On the generation of Kelvin-type waves by atmospheric disturbances

This paper considers the surface response of a semi-infinite, uniformly rotating, constant depth, homogeneous ocean to a variable atmospheric force. For a general wind and pressure system it is shown that forced Kelvin-type waves can be generated and that only the longshore wind component and the pressure can generate them. In particular a semi-infinite wind and moving pressure pattern are shown to generate Kelvin waves that travel away from the force discontinuities at the speed of shallow-water waves. The waves in the latter case exhibit a frequency shift typical of non-dispersive waves from a moving source. Some numerical values for the amplitudes of the Kelvin waves are also given.

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Solitons and other solutions of the long-short wave equation

International Journal of Numerical Methods for Heat &amp Fluid Flow ◽

10.1108/hff-07-2013-0233 ◽

2015 ◽

Vol 25 (1) ◽

pp. 129-145 ◽

Cited By ~ 10

Author(s):

Ghodrat Ebadi ◽

Aida Mojaver ◽

Sachin Kumar ◽

Anjan Biswas

Keyword(s):

Wave Equation ◽

Water Waves ◽

Soliton Solutions ◽

Expansion Method ◽

Short Wave ◽

Shallow Water Waves ◽

Content Type ◽

Integration Techniques ◽

Physical Phenomena ◽

The Waves

Purpose – The purpose of this paper is to discuss the integrability studies to the long-short wave equation that is studied in the context of shallow water waves. There are several integration tools that are applied to obtain the soliton and other solutions to the equation. The integration techniques are traveling waves, exp-function method, G′/G-expansion method and several others. Design/methodology/approach – The design of the paper is structured with an introduction to the model. First the traveling wave hypothesis approach leads to the waves of permanent form. This eventually leads to the formulation of other approaches that conforms to the expected results. Findings – The findings are a spectrum of solutions that lead to the clearer understanding of the physical phenomena of long-short waves. There are several constraint conditions that fall out naturally from the solutions. These poses the restrictions for the existence of the soliton solutions. Originality/value – The results are new and are sharp with Lie symmetry analysis and other advanced integration techniques in place. These lead to the connection between these integration approaches.

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Experiments on strong interactions between solitary waves

Journal of Fluid Mechanics ◽

10.1017/s0022112078000713 ◽

1978 ◽

Vol 85 (3) ◽

pp. 417-431 ◽

Cited By ~ 56

Author(s):

P. D. Weidman ◽

T. Maxworthy

Keyword(s):

Shallow Water ◽

Solitary Waves ◽

Water Waves ◽

Vries Equation ◽

Rectangular Channel ◽

Quantitative Agreement ◽

Strong Interactions ◽

Shallow Water Waves ◽

Qualitative And Quantitative ◽

The Waves

Experiments on the interaction between solitary shallow-water waves propagating in the same direction have been performed in a rectangular channel. Two methods were devised to compensate for the dissipation of the waves in order to compare results with Hirota's (1971) solution for the collision of solitons described by the Kortewegde Vries equation. Both qualitative and quantitative agreement with theory is obtained using the proposed corrections for wave damping.

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On Lagrangian drift in shallow-water waves on moderate shear

Journal of Fluid Mechanics ◽

10.1017/s0022112010002648 ◽

2010 ◽

Vol 660 ◽

pp. 221-239 ◽

Cited By ~ 7

Author(s):

W. R. C. PHILLIPS ◽

A. DAI ◽

K. K. TJAN

Keyword(s):

Boundary Layer ◽

Shear Flow ◽

Shallow Water ◽

Water Waves ◽

Stokes Drift ◽

Shallow Water Waves ◽

Sea Floor ◽

Langmuir Circulations ◽

A Minor ◽

The Waves

The Lagrangian drift in anO(ϵ) monochromatic wave field on a shear flow, whose characteristic velocity isO(ϵ) smaller than the phase velocity of the waves, is considered. It is found that although shear has only a minor influence on drift in deep-water waves, its influence becomes increasingly important as the depth decreases, to the point that it plays a significant role in shallow-water waves. Details of the shear flow likewise affect the drift. Because of this, two temporal cases common in coastal waters are studied, viz. stress-induced shear, as would arise were the boundary layer wind-driven, and a current-driven shear, as would arise from coastal currents. In the former, the magnitude of the drift (maximum minus minimum) in shallow-water waves is increased significantly above its counterpart, viz. the Stokes drift, in like waves in otherwise quiescent surroundings. In the latter, on the other hand, the magnitude decreases. However, while the drift at the free surface is always oriented in the direction of wave propagation in stress-driven shear, this is not always the case in current-driven shear, especially in long waves as the boundary layer grows to fill the layer. This latter finding is of particular interest vis-à-vis Langmuir circulations, which arise through an instability that requires differential drift and shear of the same sign. This means that while Langmuir circulations form near the surface and grow downwards (top down), perhaps to fill the layer, in stress-driven shear, their counterparts in current-driven flows grow from the sea floor upwards (bottom up) but can never fill the layer.

