scholarly journals AN ELECTROMAGNETIC ANALOGY FOR LONG WATER WAVES

1980 ◽  
Vol 1 (17) ◽  
pp. 48
Author(s):  
G.W. Jackson ◽  
D.L. Wilkinson
Keyword(s):  
Electromagnetic Radiation ◽  
Gravity Waves ◽  
Wave Energy ◽  
Water Waves ◽  
Wave Length ◽  
Alternative Methods ◽  
Linear Wave ◽  
Microwave Range ◽  
Nodal Lines ◽  
Long Wave

Seiching must be considered when designing mooring, berthing or navigational facilities in semi enclosed basins and harbours. The considerable oscillatory currents which may be generated along nodal lines of a seiche can result in serious surging of a vessel moored in such areas. Hydraulic modelling of the long gravity waves which form the seiche poses particular problems for the hydraulic modeller. In the prototype situation, a balance is achieved between the long wave energy entering a basin, the energy dissipation within the basin, and the wave energy which is radiated back into the ocean. The presence of the wave generator in the model may result in much of the radiated energy being reflected back into the basin, thereby distorting the seiche observed in the model. The relatively low Reynolds numbers present in the model leads to exaggerated frlctional damping of the waves and possible suppression of certain resonant modes. Scaling of long wave amplitudes in resonance' situations cannot be determined from simple Froude laws but must be based on equivalent dissipation rates in the model and the prototype (Ippen, 1966). Alternative methods of physically reproducing long wave behaviour have been studied, such- as acoustic modelling (Nakamura, 1977). However, these models also have their limitations and are largely confined to constant depth situations. It Is suggested that electromagnetic radiation in the microwave range may be used to model long gravity waves and that this method has many advantages over the various techniques which have previously been used. It can be rigorously shown that the laws which govern the propagation, reflection and dissipation of electromagnetic radiation (Maxwell's equations) are identical to the linearised equations describing the motion of long gravity waves, (Jackson and McKee, 1980). The linear wave equation gives an accurate description of long wave oscillation in basins as the wave length is of the same order as the basin dimensions and non-linear effects do not have time to develop.

Radiation of short waves from the resonantly excited capillary–gravity waves

2016 ◽  
Vol 810 ◽  
pp. 5-24 ◽  
Author(s):  
M. Hirata ◽  
S. Okino ◽  
H. Hanazaki
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Gravity Waves ◽  
Wave Pattern ◽  
Short Wave ◽  
Bond Number ◽  
Linear Wave ◽  
Cnoidal Wave ◽  
Short Waves ◽  
Long Wave

Capillary–gravity waves resonantly excited by an obstacle (Froude number: $Fr=1$) are investigated by the numerical solution of the Euler equations. The radiation of short waves from the long nonlinear waves is observed when the capillary effects are weak (Bond number: $Bo<1/3$). The upstream-advancing solitary wave radiates a short linear wave whose phase velocity is equal to the solitary waves and group velocity is faster than the solitary wave (soliton radiation). Therefore, the short wave is observed upstream of the foremost solitary wave. The downstream cnoidal wave also radiates a short wave which propagates upstream in the depression region between the obstacle and the cnoidal wave. The short wave interacts with the long wave above the obstacle, and generates a second short wave which propagates downstream. These generation processes will be repeated, and the number of wavenumber components in the depression region increases with time to generate a complicated wave pattern. The upstream soliton radiation can be predicted qualitatively by the fifth-order forced Korteweg–de Vries equation, but the equation overestimates the wavelength since it is based on a long-wave approximation. At a large Bond number of $Bo=2/3$, the wave pattern has the rotation symmetry against the pattern at $Bo=0$, and the depression solitary waves propagate downstream.

Solutions, group analysis and conservation laws of the (2+1)-dimensional time fractional ZK–mZK–BBM equation for gravity waves

2020 ◽  
pp. 2150140
Author(s):  
Changna Lu ◽  
Shengxiang Chang ◽  
Zongguo Zhang ◽  
Hongwei Yang
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Inverse Method ◽  
Fractional Derivatives ◽  
Group Analysis ◽  
Bilinear Method ◽  
Wave Regime ◽  
Long Wave ◽  
Conservation Theorem ◽  
Bbm Equation

