Stable deep-water waves propagating in one and two dimensions

Diane Henderson1,∗ and Harvey Segur2

1 218 McAllister Building, Department of Mathematics, Penn State University, University Park PA 16802, USA2 Department of Applied Mathematics, University of Colorado, Boulder Colorado, 80309–0526, USA

This work is dedicated to the memory of Joseph Hammack.

Deep-water, narrow-banded wavetrains of uniform amplitude propagating in one horizontal dimension were shown to beunstable to small perturbations with nearly the same frequency and direction about 40 years ago. More recently, interactions oftwo narrow-banded wavetrains of uniform amplitude propagating in arbitrary directions were shown to be similarly unstable.In both cases, the instabilities have been described by either a single nonlinear Schrodinger (NLS) equation or by two coupledNLS equations. However, the inclusion in these equations of any amount of damping (of a particular type) stabilizes theinstabilities. Experiments show that the evolution of perturbed wavetrains in both cases is more accurately described by theNLS models that include damping when the amplitudes are small or moderate. When the amplitude of either the underlyingwaves or the perturbations is large, neither the undamped NLS model nor the damped NLS model accurately predicts theobserved behaviour, which includes frequency downshifting.

1 Introduction

In a landmark paper, Benjamin & Feir [1] showed that a uniform train of plane waves of moderate amplitude in deep waterwithout dissipation is unstable to a small perturbation of other waves travelling in the same direction with nearly the samefrequency. This modulational instability is a finite-amplitude effect, in the sense that the unperturbed wave train (the carrierwave) must have finite amplitude, and the growth rate of the instability is proportional to the square of that amplitude, at leastfor small amplitudes. More recently, several authors, including [2, 5–7], showed that deep-water waves with two-dimensionalsurface patterns (so the velocity fields are three-dimensional), resulting from the interaction of two uniform-amplitude wave-trains are similarly unstable. These instability results occur in the undamped case only. If damping effects are included in themodel equations, then these instabilities are stabilized as shown by [3,6,9]. Here we summarize comparisons of the undampedand damped nonlinear Schrodinger (NLS) models with experiments and verify that any amount of damping, of the right type,stabilizes the instability when the wave amplitudes are small or moderate. Our experiments (not shown here) also show thatthe question of modulational stability is not central for large amplitude patterns, for which downshifting occurs so that boththe undamped and the damped NLS-type models are inadequate.

2 Stabilizing the instability

Zakharov [10] derived the (undamped) nonlinear Schrodinger equation to describe approximately the slow evolution of thecomplex envelope, ψ, of a nearly monochromatic train of plane waves of moderate amplitude in deep water. Other authorsincluded the effects of damping so that an NLS equation in two spatial dimensions that includes linear damping is

i(∂t − δ)ψ + α∂2xψ + β∂2

yψ + γ|ψ|2ψ = 0, (1)

where {α, β, γ, δ > 0} are real-valued constants. Segur et al [3] showed that in contrast to the undamped, instability result,the uniform wavetrain solution of this damped NLS equation (1) is stable to long-wave perturbations. Their experimentsconfirmed this result for waves with moderate amplitudes. An example is given in Figure 1, which shows results from anexperiment on a (3.33 Hz) wavetrain (carrier wave), with a long-wave perturbation imposed, propagating in one horizontaldimension. The figure shows a time series measured at a fixed location and its corresponding Fourier Transform. The imposedlong-wave perturbation is apparent in the time series as a modulated envelope and in the Fourier Transform as sideband peakson either side of the carrier wave. The carrier wave is predicted to be (i) unstable by the undamped (δ = 0) version of (1), (ii)stable by the damped (δ > 0) version, and (iii) unstable by an undamped version that also includes higher-order terms. Thesepredictions are depicted in Figures 1(c) and (d), which show results of numerical integrations of (i), (ii), and (iii). All of thenumerics used measured amplitudes and phases for initial data (obtained from Figure 1(a) and (b)). The damping rate used inmodel (ii) was determined by measurements of the energy decay in the time series. The higher-order NLS used in model (iii)

∗ Corresponding author E-mail: dmh@math.psu.edu, Phone: +1 814 865 7527, Fax: +1 814 865 3735

PAMM · Proc. Appl. Math. Mech. 7, 1100401–1100402 (2007) / DOI 10.1002/pamm.200700393

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

0 5 10 15 20

time �s�

�0.2

�0.1

0

0.1

0.2

0.3

(a)0 2 3 6 8 10

frequency �Hz �

0.01

0.1

(b)0 100 200 300 400 500 600

X �cm �

0.1

0.2

2�a�

1��c

m�

(c)0 100 200 300 400 500 600

X �cm �

0.1

0.2

2�a

1��c

m�

(d)

