A detailed quantitative comparison of fully nonlinear computations with the measurements of unidirectional wave groups is presented. Computational results on evolving wave groups were compared with previous available experiments. The local surface elevation variation, the evolution of envelope shapes, the velocity of propagation of the steepest crests in the group and their relation to the height of the crests were obtained numerically and experimentally. Conditions corresponding to incipient wave breaking were investigated in greater detail. The results shed additional light on mechanisms leading to the breaking of steep waves, as well as on the crucial importance of exact matching between initial conditions in computations and experiments.

Over the past few decades, rogue, or freak, waves have attracted considerable
interest due to their destructive impact on offshore structures and ships

In the present study, we aim to extend the numerical analysis of the
conditions prevailing at the inception of breaking of the steepest crest in
the PB-like wave train by carrying out fully nonlinear simulations. The
simulations were performed using the conformal mapping method approach
developed by

In Sect.

The CS numerical scheme enables computation of a unidirectional wave field in
time, starting from a given spatial distribution as an initial condition,
under the assumption of potential flow. In laboratory experiments, however,
waves are generated by a wavemaker usually placed at one end of the
experimental facility. The experimental data are commonly accumulated only at
preselected fixed locations within the tank where the sensors are placed.
Quantitative comparison of numerical computations with experimental results
gained in those experiments may thus constitute a complicated task. For a
narrow-banded wave field this problem has been considered by

As demonstrated in

The present study has been carried out in the 18 m long, 1.2

Since the nonlinear numerical solver in the present study requires an initial condition as a given spatial distribution at a certain instant, the following procedure to determine the appropriate initial spatial distribution was adopted.

Using Eq. (

The temporal evolution of the initial wave field presented in the bottom
panel of Fig.

The initial surface elevation is given as a function of the physical variable

The prescribed initial variation of the surface elevation

For time integration, a fourth-order Runge–Kutta scheme was used. We refer
the reader to

The CS numerical scheme allows for the computation of the velocity potential
as a function of two parameters:

In Fig.

The temporal evolution of the surface elevation

The temporal evolution of the horizontal fluid velocity in a moving
reference frame; wave parameters as in Fig.

The spatial variation of the velocity of the fluid at the surface

Measurements in a wave tank are routinely performed using wave gauges spread
along the facility. To facilitate the direct comparison between numerical and
experimental results, we need to first determine the location of the
wavemaker in our numerical simulations. Then, we examine “vertical” cross
sections of the data as presented in Fig.

A closer look at the surface elevation variation with time is presented in
Fig.

In Fig.

Surface elevation variation at fixed values of

The computed temporal variation of the surface elevation at various
locations relative to the wavemaker in a moving (with

The computed variation with time of the normalized maximum crest
elevation relative to the background wave amplitude

It transpires from the comparison of the two curves in
Fig.

The experiments were carried out with the goal of enabling quantitative
comparison of the numerical results with experiments. The wavemaker driving
signal was designed to generate surface elevation variation in time
corresponding to the lowest curve in Fig.

An example of the sequence of the experimentally recorded wave trains for

It is impractical to carry out direct comparison of the fast varying surface
elevation records measured in the experiments as presented in
Fig.

Measured surface elevation

Comparison between the envelopes of the measured and numerical
surface elevations

Figure

The computed temporal variation of the maximum crest heights, of the
velocities of the steepest crests (

Variation along the tank of the maximum crest heights, propagation velocity of the highest crests, and the water particle velocity at those crests. Red lines denote numerical results, blue symbols experiments. The locations where breaking was observed in the experiments are marked.

Important parameters of the wave train in the course of its propagation along
the tank obtained in the simulations are plotted in
Figs.

The evolution of the steepest crest heights along the tank, as plotted in
Fig.

The bottom curve in Fig.

At distances exceeding about 7

It was suggested in

In this respect it should be stressed that the velocities

Frequency spectra at selected locations. Red line – numerical simulations; blue line – experimental data.

Wavenumber spectra at selected times.

The amplitude spectra of the wave train are plotted in
Figs.

In the present study, fully nonlinear numerical simulations of the evolution
of a unidirectional nonlinear wave train with an initial shape of a Peregrine
breather were qualitatively and quantitatively compared with the experimental
results. To the best of our knowledge, this is the first attempt to carry out
direct comparison of the results of fully nonlinear simulations of a
deterministic wave train with experiments. The simulations were carried out
using a conformal mapping approach as detailed in

The initial condition in experiments thus corresponds to surface elevation
variation with time at a prescribed wavemaker location.

In the present simulations, the initial spatial distribution of the surface
elevation is based on the PB analytical solution. In order to determine in
the numerical solution the measurable temporal variation of the surface
elevation at any given location along the tank, the initial spatial
distribution in the present study was centered upstream of the wavemaker; see
Fig.

Several important points regarding PB were highlighted in this study. The
solution (Eq.

Two different approaches were suggested to deal with the problems outlined in
the previous paragraph. In numerical simulations, the truncated wave train
with the spatial extension that contains an integer number of carrier
wavelengths is often used as the initial condition

The computational results indeed are in a good qualitative and, to a large
extent, quantitative agreement with the current experiments, as well as with
those of

A different approach was suggested by Saket et al., 2015, ArXiv 1508.07702, and Barthelemy et al., 2015, ArXiv 1508.06002, after this manuscript was submitted.

This combined numerical and experimental study of nonlinear wave trains also
clarifies the limitations of possible agreement between fully nonlinear
solution and experiment. We note that while the periodicity in the time
domain is possible for propagating and evolving unidirectional waves, they
are, strictly speaking, aperiodic in space. This point adds an additional
aspect to essential differences that exist between the spatial and temporal
formulations of the wave evolution problem, as discussed above. We therefore
believe that all nonlinear solutions based on spatially periodic boundary
conditions, as in the method adopted here, as well as in a variety of
alternative methods that employ spatial discrete Fourier decomposition,
contain intrinsic inaccuracy. These numerical solutions thus can only provide
approximate results and require careful experiments to verify their validity.
The present study shows that the fully nonlinear solution, although flawed,
yields better agreement with experiments than the application of the spatial
version of the modified nonlinear Schrödinger (Dysthe) equation limited
to the third order that does not require spatial periodicity
(

The support of this study by a grant no. 20102019 from the US-Israel Binational Science Foundation is gratefully acknowledged. The authors wish to thank Andrey Zavadsky for his valuable assistance. Edited by: V. Shrira Reviewed by: two anonymous referees