Based on the investigation of (2+1)-dimensional ZK–mZK–BBM equation, it describes the gravity water waves in a long-wave regime. With the help of the semi-inverse method and the variational method, the time fractional ZK–mZK–BBM equation is derived in the sense of Riemann–Liouville fractional derivatives, which opens a new window for understanding the features of gravity water waves. Further, the symmetry of the (2+1)-dimensional time fractional ZK–mZK–BBM equation is studied by fractional order symmetry. Meanwhile, based on the new conservation theorem, the conserved laws of (2+1)-dimensional time fractional ZK–mZK–BBM equation are constructed. Finally, we show how to derive the solutions of the time fractional ZK–mZK–BBM equation by a bilinear method and the radial basis functions (RBFs) meshless approach.

scholarly journals New solutions for the propagation of long water waves over variable depth

1994 ◽  
Vol 278 ◽  
pp. 391-406 ◽  
Author(s):  
Yinglong Zhang ◽  
Songping Zhu
Keyword(s):  
Wave Propagation ◽  
Gravity Waves ◽  
Analytical Solutions ◽  
Water Waves ◽  
Numerical Solutions ◽  
Practical Significance ◽  
Variable Depth ◽  
Surface Gravity Waves ◽  
Long Wave ◽  
Variable Water

Based on the linearized long-wave equation, two new analytical solutions are obtained respectively for the propagation of long surface gravity waves around a conical island and over a paraboloidal shoal. Having been intensively studied during the last two decades, these two problems have practical significance and are physically revealing for wave propagation over variable water depth. The newly derived analytical solutions are compared with several previously obtained numerical solutions and the accuracy of those numerical solutions is discussed. The analytical method has the potential to be used to find solutions for wave propagation over more natural bottom topographies.

scholarly journals Excitation Forces Estimation for Non-linear Wave Energy Converters: A Neural Network Approach

2020 ◽  
Vol 53 (2) ◽  
pp. 12334-12339
Author(s):  
M. Bonfanti ◽  
F. Carapellese ◽  
S.A. Sirigu ◽  
G. Bracco ◽  
G. Mattiazzo
Keyword(s):  
Neural Network ◽  
Wave Energy ◽  
Linear Wave ◽  
Network Approach ◽  
Neural Network Approach ◽  
Wave Energy Converters ◽  
Non Linear ◽  
Energy Converters

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

On stability of parallel flow of an incompressible fluid of variable density and viscosity

1962 ◽  
Vol 58 (4) ◽  
pp. 646-661 ◽  
Author(s):  
P. G. Drazin
Keyword(s):  
Incompressible Fluid ◽  
Water Waves ◽  
Salt Water ◽  
Reynolds Numbers ◽  
Variable Density ◽  
Stability Equation ◽  
Wave Disturbances ◽  
Long Wave ◽  
Vertical Variations ◽  
The Stability

ABSTRACTSome aspects of generation of water waves by wind and of turbulence in a heterogeneous fluid may be described by the theory of hydrodynamic stability. The technical difficulties of these problems of instability have led to obscurities in the literature, some of which are elucidated in this paper. The stability equation for a basic steady parallel horizontal flow under the influence of gravity is derived carefully, the undisturbed fluid having vertical variations of density and viscosity. Methods of solution of the equation for large Reynolds numbers and for long-wave disturbances are described. These methods are applied to simple models of wind blowing over water and of fresh water flowing over salt water.

On periodic water-waves and their convergence to solitary waves in the long-wave limit

1981 ◽  
Vol 303 (1481) ◽  
pp. 633-669 ◽  
Keyword(s):  
Integral Equation ◽  
Solitary Waves ◽  
Water Waves ◽  
Periodic Problem ◽  
Existence Theory ◽  
Periodic Waves ◽  
Equation Formulation ◽  
Long Wave ◽  
Integral Equation Formulation ◽  
Wave Limit

A detailed discussion of Nekrasov’s approach to the steady water-wave problems leads to a new integral equation formulation of the periodic problem. This development allows the adaptation of the methods of Amick & Toland (1981) to show the convergence of periodic waves to solitary waves in the long-wave limit. In addition, it is shown how the classical integral equation formulation due to Nekrasov leads, via the Maximum Principle, to new results about qualitative features of periodic waves for which there has long been a global existence theory (Krasovskii 1961, Keady & Norbury 1978).

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

Steep gravity waves in water of arbitrary uniform depth

1977 ◽  
Vol 286 (1335) ◽  
pp. 183-230 ◽  
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Perturbation Parameter ◽  
Intermediate Energy ◽  
Finite Amplitude ◽  
Wave Profile ◽  
Accurate Calculation ◽  
Theoretical Developments ◽  
Deep Water Waves

Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Padé approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Padé-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.

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