Fig. 1 Time series (a) and corresponding Fourier Transform (b) for a perturbed wavetrain. Predictions and measurements (dots) of theevolution of the Fourier amplitudes of the left-hand (c) and right-hand (d) sidebands as a function of distance down the tank. Predictionsare from (1) with δ = 0.026/sec, (0.11/m) – solid curves, (1) with δ = 0 – dashed curves, (1) with higher-order terms included – dottedcurves. Each dot corresponds to Fourier amplitudes from one set of measurements at a fixed spatial location, like those in (b). The measuredvalues are multiplied by an exponential growth term to factor out damping.

is the one given in [8]. The curves show the Fourier amplitudes of the left- and right-hand sidebands as they evolve a distanceX from the initial gage location. Also shown are the measured Fourier amplitudes. Both undamped models (which areindistinguishable in this experiment) predict unbounded growth within the length of the wavetank. However, the perturbationdid not grow; the wavetrain remained stable, in agreement with the damped-NLS equation, which predicts stabilization.

A similar result holds for the interaction of wavetrains. It was shown in [2, 5–7, 9], that deep-water waves with two-dimensional surface patterns (so the velocity fields are three-dimensional), resulting from the interaction of two uniform-amplitude wavetrains are modulationally unstable within the context of undamped coupled NLS equations. However, as in thecase of a single uniform wavetrain, the uniform pattern solution of coupled NLS equations is stable if the damping rates arenon-zero as shown by [6, 9].

Figure 2 shows a photograph from [9] of an experiment using two symmetric wavetrains, forming a pattern of rectangularcells dilineated by nodal lines of no surface displacement parallel to the direction of propagation (which is from the top to thebottom of the photo). Gages that measured surface displacement were placed in one of these nodal lines and in the antinodalregions on either side. (See [4] for details of the experimental facility and [9] for a comparison with theory.) In this experimentthe pattern was seeded with a long-wave perturbation. Figure 2 also shows predictions of the Fourier amplitudes of the leftand right hand sidebands in the antinodal regions due to the perturbation, using measured initial amplitudes and phases in theundamped and the damped versions of coupled NLS equations. The damped version, which predicts stability of the pattern,more accurately predicts the measured evolution of the perturbation.

In conclusion, dissipation stabilizes the modulational instability in single and coupled NLS-type models, and is required inthese models to describe accurately the evolution of small to moderate amplitude wave patterns.

0 50 100 150

x �cm �

0.1

0.2

0.3

a�

1�c

m�

(a)

0 50 100 150

x �cm �

0.1

0.2

0.3

a1�c

m�

(b)

Fig. 2 Photo of a symmetric pattern of per-turbed interacting wavetrains. Predictions andmeasurements or the Fourier amplitudes of the(a) left- and (b) right-hand sidebands. Pre-dictions are from the undamped (dashed) anddamped (solid) versions of coupled NLS equa-tions. The undamped results are multiplied byan exponential decay to account for comparisonwith data.

We thank the National Science Foundation for support through grants NSF-DMS-FRG-0139847, NSF-DMS-FRG 0139742,NSF-DMS 0708352 and NSF-DMS 0709415.

References

[1] T.B. Benjamin, J.E. Feir, J. Fluid Mech. 27, 417 (1967).[2] A.K. Dar and K.P. Das, Phys Fluids, A3, 3021, (1991).[3] H. Segur, D. Henderson, J. Carter, J. Hammack, C-M Li, D. Pheiff, K. Socha, J. Fluid Mech., 539, 229, (2005).[4] J. L. Hammack, D. M. Henderson, H. Segur, J. Fluid Mech., 532, 1, (2005).[5] M. Onorato, A.R. Osborne, M. Serio, Phys. Rev. Lett., 96, 014503-1, (2006).[6] W. Craig, D.M. Henderson, M. Oscamou, H. Segur, Math. Comput. Simul., doi:10.1016/j.matcom (2006).[7] P.K. Shukla, I. Kourakis, B. Eliasson, M. Marklund, L. Stenflo, Phys. Rev. Lett., 97, 094501-1 (2006).[8] E. Lo, C.C. Mei, J. Fluid Mech., 150, 395, (1985).[9] H. Segur, D.M. Hendrson, Euro. Phys. J. - Special Topics, 147, 25, (2007).

[10] V.E. Zakharov, J. Appl. Mech. Tech. Phys., 2, 190, (1968).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Minisymposia – 10 Fluid Mechanics 1100402