The instability of Wilton ripples

Trichtchenko, Olga; Deconinck, Bernard; Wilkening, Jon

doi:10.1016/j.wavemoti.2016.06.004

Cited by 25 publications

(55 citation statements)

References 32 publications

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“…This depth was picked for illustrative purposes only and it can be compared with the results in Ref. for gravity‐capillary waves. For illustrative purposes, we pick the flexural rigidity parameter so that the resonant mode is

K = 7

(ie,

D \approx 1.65 \times 10^{- 5}

) as presented in Figure , which shows 7 secondary minima and the resonant mode

K = 10

(ie,

D \approx 8.11 \times 10^{- 6}

) as shown in Figure where we see 10 secondary minima in the bottom left part of the plot of the normalised wave profile.…”

Section: More General Resultsmentioning

confidence: 99%

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Stability of periodic traveling flexural‐gravity waves in two dimensions

et al. 2018

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In this work, we solve the Euler's equations for periodic waves traveling under a sheet of ice. These waves are referred to as flexural‐gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrödinger equation that describes the modulational instability of periodic traveling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier–Floquet–Hill method to show they agree qualitatively. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyze high‐frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice.

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K = 7

(ie,

D \approx 1.65 \times 10^{- 5}

) as presented in Figure , which shows 7 secondary minima and the resonant mode

K = 10

(ie,

D \approx 8.11 \times 10^{- 6}

) as shown in Figure where we see 10 secondary minima in the bottom left part of the plot of the normalised wave profile.…”

Section: More General Resultsmentioning

confidence: 99%

“…The red is the nonlinear model (solid line, labeled NL) and the blue is the linear model (dashed line, labeled LIN). The black line represents (22). The gray area is the unstable region (focusing NLS regime) and the white area is the stable region.…”

Section: Asymptotic Analysismentioning

confidence: 99%

Stability of periodic traveling flexural‐gravity waves in two dimensions

et al. 2018

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“…In Section 3.2.2, we will show compactness of (Θ, Γ) given appropriate choice of domain, as the bifurcation theorem we shall apply to (27), (28) requires an "identity plus compact" formulation.…”

Section: 4mentioning

confidence: 99%

“…However, as in [9] (and what was used in our "identity plus compact" formulation (27), (28)), we can write W * as the sum…”

Section: 1mentioning

confidence: 99%

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Periodic traveling interfacial hydroelastic waves with or without mass

Akers

Ambrose

Sulon

2017

Z. Angew. Math. Phys.

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Abstract. We study the motion of an interface between two irrotational, incompressible fluids, with elastic bending forces present; this is the hydroelastic wave problem. We prove a global bifurcation theorem for the existence of families of spatially periodic traveling waves on infinite depth. Our traveling wave formulation uses a parameterized curve, in which the waves are able to have multi-valued height. This formulation and the presence of the elastic bending terms allows for the application of an abstract global bifurcation theorem of "identity plus compact" type. We furthermore perform numerical computations of these families of traveling waves, finding that, depending on the choice of parameters, the curves of traveling waves can either be unbounded, reconnect to trivial solutions, or end with a wave which has a self-intersection. Our analytical and computational methods are able to treat in a unified way the cases of positive or zero mass density along the sheet, the cases of single-valued or multi-valued height, and the cases of single-fluid or interfacial waves.

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A Method for Identifying Stability Regimes Using Roots of a Reduced-Order Polynomial

Trichtchenko

2019

Nonlinear Water Waves

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For dispersive Hamiltonian partial differential equations of order 2N + 1, N ∈ Z, there are two criteria to analyse to examine the stability of small-amplitude, periodic travelling wave solutions to highfrequency perturbations. The first necessary condition for instability is given via the dispersion relation. The second criterion for instability is the signature of the eigenvalues of the spectral stability problem given by the sign of the Hamiltonian. In this work, we show how to combine these two conditions for instability into a polynomial of degree N . If the polynomial contains no real roots, then the travelling wave solutions are stable. We present the method for deriving the polynomial and analyse its roots using Sturm's theory via an example.

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The instability of Wilton ripples

Cited by 25 publications

References 32 publications

Stability of periodic traveling flexural‐gravity waves in two dimensions

Stability of periodic traveling flexural‐gravity waves in two dimensions

Periodic traveling interfacial hydroelastic waves with or without mass

A Method for Identifying Stability Regimes Using Roots of a Reduced-Order Polynomial

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