Trimodal Steady Water Waves

Ehrnström, Mats; Wahlén, Erik

doi:10.1007/s00205-014-0812-3

Cited by 19 publications

(39 citation statements)

References 29 publications

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“…This fact in combination with the weak regularity of the density gradient reduces the number of possible bifurcation parameters. For this reason the best (probably only) choice for a bifurcation parameter is the wavelength λ (the wavelength has also been used in [24] as one of the bifurcation parameters). It is worth pointing out that this choice provides a remarkable identity in (see (4.32) below) that leads us to a very simple and elegant dispersion relation, cf.…”

Section: Introductionmentioning

confidence: 99%

Stratified periodic water waves with singular density gradients

Escher

Knopf

Lienstromberg

et al. 2020

Annali di Matematica

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We consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each crest and trough. Our analysis uses, to a large extent, the availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis.2010 Mathematics Subject Classification. 35Q35; 35B32; 76B70; 76B47.

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Section: Introductionmentioning

confidence: 99%

Stratified periodic water waves with singular density gradients

Escher

Knopf

Lienstromberg

et al. 2020

Annali di Matematica

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“…To analytically capture the larger dimension of the space of solutions nearby the trivial ones, one requires an additional free parameter in addition to the wavespeed, used in the one-dimensional bifurcation. In line with [14] we choose to use the period as this extra parameter, while holding the surface tension fixed. The result, presented in Theorem 4.1, depends on the resonances between the two frequencies appearing in the nullspace: if one of the wavenumbers is a multiple of the other, one obtains a slit disk of solutions, excluding bifurcation straight in the direction of the higher wavenumber; if not, one obtains a full open disk of solutions, see Figure 2.…”

Section: Introductionmentioning

confidence: 99%

“…More generally, Wilton ripples, as these kinds of waves are sometimes called, have earlier been found to exist for the Euler equations with surface tension [26,30], and their spectral stability has been numerically investigated in [31]. They also exist in the presence of vorticity [25], even without capillarity [9,14]. In that case, one may even construct arbitrary large kernels [1,10], and corresponding multi-dimensional solution sets [23].…”

Section: Introductionmentioning

confidence: 99%

On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

et al. 2019

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We study the bifurcation of periodic travelling waves of the capillary-gravity Whitham equation. This is a nonlinear pseudodifferential equation that combines the canonical shallow water nonlinearity with the exact (unidirectional) dispersion for finite-depth capillarygravity waves. Starting from the line of zero solutions, we give a complete description of all small periodic solutions, unimodal as well bimodal, using simple and double bifurcation via Lyapunov-Schmidt reductions. Included in this study is the resonant case when one wavenumber divides another. Some bifurcation formulas are studied, enabling us, in almost all cases, to continue the unimodal bifurcation curves into global curves. By characterizing the range of the surface tension parameter for which the integral kernel corresponding to the linear dispersion operator is completely monotone (and therefore positive and convex; the threshold value for this to happen turns out to be T = 4 π 2 , not the critical Bond number 1 3 ), we are able to say something about the nodal properties of solutions, even in the presence of surface tension. Finally, we present a few general results for the equation and discuss, in detail, the complete bifurcation diagram as far as it is known from analytical and numerical evidence. Interestingly, we find, analytically, secondary bifurcation curves connecting different branches of solutions; and, numerically, that all supercritical waves preserve their basic nodal structure, converging asymptotically in L 2 (S) (but not in L ∞ ) towards one of the two constant solution curves. 4 π 2 , we are able in Theorem 2.6 to show that the kernel is completely monotone, a delicate structural property shared by the kernel for the linear dispersion in the pure gravity case (not its inverse). Moreover, we

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“…In [1] the authors, using different bifurcation parameters, construct bimodal waves that are different from those obtained in [11] and [12]. Later, in 2015, Ehrnström and Wahlén [13] established existence of trimodal waves. These are periodical waves of small amplitude with the first approximation given by a combination of three basic modes (cos-functions with different periods).…”

Section: Steady Waves Over Streams With Counter-currentsmentioning

confidence: 99%

“…In the papers [11], [1], [13] the authors construct symmetric small-amplitude and periodic waves for which the linear approximation is given by a combinations of two or three co-sinus functions with different wavelengths. In [11] they state a natural question: are there waves for which the linear approximation is given by any number of basic modes?…”

Section: N-modal Wavesmentioning

confidence: 99%

Small-amplitude steady water waves with vorticity

Lokharu¹

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The problem of describing two-dimensional traveling water waves is considered. The water region is of finite depth and the interface between the region and the air is given by the graph of a function. We assume the flow to be incompressible and neglect the effects of surface tension. However we assume the flow to be rotational so that the vorticity distribution is a given function depending on the values of the stream function of the flow. The presence of vorticity increases the complexity of the problem and also leads to a wider class of solutions.First we study unidirectional waves with vorticity and verify the Benjamin-Lighthill conjecture for flows whose Bernoulli constant is close to the critical one. For this purpose it is shown that every wave, whose slope is bounded by a fixed constant, is either a Stokes or a solitary wave. It is proved that the whole set of these waves is uniquely parametrised (up to translation) by the flow force which varies between its values for the supercritical and subcritical shear flows of constant depth. We also study largeamplitude unidirectional waves for which we prove bounds for the free-surface profile and for Bernoulli's constant.Second, we consider small-amplitude waves with counter currents. Such flows admit layers, where the fluid flows in different directions. In this case we prove that the initial nonlinear free-boundary problem can be reduced to a finite-dimensional Hamiltonian system with a stable equilibrium point corresponding to a uniform stream. As an application of this result, we prove the existence of non-symmetric wave profiles. Furthermore, using a different method, we prove the existence of periodic waves with an arbitrary number of crests per period.i ii Populärvetenskaplig sammanfattningI denna avhandling betraktar vi problemet att beskriva tvådimensionella vandrande vattenvågor. Vattenregionen antas ha ändligt djup och ytan mellan området och luften beskrivs av grafen till en funktion. Vi antar att flödet är inkompressibelt and bortser från ytspänningseffekter. Vidare antar vi att flödet är inte är rotationsfritt, så att vorticiteten beskrivs av en given funktion som beror på värden av strömmningsfunktionen för flödet. Existensen av vorticitet ökar komplexiteten hos problemet och leder också till en större lösningsklass.Först studerar vi enkelriktade vågor med vorticitet och verifierar Benjamin-Lighthillkonjekturen för flöden vars Bernoullikonstant ligger nära den kritiska nivån. Av denna anledning visas att varje våg vars lutning är begränsad av en fix konstant är antingen en Stokesvåg eller en solitär våg. Hela mängden av dessa vågor kan på ett entydigt sätt parametriseras (upp till translationer) av flödeskraften som varierar mellan värden för superkritiska och subkritiska skjuvflöden på konstant djup. Vi studerar också enkelriktade vågor med hög amplitud där vi visar begränsningar för den fria ytprofilen och för Bernoullikonstanten.Sedan betraktar vi lågamplitudsvågor med motströmmar. Sådana flöden tillåter lager där vätskan flödar i olika...

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Trimodal Steady Water Waves

Cited by 19 publications

References 29 publications

Stratified periodic water waves with singular density gradients

Stratified periodic water waves with singular density gradients

On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

Small-amplitude steady water waves with vorticity

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