^{1,2,a)}, G. F. Clauss

^{3}, M. Klein

^{3,b)}and M. Onorato

^{4,c)}

### Abstract

The problem of existence of stable nonlinear groups of gravity waves in deep water is considered by means of laboratory and numerical simulations with the focus on strongly nonlinear waves. Wave groups with steepness up to A cr ω m 2/g ≈ 0.30 are reproduced in laboratory experiments (A cr is the wave crest amplitude, ω m is the mean angular frequency, and g is the gravity acceleration). We show that the groups remain stable and exhibit neither noticeable radiation nor structural transformation for more than 60 wavelengths or about 15-30 group lengths. These solitary wave patterns differ from the conventional envelope solitons, as only a few individual waves are contained in the group. Very good agreement is obtained between the laboratory results and numerical simulations of the potential Euler equations. The envelope soliton solution of the nonlinear Schrödinger equation is shown to be a reasonable first approximation for specifying the wave-maker driving signal. The short intense envelope solitons possess vertical asymmetry similar to regular Stokes waves with the same frequency and crest amplitude. Nonlinearity is found to have remarkably stronger effect on the speed of envelope solitons in comparison to the nonlinear correction to the Stokes wave velocity.

The research has received funding from the EC's Seventh Framework Programme FP7-SST-2008-RTD-1 project EXTREME SEAS – Design for Ship Safety in Extreme Seas (http://www.mar.ist.utl.pt/extremeseas/) under Grant Agreement No. 234175. A.S. acknowledges partial support from the EC's Seventh Framework Programme FP7-PEOPLE-2009-IIF under Grant Agreement No. 254389/909389, and also RFBR 11-02-00483 and 12-05-33087. M.O. has been funded by the Office of Naval Research (ONR) Grant No. N000141010991. M.O. acknowledges Dr. B. Giulinico for discussions. A.S. is grateful to colleagues in the Department of Mathematics and EPSAM at Keele for the warm hospitality. We thank the anonymous referee who suggested a 3D instability as possible mechanism which can limit the solitary group amplitude in laboratory experiments.

I. INTRODUCTION

II. WEAKLY NONLINEAR THEORY FOR ENVELOPE SOLITONS

III. NUMERICAL SIMULATIONS

IV. LABORATORY TESTS

V. CONCLUSIONS

### Key Topics

- Hydrodynamic waves
- 65.0
- Acoustic standing waves
- 19.0
- Laboratory procedures
- 16.0
- Hydrodynamic solitary waves
- 14.0
- Time series analysis
- 10.0

## Figures

Evolution of the potential, kinetic, and total energies (see the legend) in numerical simulations; variations with respect to the corresponding initial values. Experiment No. 9 from Table I is shown (k 0 A 0 = 0.3).

Evolution of the potential, kinetic, and total energies (see the legend) in numerical simulations; variations with respect to the corresponding initial values. Experiment No. 9 from Table I is shown (k 0 A 0 = 0.3).

Evolution of some features of the stationary wave group as function of time. The steepness of wave troughs and wave crests is shown by dashed and thin solid lines, respectively; estimation of the steepness on the basis of the wave height is given by thick solid line; dots show maximum local slope of the surface displacement. Cases k 0 A 0 = 0.2 (a) and k 0 A 0 = 0.3 (b) (experiment Nos. 3 and 9 from Table I , respectively).

Evolution of some features of the stationary wave group as function of time. The steepness of wave troughs and wave crests is shown by dashed and thin solid lines, respectively; estimation of the steepness on the basis of the wave height is given by thick solid line; dots show maximum local slope of the surface displacement. Cases k 0 A 0 = 0.2 (a) and k 0 A 0 = 0.3 (b) (experiment Nos. 3 and 9 from Table I , respectively).

Envelopes of the stationary wave groups which are observed in numerical simulations, for values k 0 A 0 = 0.15, 0.16, 0.20, 0.22, 0.23, 0.25, 0.28, 0.29, 0.30, 0.31, 0.32 (a), and the wave envelope vertical asymmetry A cr /A tr as function of dimensionless crest amplitude (stars) (b). Dashed line in (b) shows the asymmetry of the uniform Stokes wave, when the strongly nonlinear correction to the frequency is taken into account.

Envelopes of the stationary wave groups which are observed in numerical simulations, for values k 0 A 0 = 0.15, 0.16, 0.20, 0.22, 0.23, 0.25, 0.28, 0.29, 0.30, 0.31, 0.32 (a), and the wave envelope vertical asymmetry A cr /A tr as function of dimensionless crest amplitude (stars) (b). Dashed line in (b) shows the asymmetry of the uniform Stokes wave, when the strongly nonlinear correction to the frequency is taken into account.

