in this paper can be used to provide an incident wave model with analytical expression for solving the problems of diffraction and response of floating bodies in time domain.

Numerical study of nonlinear interactions of bi-chromatic progressive deep-water waves

This paper proposes a 3-D non-hydrostatic free surface flow model with a newly proposed general boundary-fitted grid system to simulate the nonlinear interactions of the bi-chromatic deep-water gravity waves.First,the monochromatic bidirectional and bi-chromatic bidirectional waves of small wave steepness are successively simulated to verify the abilities of the numerical model.Then,a series of bi-chromatic progressive waves of moderate wave steepness and different crossing angles are simulated and analyzed in detail.It is found that if the crossing angle is close to or smaller than the resonant angle,apparent discrepancies are observed among the numerical results,the linear wave theory,and the steady third-order theory.Otherwise,the three solutions coincide well.Through analysis,it is concluded that the discrepancies are caused by the third-order quasi-resonant interactions between the bi-chromatic progressive waves.Such interactions not only could modify the wave spectrum,but could also change the wave shape patterns.

NUMERICAL SOLUTION OF A NONLINEAR REACTION-DIFFUSION EQUATION

A nonlinear reaction-diffusion equation is studied numerically by a Petrov-Galerkin finite element method, which has been proved to be 2nd-order accurate in time and 4th-order in space. The comparison between the exact and numerical solutions of progressive waves shows that this numerical scheme is quite accurate, stable andefflcient. It is also shown that any local disturbance will spread, have a full growth and finally form two progressive waves propagating in both directions. The shape and the speed of the long term progressive waves are determined by the system itself, and do not depend on the details of the initial values.

Analytic approximation to nonlinear hydroelastic waves traveling in a thin elastic plate floating on a fluid

An analytic approximation method known as the homotopy analysis method(HAM)is applied to study the nonlinear hydroelastic progressive waves traveling in an infinite elastic plate such as an ice sheet or a very large floating structure(VLFS)on the surface of deep water.A convergent analytical series solution for the plate deflection is derived by choosing the optimal convergencecontrol parameter.Based on the analytical solution the efects of diferent parameters are considered.We find that the plate deflection becomes lower with an increasing Young’s modulus of the plate.The displacement tends to be flattened at the crest and be sharpened at the trough as the thickness of the plate increases,and the larger density of the plate also causes analogous results.Furthermore,it is shown that the hydroelastic response of the plate is greatly afected by the high-amplitude incident wave.The results obtained can help enrich our understanding of the nonlinear hydroelastic response of an ice sheet or a VLFS on the water surface.

PROGRESSING WAVE SOLUTIONS TO QUASI-LINEAR SYSTEMS MIXED PROBLEMS

The author studies the technique of paradifferential operator defined on a space of conormaldistribution with three indeces,and then use this technique to prove that a progressing wavewhich hits the boundary is reflected according to the usual law.