LXXII. <i>On ripples</i>

Wilton, Jayne

doi:10.1080/14786440508635350

Cited by 214 publications

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“…In the hydroelastic problem, generalised hydraulic fall solutions are also found to exist for small E b , where the upstream Froude number intersects the upstream linear dispersion relation twice. The resonance between the two modes is similar to the resonance in the gravity-capillary case (see, for example, Wilton 1915;Vanden-Broeck 2002), and thus waves of two different wavelengths travelling at the same speed can appear on ). However, the accurate computation of these waves is difficult, and we do not present results in this region due to the computational limitations.…”

Section: Fully Nonlinear Resultsmentioning

confidence: 65%

Hydraulic falls under a floating ice plate due to submerged obstructions

Page

Părău

2014

J. Fluid Mech.

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Steady two-dimensional nonlinear flexural-gravity hydraulic falls past a submerged obstruction on the bottom of a channel are considered. The fluid is assumed to be ideal and is covered above by a thin ice plate. Cosserat theory is used to model the sheet of ice as a thin elastic shell, and boundary integral equation techniques are then employed to find critical flow solutions. By utilising a second obstruction, solutions with a train of waves trapped between two obstructions are investigated.

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Section: Fully Nonlinear Resultsmentioning

confidence: 65%

Hydraulic falls under a floating ice plate due to submerged obstructions

Page

Părău

2014

J. Fluid Mech.

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“…Relations (4.5)-(4.8) are often referred to as the Stokes expansion. Stokes' work was later generalized to the complete GC system by Wilton [11], Schwartz & Vanden-Broeck [12], Chen & Saffman [13] and others. As we shall see ef 1 (x, y) and eh 1 (x) are the classical linear approximations and the terms of order e 2 , e 3 , .…”

Section: (A) Analytical Solutionsmentioning

confidence: 99%

Two-dimensional generalized solitary waves and periodic waves under an ice sheet

Vanden‐Broeck

Părău

2011

Phil. Trans. R. Soc. A.

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“…When there is no nonzero solution, nonlinear waves do not bifurcate from uniform horizontal streams; when there is only one solution, a sheet of solutions representing a parametrized family of bifurcations from simple eigenvalues occurs; when there are two independent solutions, there bifurcate three sheets of small-amplitude periodic waves. The latter corresponds to the presence of secondary bifurcations from curves of "special" solutions, the hydroelastic analogue of what are known as Wilton ripples [15], as described in the Abstract and in Section 1.2. To quote from [6], "Waves characterized by two dominant modes are often called Wilton's ripples in the literature in reference to Wilton's paper (1915).…”

Section: Introductionmentioning

confidence: 99%

Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves

Baldi¹,

Toland²

2010

Interfaces Free Bound.

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The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic extensible rod, and the fluid beneath is supposed to be in steady two-dimensional irrotational motion under gravity. When the wavelength has been normalized to be 2π , and when gravity and the density of the undeformed membrane are prescribed, there are two free parameters in the problem: the speed of the wave and the drift velocity of the membrane.It is observed that the problem, when linearized about uniform horizontal flow, has at most two independent solutions for any values of the parameters. When the linearized problem has only one normalized solution, it is shown that the full nonlinear problem has a sheet of solutions consisting of a family of curves bifurcating from simple eigenvalues. Here one of the problem's parameters is used to index a family of bifurcation problems in which the other is the bifurcation parameter.When the linearized problem has two solutions, with wave numbers k and l such that max{k, l}/min{k, l} / ∈ Z, it is shown that there are three two-dimensional sheets of bifurcating solutions. One consists of "special" solutions with minimal period 2π/k; another consists of "special" solutions with minimal period 2π/ l; and the third, apart from those on the curves where it intersects the "special" sheets, consists of "general" solutions with minimal period 2π .The two sheets of "special" solutions are rather similar to those that occur when the linearized problem has only one solution. However, points where the first sheet or the second sheet intersects the third sheet are period-multiplying (or symmetry-breaking) secondary bifurcation points on primary branches of "special" solutions. This phenomenon is analogous to that of Wilton ripples, which arises in the classical water-wave problem when the surface tension has special values. In the case of Wilton ripples, the coefficient of surface tension and the wave speed are the problem's two parameters. In the present context, there are two speed parameters, meaning that the membrane elasticity does not need to be highly specified for this symmetry-breaking phenomenon to occur.

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LXXII. On ripples

Cited by 214 publications

References 0 publications

Hydraulic falls under a floating ice plate due to submerged obstructions

Hydraulic falls under a floating ice plate due to submerged obstructions

Two-dimensional generalized solitary waves and periodic waves under an ice sheet

Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves

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