The stationary wave group generated from the initial condition characterized by k 0 A 0 = 0.30 (experiment No. 9 from Table I ). Surface elevations (solid lines) and wave envelopes (dashed lines) are given in panels (a) and (b) as functions of coordinate and time, respectively. The wavenumber spectrum and frequency spectrum at the moments of maximum wave crest (solid lines) and the deepest trough (dashed lines) are shown in panels (c) and (d).

The stationary wave group generated from the initial condition characterized by k 0 A 0 = 0.30 (experiment No. 9 from Table I ). Surface elevations (solid lines) and wave envelopes (dashed lines) are given in panels (a) and (b) as functions of coordinate and time, respectively. The wavenumber spectrum and frequency spectrum at the moments of maximum wave crest (solid lines) and the deepest trough (dashed lines) are shown in panels (c) and (d).

Test setup – side view on the seakeeping basin with the wave generator on the left, the damping slope on the right, as well as the positions of the ten wave gauges installed for this test campaign.

Test setup – side view on the seakeeping basin with the wave generator on the left, the damping slope on the right, as well as the positions of the ten wave gauges installed for this test campaign.

Time series of the surface elevation at different distances, measured in the laboratory tank: an unstable wave group (a) (experiment No. 30.29) and stationary wave group (b) (experiment No. 30.16). Both the cases correspond to k 0 A 0 = 0.3, but to different carrier wave frequencies and different methods of signal generation.

Time series of the surface elevation at different distances, measured in the laboratory tank: an unstable wave group (a) (experiment No. 30.29) and stationary wave group (b) (experiment No. 30.16). Both the cases correspond to k 0 A 0 = 0.3, but to different carrier wave frequencies and different methods of signal generation.

Stationary wave groups: comparison between laboratory and numerical results. The sequence of thin solid lines is the time series of surface elevations registered by 10 gauges, when plotted in co-moving references. The panels show results of experiments 29.14 and 30.07, k 0 A 0 = 0.20 (a), experiments 30.13 and 30.16, k 0 A 0 = 0.30 (b), and experiment 30.37, k 0 A 0 = 0.35 (c). The series from gauge 3 is given by a thicker line. The enveloping curves (dashed lines) are obtained in the strongly numerical simulations of the Euler equations with appropriate amplitudes of the initial condition: k 0 A 0 = 0.15 (a), k 0 A 0 = 0.23 (b), and k 0 A 0 = 0.29 (c) (simulations 1, 5, and 8 from Table I , respectively). Horizontal dotted lines mark the scaled envelope crest amplitudes, which are A cr ω 2 m /g ≈ 0.150, 0.235, and 0.301 for cases (a), (b), and (c), respectively.

Stationary wave groups: comparison between laboratory and numerical results. The sequence of thin solid lines is the time series of surface elevations registered by 10 gauges, when plotted in co-moving references. The panels show results of experiments 29.14 and 30.07, k 0 A 0 = 0.20 (a), experiments 30.13 and 30.16, k 0 A 0 = 0.30 (b), and experiment 30.37, k 0 A 0 = 0.35 (c). The series from gauge 3 is given by a thicker line. The enveloping curves (dashed lines) are obtained in the strongly numerical simulations of the Euler equations with appropriate amplitudes of the initial condition: k 0 A 0 = 0.15 (a), k 0 A 0 = 0.23 (b), and k 0 A 0 = 0.29 (c) (simulations 1, 5, and 8 from Table I , respectively). Horizontal dotted lines mark the scaled envelope crest amplitudes, which are A cr ω 2 m /g ≈ 0.150, 0.235, and 0.301 for cases (a), (b), and (c), respectively.

Scaled frequency Fourier amplitude spectrum for experiment No. 30.16 in linear (a) and semi-logarithmic (b) coordinates. The lines show results from all 10 gauges.

Scaled frequency Fourier amplitude spectrum for experiment No. 30.16 in linear (a) and semi-logarithmic (b) coordinates. The lines show results from all 10 gauges.

Velocities of stationary wave groups, observed in numerical simulations (stars), and of the selected “best” wave groups measured in laboratory experiments (circles). The speed of the uniform Stokes wave is given by the dashed line for the reference.

Velocities of stationary wave groups, observed in numerical simulations (stars), and of the selected “best” wave groups measured in laboratory experiments (circles). The speed of the uniform Stokes wave is given by the dashed line for the reference.

## Tables

Characteristics of stationary wave groups observed in numerical simulations.

Characteristics of stationary wave groups observed in numerical simulations.

Initial conditions for selected laboratory experiments.

Initial conditions for selected laboratory experiments.

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