The bifurcation and secondary bifurcation of capillary-gravity waves

Toland, John; Jones, Mark

doi:10.1098/rspa.1985.0063

Cited by 50 publications

(26 citation statements)

References 8 publications

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“…Reeder & Shinbrot [9] analysed the problem with more rigour but again they confined themselves to Wilton ripples. Capillary-gravity waves on deep water have been treated formally by Chen & Saffman [2] and rigorously by Jones & Toland [7,11].…”

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“…Introduction 1.1 INTRODUCTORY REMARKS. Recently a number of studies (Chen & Saffman [2], Jones & Toland [7,11], Hogan [5]) have been made of periodic capillary-gravity waves which form the free surface of an ideal fluid contained in a channel of infinite depth. However, little work appears to have been done on the corresponding problem when the depth is finite.…”

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Small amplitude capillary-gravity waves in a channel of finite depth

Jones

1989

Glasgow Math. J.

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Introductory Remarks. Recently a number of studies (Chen & Saffman [2], Jones & Toland [7,11], Hogan [5]) have been made of periodic capillary-gravity waves which form the free surface of an ideal fluid contained in a channel of infinite depth. However, little work appears to have been done on the corresponding problem when the depth is finite. The most significant contributions appear to be those of Reeder & Shinbrot [9], Barakat & Houston [1] and Nayfeh [8] all of whom confined themselves to Wilton ripples (see §1.3). Yet there are sound reasons why such a study should be made. For quite apart from the unsolved problem regarding the type of capillary-gravity waves which may occur at finite depths, the consideration of the finite depth problem may be regarded as a first step in the study of solitary capillary-gravity waves. In this paper, a new integral equation for the infinite depth problem, due to J. F. Toland and the author, is adapted to be of use in tackling the finite depth problem. Using this we obtain results for the exact equations of motion which answer rigorously the questions of existence and multiplicity of small amplitude solutions of the periodic capillary-gravity wave problem of finite depth.

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Small amplitude capillary-gravity waves in a channel of finite depth

Jones

1989

Glasgow Math. J.

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“…Therefore the general solutions on this sheet represent a symmetry-breaking (or period-multiplying) secondary bifurcation on the curves of special solutions. This is the hydroelastic analogue of Wilton ripples, a type of water wave which arises in the presence of surface tension [8,14,15]. In Wilton-ripple theory there are also two parameters, the wave speed and the surface tension coefficient (which measures surface elasticity).…”

Section: Bifurcation Picturementioning

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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“…This fact is essentially known early in this century (Wilton [24] ) . The mathematical proof of the existence of the secondary branches is given in [12,23] . A slightly different proof was given later in [16] .…”

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confidence: 99%

“…The problem is known to be a bifurcation problem ( [12,13,16,22,23]). In ® p is the nondimensional gravity constant ; ® q is the nondimensional capillarity constant ; o 77 Is the aspect ratio of the flow.…”

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Remarks on the Bifurcation of Two Dimensional Capillary-Gravity Waves of Finite Depth

Okamoto¹,

Shoji²

1994

Publ. Res. Inst. Math. Sci.

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We consider the problem to determine the two dimensional water waves of permanent configuration. The problem is a bifurcation problem ([16]). In this paper, we define a mapping G between two Hilbert spaces and prove that the solutions of the above problem are in one-to-one correspondence to the zeros of the mapping G. Our most important contribution in this paper is to clarify the role of the aspect ratio, i.e., the ratio between the mean depth of the flow and the wave length. In particular, we prove that there is no degenerate bifurcation point, whatever the aspect ratio may be.

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The bifurcation and secondary bifurcation of capillary-gravity waves

Cited by 50 publications

References 8 publications

Small amplitude capillary-gravity waves in a channel of finite depth

Small amplitude capillary-gravity waves in a channel of finite depth

Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves

Remarks on the Bifurcation of Two Dimensional Capillary-Gravity Waves of Finite Depth

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