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Propagation of long waves due to atmospheric disturbances on a rotating sea

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1956.0010 ◽

1956 ◽

Vol 233 (1195) ◽

pp. 556-569 ◽

Cited By ~ 11

Keyword(s):

Steady State ◽

Long Waves ◽

Pressure Gradients ◽

Constant Amplitude ◽

Rotating System ◽

Dispersive Waves ◽

Atmospheric Disturbances ◽

The Waves ◽

Steady State Amplitude ◽

Maximum Group

Long waves in shallow water in a non-rotating system are not dispersive but in a rotating system they are. This paper investigates the generation and propagation of these dispersive waves in an infinite sea. The mode of generation is by air-pressure gradients or wind stresses applied to the surface. Bottom friction is neglected. The surface elevation due to a stationary force of constant amplitude suddenly applied and maintained at t = 0 over one-half of an infinite sea is shown to approach, through a series of oscillations approximating more or less to an inertia period, a steady-state amplitude decreasing with distance from the generating area, The longitudinal and transverse velocities are also given. The time elapsed from the initial disturbance at a point to the first maximum of elevation decreases with the distance of the point from the edge of the generating area. A generating area whose edge moves forward with the maximum group velocity of the waves is shown to lead to an elevation of ever-increasing height. The effect of a barrier placed at right angles to the direction of propagation is also briefly considered.

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The interaction of large amplitude shallow-water waves with an ambient shear flow: non-critical flows

Journal of Fluid Mechanics ◽

10.1017/s0022112072002836 ◽

1972 ◽

Vol 56 (2) ◽

pp. 241-255 ◽

Cited By ~ 17

Author(s):

P. A. Blythe ◽

Y. Kazakia ◽

E. Varley

Keyword(s):

Gravity Waves ◽

Water Waves ◽

Large Amplitude ◽

Vertical Direction ◽

Shallow Water Waves ◽

Flow Equations ◽

Hydraulic Flow ◽

New Class ◽

Simple Waves ◽

The Waves

This paper describes the behaviour of large amplitude, long gravity waves as they move over a horizontal bed into a region where the flow is steady but sheared in a vertical direction. A new class of exact solutions to the nonlinear hydraulic flow equations is derived. These solutions describe progressing waves and are sufficiently general to allow both the shape of the free surface at any instant and the shear profile of the undisturbed flow to be specified. The waves are examples of neutrally stable disturbances in the sense that they neither grow nor decay in amplitude, although, like simple waves on an unsheared flow, they can break.

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The propagation of continental shelf waves

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1968.0115 ◽

1968 ◽

Vol 305 (1481) ◽

pp. 235-250 ◽

Cited By ~ 77

Keyword(s):

Continental Shelf ◽

Kelvin Waves ◽

Negative Group ◽

Constant Depth ◽

Phase Velocities ◽

Shelf Waves ◽

Limiting Case ◽

The Waves ◽

Continental Shelf Waves ◽

Negative Group Velocity

Taking the case of a continental shelf of exponential slope, and assuming zero horizontal divergence, the authors derive a simple theory of free shelf waves, which, however, is more general than previous theories in that shorter as well as longer waves (in comparison with previous work) are taken into account. The properties of the waves are discussed, and the dispersion curves for each mode are obtained. Although the phase velocities of shelf waves are always in the same sense as those of Kelvin waves, there is a negative group velocity for a range of wavelengths, indicating that energy can propagate in the opposite sense. A similar approach is used to derive a theory for free waves propagating on a shelf between two regions of constant depth. The limiting case of a shelf of zero width is also considered, and is compared with a limiting case of the ‘double-Kelvin’ waves discovered by Longuet-Higgins (1967).

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Soliton interaction as a possible model for extreme waves in shallow water

Nonlinear Processes in Geophysics ◽

10.5194/npg-10-503-2003 ◽

2003 ◽

Vol 10 (6) ◽

pp. 503-510 ◽

Cited By ~ 54

Author(s):

P. Peterson ◽

T. Soomere ◽

J. Engelbrecht ◽

E. van Groesen

Keyword(s):

Shallow Water ◽

Water Waves ◽

Wave System ◽

Soliton Interaction ◽

Resonance Case ◽

Extreme Waves ◽

Shallow Water Waves ◽

Near Resonance ◽

Petviashvili Equation ◽

The Waves

Abstract. Interaction of two long-crested shallow water waves is analysed in the framework of the two-soliton solution of the Kadomtsev-Petviashvili equation. The wave system is decomposed into the incoming waves and the interaction soliton that represents the particularly high wave hump in the crossing area of the waves. Shown is that extreme surface elevations up to four times exceeding the amplitude of the incoming waves typically cover a very small area but in the near-resonance case they may have considerable extension. An application of the proposed mechanism to fast ferries wash is discussed.

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