# Extreme wave phenomena in down-stream running modulated waves

Andonowati<EMAIL_ADDRESS>N. Karjanto  E. van Groesen  Department of
Mathematics, Institut Teknologi Bandung, Jl. Ganesha 10 Bandung, 40123,
Indonesia  Department of Applied Mathematics, University of Twente, The
Netherlands

Received 1 June 2005; received in revised form 1 March 2006; accepted 12 April
2006
Available online 27 June 2006

###### Abstract

Modulational, Benjamin-Feir, instability is studied for the down-stream
evolution of surface gravity waves. An explicit solution, the soliton on
finite background, of the NLS equation in physical space is used to study
various phenomena in detail. It is shown that for sufficiently long modulation
lengths, at a unique position where the largest waves appear, phase
singularities are present in the time signal. These singularities are related
to wave dislocations and lead to a discrimination between successive ‘extreme’
waves and much smaller intermittent waves. Energy flow in opposite directions
through successive dislocations at which waves merge and split, causes the
large amplitude difference. The envelope of the time signal at that point is
shown to have a simple phase plane representation, and will be described by a
symmetry breaking unfolding of the steady state solutions of NLS. The results
are used together with the maximal temporal amplitude MTA, to design a
strategy for the generation of extreme (freak, rogue) waves in hydrodynamic
laboratories.
© 2006 Elsevier Inc. All rights reserved.

###### keywords:

Extreme waves , Freak waves , Rogue waves , Phase singularity , Wave
dislocation , Benjamin-Feir instability , Maximal temporal amplitude , Soliton
on finite background , Wave generation , Hydrodynamic laboratory

###### MSC:

[2008] 35Q51 , 76B25 , 35Q55 , 35B30 , 76E30 , 76B15

##  1 Introduction

Weakly nonlinear, dispersive evolution equations describe phenomena in such
areas as nonlinear optics and gravity-driven surface waves on a layer of
fluid. When the diverging effect of dispersion and the converging effect of
nonlinearity balance each other, phenomena like solitons and soliton-like
interactions appear for exponentially confined waves, their periodic
equivalents as well as for the collective behaviour of waves in wave groups
[1, 2]. Amplitude increase and large deformation of wave profiles are
characteristic and relatively easy observable phenomena. However, the details
of the processes and the interaction between them remain rather difficult,
despite many investigations. In fact, interesting phenomena can be studied in
detail theoretically only in simplified models. Fortunately, the important
phenomena seem to be generic enough to be captured by these models, so that
what we learn from these models is also useful to describe important aspects
of more realistic situations. This paper will contribute to this type of
research in the following way.

Taking the well-known model for the collective behaviour of wave groups that
is called the NLS-(NonLinear Schrödinger) equation, we will study in much
detail one specific solution. This solution, the so-called soliton of finite
background (SFB), can be written down explicitly, and we will extract from it
detailed information about such processes like (extreme) amplitude
amplification, phase singularity and wave dislocation, a generic form of
extreme waves, and spectral evolution and interaction of wave modes. Some of
the properties to be described in this paper will be known at least
qualitatively, but to our knowledge the explicit description of the properties
given below in the physical setting leads to some surprising and illuminating
results, in particular about the wave dislocation and properties of the
extreme signal.

The direct motivation of this paper stems from the need of hydrodynamic
laboratories to generate ‘extreme’ waves, often also called ‘freak’ or ‘rogue’
waves. In such waves, ships and marine constructions can be tested under
extreme sea-conditions. Technical restrictions on the wavemaker make it
impossible to generate such large waves directly; the nonlinearity in the
physical phenomenon when the wave is running downstream is exploited to obtain
the large waves. Hence, to generate large waves at a certain position in a
long wave tank, the motion of a wave maker at one end of the wave tank will
generate moderately small waves that start to travel downstream along the tank
over an initially still water level. We can view this evolution as a
signalling problem: the wave elevation at each position defines a time-signal,
so that the down-stream evolution of waves correspond to the subsequent
signals as a response to the time signal at the position of the wave maker.
Due to the non-linear effects determined by nature, large deformations may
appear and large and steep waves can emerge within the wave tank. Such strong
non-linear effects and the appearance of large amplitude amplification in
generated waves can be seen in experimental results [3, 4] as well as
numerical results [5, 6, 7, 8]. It is of the laboratory’s interest to generate
extreme waves in a deterministic way based on reliable theoretical
predictions, for which we will use the SFB solution of the NLS model in this
paper.

