Steady capillary-gravity waves on deep water

The properties of capillary-gravity waves of permanent form on deep water are studied. Two different formulations to the problem are given. The theory of simple bifurcation is reviewed. For small amplitude waves a formal perturbation series is used. The Wilton ripple phenomenon is reexamined and shown to be associated with a bifurcation in which a wave of permanent form can double its period. It is shown further that Wilton's ripples are a special case of a more general phenomenon in which bifurcation into subharmonics and factorial higher harmonics can occur. Numerical procedures for the calculation of waves of finite amplitude are developed. Bifurcation and limit lines are calculated. Pure and combination waves are continued to maximum amplitude. It is found that the height is limited in all cases by the surface enclosing one or more bubbles. Results for the shape of gravity waves are obtained by solving an integra-differential equation. It is found that the family of solutions giving the waveheight or equivalent parameter has bifurcation points. Two bifurcation points and the branches emanating from them are found specifically, corresponding to a doubling and tripling of the wavelength. Solutions on the new branches are calculated.

NONLINEAR GRAVITY-WAVES IN DEEP-WATER

In consideration of the problem on the boundary condition of nonlinear free water wave, coordinate transform is used to handle the free boundary. Supposing the solution form be the traveling wave, the ordinary differential equations of the one-order autonomous system with two variables are caused, then expanding the nonlinear terms at the equilibrium point with the Taylor expansion, we obtained the solution to traveling wave. The linear approximate equation near the equilibrium point is the small amplitude wave. A new nonlinear periodic traveling wave and nonlinear dispersion relation are shown when expanding to the second-order terms. A conclusion that the expansion of dispersion relation does not contain any odd-power terms of wave steepness and because of the nonlinear effort an oscillate structure is produced in the vertical direction is drawn.

Theory of small aspect ratio waves in deep water

In the limit of small values of the aspect ratio parameter (or wave steepness) which measures the amplitude of a surface wave in units of its wave-length, a model equation is derived from the Euler system in infinite depth (deep water) without potential flow assumption. The resulting equation is shown to sustain periodic waves which on the one side tend to the proper linear limit at small amplitudes, on the other side possess a threshold amplitude where wave crest peaking is achieved. An explicit expression of the crest angle at wave breaking is found in terms of the wave velocity. By numerical simulations, stable soliton-like solutions (experiencing elastic interactions) propagate in a given velocities range on the edge of which they tend to the peakon solution. (c) 2005 Elsevier B.V. All rights reserved.

An integrable evolution equation for surface waves in deep water

In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative method, an asymptotic model for small wave steepness ratio is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical results is performed.

Theory and calculation of the nonlinear energy transfer between sea waves in deep water /

"May 1982."

On the solution of the problem of scattering of surface water waves by a sharp discontinuity in the surface boundary conditions

Closed-form analytical expressions are derived for the reflection and transmission coefficients for the problem of scattering of surface water waves by a sharp discontinuity in the surface-boundary-conditions, for the case of deep water. The method involves the use of the Havelock-type expansion of the velocity potential along with an analysis to solve a Carleman-type singular integral equation over a semi-infinite range. This method of solution is an alternative to the Wiener-Hopf technique used previously.

Instability and evolution of nonlinearly interacting water waves

We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schrodinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term evolution of the latter by means of computer simulations of the governing nonlinear equations and demonstrate the formation of localized coherent wave envelopes. Our results should be useful for understanding the formation and nonlinear propagation characteristics of large-amplitude freak waves in deep water.

The modulational instability in deep water under the action of wind and dissipation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A multimodal physiotherapy programme plus deep water running for improving cancer-related fatigue and quality of life in breast cancer survivors

