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).
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).
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.
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.
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.
Characteristics of stationary wave groups observed in numerical simulations.
Initial conditions for selected laboratory experiments.
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