The reflection of a solitary wave by a vertical wall

Byatt-Smith, J. G. B.

doi:10.1017/s0022112088003349

Cited by 42 publications

(33 citation statements)

References 26 publications

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“…Another striking feature of the solitary wave reflection/collision was added by Su & Mirie (1980), who observed dispersive trailing waves behind the solitary waves after the collision from their thirdorder perturbation solution for the Boussinesq equation. Byatt-Smith (1988, 1989 made similar observations by solving two weakly coupled KdV equations. He also noticed slight changes in the heights of the solitary waves after interactions.…”

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confidence: 54%

Contact line dynamics and boundary layer flow during reflection of a solitary wave

2012

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In this paper we present a set of wave flume experiments for a solitary wave reflecting off a vertical wall. A particle tracking velocimetry (PTV) technique is used to measure free-surface velocity and the velocity field in the vicinity of the moving contact line. We observe that the free surface undergoes the so-called rolling motion as the contact line moves up and down the vertical wall, and fluid particles on the free surface almost always flow toward the wall except at the end of the reflection process. As the contact line descends along the wall, wall boundary layer flows move in a downward direction and therefore the boundary layer acts like a conduit through which the surface-rolling-induced flow escapes from the meniscus. However, during the last phase of the reflection process flow reversal occurs inside the wall boundary layer. An approximate analytical solution is developed to explain the flow reversal feature. Very good agreement between the approximate theory and measured data is obtained. Because of the flow reversal, boundary layer flows collide with the surfacerolling-induced flows. The collision gives rise to a jet ejecting from the meniscus into the water body, which later evolves into a small eddy. It is noticed that the fluid particles in different regions such as the free stream, the free-surface boundary layer and the wall boundary layer, can be transported to other regions by passing through the meniscus.

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confidence: 54%

Contact line dynamics and boundary layer flow during reflection of a solitary wave

2012

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“…Furthermore, since these papers consider specifically the head-on collision of solitary waves, they are concerned with events which occur on relatively short time scales (i.e., time scales of O( 1 ) in our scaling). As noted in [5, p. 503], these expansions are not uniformly valid in time, and it is not clear whether or not their solutions could be controlled over time scales of O( It is worth noting that, in spite of the differences between our approach and those discussed here, Byatt-Smith [5] also finds that corrections to the amplitude of the solitary wave evolve according to the linearized KdV equation. An alternative approach to improve the KdV approximation to water waves is to work directly with a Boussinesq approximation to the water wave problem, as done by Bona and Chen in [2].…”

Section: Some Numericsmentioning

confidence: 81%

“…For the actual water wave equations, there have been a number of studies of corrections to the KdV approximation to water waves spanning the spectrum from nonrigorous asymptotic expansions [4], [5], [24], [2] to numerical solutions of the equations of motion and comparison with the KdV predictions [3], [25], [9] to experimental investigations [16], [7]. We concentrate here on the theoretical studies since they have the closest connection to our work.…”

Section: Some Numericsmentioning

confidence: 99%

Higher Order Modulation Equations for a Boussinesq Equation

Wayne¹,

Wright²

2002

SIAM J. Appl. Dyn. Syst.

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Abstract. In order to investigate corrections to the common KdV approximation to long waves, we derive modulation equations for the evolution of long wavelength initial data for a Boussinesq equation. The equations governing the corrections to the KdV approximation are explicitly solvable, and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone. We also present the results of numerical experiments which show that the error estimates we derive are essentially optimal.Key words. Boussinesq equation, KdV approximation, modulation equations, water wave problem AMS subject classifications. 76B15, 35Q51, 35Q53PII. S11111111024112981. Introduction. Modulation, or amplitude, equations are approximate, often explicitly solvable, model equations derived-usually through asymptotic analysis and the method of multiple time scales-to model more complicated physical situations. Although these equations have been used for over a century, only lately has there been an attempt to rigorously relate solutions of the modulation equations to the original physical problem. [23], the validity of Korteweg-de Vries (KdV) equations as a leading order approximation to the evolution of long wavelength water waves and to a number of other dispersive partial differential equations has been established.While the KdV approximation is extremely useful due to its simplicity and the fact that the KdV equation can be explicitly solved by the inverse scattering transform, both experimentally and numerically one observes departures from the predictions of the KdV equation. Our goal in this paper is to derive modulation equations which govern corrections to the KdV model. In the present paper, we will not work with the full water wave problem but rather will study modulation equations for long wavelength solutions of the Boussinesq equation:

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“…This anomalous result is explained in Section 3.2 (see p. 17). Byatt-Smith (1987b, 1988 considered the interaction of solitary-wave solutions of the BenjaminBona-Mabony (BBM) equation and the reflection of a solitary wave from a wall by the same perturbation technique based on inverse scattering. Kkhenassamy & Olver (1992) considered the extended KdV equation of (1.1) and showed that exact solitary-wave solutions (waves with sech 2 profiles) exist only for the special parameter combination c 2 + c 3 = 30c 4 , 3c 3 = c,.…”

Section: Introductionmentioning

confidence: 99%

Soliton interaction for the extended Korteweg-de Vries equation

1996

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Soliton interactions for the extended Korteweg-de Vries (KdV) equation are examined. It is shown that the extended KdV equation can be transformed (to its order of approximation) to a higher-order member of the KdV hierarchy of integrable equations. This transformation is used to derive the higher-order, two-soliton solution for the extended KdV equation. Hence it follows that the higher-order solitary-wave collisions are elastic, to the order of approximation of the extended KdV equation. In addition, the higher-order corrections to the phase shifts are found. To examine the exact nature of higher-order, solitary-wave collisions, numerical results for various special cases (including surface waves on shallow water) of the extended KdV equation are presented. The numerical results show evidence of inelastic behaviour well beyond the order of approximation of the extended KdV equation; after collision, a dispersive wavetrain of extremely small amplitude is found behind the smaller, higher-order solitary wave.

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The reflection of a solitary wave by a vertical wall

Cited by 42 publications

References 26 publications

Contact line dynamics and boundary layer flow during reflection of a solitary wave

Contact line dynamics and boundary layer flow during reflection of a solitary wave

Higher Order Modulation Equations for a Boussinesq Equation

Soliton interaction for the extended Korteweg-de Vries equation

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