The aim of the study was to assess the feasibility and effectiveness of aquatic‐based exercise in the form of deep water running ( DWR ) as part of a multimodal physiotherapy programme ( MMPP ) for breast cancer survivors. A controlled clinical trial was conducted in 42 primary breast cancer survivors recruited from community‐based P rimary C are C entres. Patients in the experimental group received a MMPP incorporating DWR , 3 times a week, for an 8‐week period. The control group received a leaflet containing instructions to continue with normal activities. Statistically significant improvements and intergroup effect size were found for the experimental group for P iper F atigue S cale‐ R evised total score ( d = 0.7, P = 0.001), as well as behavioural/severity ( d = 0.6, P = 0.05), affective/meaning ( d = 1.0, P = 0.001) and sensory ( d = 0.3, P = 0.03) domains. Statistically significant differences between the experimental and control groups were also found for general health ( d = 0.5, P < 0.05) and quality of life ( d = 1.3, P < 0.05). All participants attended over 80% of sessions, with no major adverse events reported. The results of this study suggest MMPP incorporating DWR decreases cancer‐related fatigue and improves general health and quality of life in breast cancer survivors. Further, the high level of adherence and lack of adverse events indicate such a programme is safe and feasible.

Theory of Non-linear Waves in Multi-dimensions: with Special Reference to Surface Water Waves

The surface water waves are "modal" waves in which the "physical space" (t, x, y, z) is the product of a propagation space (t, x, y) and a cross space, the z-axis in the vertical direction. We have derived a new set of equations for the long waves in shallow water in the propagation space. When the ratio of the amplitude of the disturbance to the depth of the water is small, these equations reduce to the equations derived by Whitham (1967) by the variational principle. Then we have derived a single equation in (t, x, y)-space which is a generalization of the fourth order Boussinesq equation for one-dimensional waves. In the neighbourhood of a wave froat, this equation reduces to the multidimensional generalization of the KdV equation derived by Shen & Keller (1973). We have also included a systematic discussion of the orders of the various non-dimensional parameters. This is followed by a presentation of a general theory of approximating a system of quasi-linear equations following one of the modes. When we apply this general method to the surface water wave equations in the propagation space, we get the Shen-Keller equation.

A Note on Diffraction of Water Waves by a Nearly Vertical Barrier

A simplified perturbational analysis is employed, together with the application of Green's theorem, to determine the first-order corrections to the reflection and transmission coefficients in the problem of diffraction of surface water waves by a nearly vertical barrier in two basically important cases: (i) when the barrier is partially immersed and (ii) when the barrier is completely submerged. The present analysis produces the desired results fairly easily and relatively quickly as compared with the known integral equation approach to this class of diffraction problems.

Solution of a Boundary-Value Problem associated with Diffraction of Water Waves by a Nearly Vertical Barrier

An exact solution is derived for a boundary-value problem for Laplace's equation which is a generalization of the one occurring in the course of solution of the problem of diffraction of surface water waves by a nearly vertical submerged barrier. The method of solution involves the use of complex function theory, the Schwarz reflection principle, and reduction to a system of two uncoupled Riemann-Hilbert problems. Known results, representing the reflection and transmission coefficients of the water wave problem involving a nearly vertical barrier, are derived in terms of the shape function.

A New Approach to the Problem of Scattering of Water Waves by Vertical Barriers

Using a modified Green's function technique the two well-known basic problems of scattering of surface water waves by vertical barriers are reduced to the problem of solving a pair of uncoupled integral equations involving the “jump” and “sum” of the limiting values of the velocity potential on the two sides of the barriers in each case. These integral equations are then solved, in closed form, by the aid of an integral transform technique involving a general trigonometric kernel as applicable to the problems associated with a radiation condition.

Obliquely incident water waves against a vertical cliff

Utilizing the commutativity property of the Cartesian coordinate differential operators arising in the boundary conditions associated with the propagation of surface water waves against a vertical cliff, under the assumptions of linearized theory, the problem of obliquely incident surface waves is considered for solution. The case of normal incidence, handled by previous workers follow as a particular limiting case of the present problem, which exhibits a source/sink type behavior of the velocity potential at the shore-line. An independent method of attack is also presented to handle the case of normal incidence.

On the incoming water waves against a vertical cliff

The problems of obliquely incident surface water waves against a vertical cliff have been handled in both the cases of water of infinite as well as finite depth by straightforward uses of appropriate Havelock-type expansion theorems. The logarithmic singularity along the shore-line has been incorporated in a direct manner, by suitably representing the Dirac's delta function.