Different from the deterministic wave generation in a laboratory, in real
seas, many investigations on extreme (freak) waves in random sea states have
been carried out in the past years. Then an extreme wave is understood [9] as
a large wave with wave height $H$ exceeding the significant wave height
$H_{s}$ by a factor of $2.2$. Most of these studies are for initial value
problems: different from the signalling problem to be considered here, then
the temporal evolution of an initial wave elevation over a spatial domain is
investigated.

In all such investigations, extreme waves, often occurring in wave group
structures, are of most interest and discussed in several papers, such as [10,
11, 12, 13, 14]. The nonlinear self-focusing phenomenon is described to result
in very steep waves of high amplitude that arise intermittently within wave
group structures and that govern the dynamics of extreme wave occurrences [15,
16]. Modulational instability of Benjamin-Feir type [17] is often described as
the cause of self-focusing that leads to the generation of extreme waves. In
this paper, we will investigate this effect in more detail. To facilitate the
understanding of the phenomenon, we refer to Fig. 1.

Figure 1: A snapshot of a wavefield of waves generated at the left, while
travelling to the right; the physical unit along the axis is in meter, and the
wavelength is approximately equal to the depth of the layer. The dash-dotted
line denotes the envelope of the wave groups, which consist of about six
waves, while the dashed line denotes the MTA, the maximum temporal amplitude,
which is the largest possible wave elevation at a fixed position. See the text
for a more extensive description.

This figure shows, in real dimensions, at a certain time a spatial view of
(first order) waves with a wavelength of approximately the depth of the layer
of fluid, about $5$ m. The waves are generated at the left, at $x=0$ in the
figure, and travel to the right. A number of waves (about six waves in the
figure) of varying amplitude, form a wave group, which has the envelope as
depicted by the dashed line in the figure. These wave groups are seen to
change substantially depending on the distance from the wavemaker: from small
modulations close to the wave maker, to severely amplified and distorted wave
groups near the point where the waves of maximal amplitude appear (in the
figure at about $200$ m), after which the groups decrease again for increasing
distance. This plot of the waves will change at each moment, the waves
translate with the phase velocity and the envelope with the group velocity.
The maximal wave height at a certain position is given by the dotted line,
which denotes the maximal temporal amplitude MTA. The MTA, introduced
originally in [18], is a useful concept in the study of the long-time
behaviour of wavefields; see also [19, 20] where it was used to study the
phenomenon of wave breaking.

Fig. 1 shows wave groups that will be studied in this paper. These waves
evolve from a modulated monochromatic wave. In fact, we will use a fully
nonlinear extension of the linear modulational (Benjamin-Feir) instability of
a monochromatic wave which is described by a specific explicit solution of
NLS, called the soliton on finite background (SFB) mentioned above. When
translated to physical variables, the SFB shows additional phenomena compared
to its description as an amplitude in the NLS-setting. Actually, the SFB is
not an isolated solution, but is embedded in a family of SFB-solutions. This
family depends on three essential parameters: the frequency $\omega$ of the
monochromatic ‘carrier’ wave, the modulation-length $\nu$, which together with
the frequency $\omega$ determines the number of waves in one wave group, and
the maximal amplitude which is related to the asymptotic value of the
monochromatic wave amplitude. SFB solutions are described in detail in [1] and
have already been considered in [13, 21] as a possible description of large
amplitude increase of surface waves leading to ‘freak’, or ‘rogue’ waves. In
this paper, we will add substantially to these previous investigations.

Actually, besides SFB, large amplitude phenomena are also present in two
different but closely related explicit solutions of NLS, the class of Ma-
solutions and the single polynomial solution of Peregrine; see [22, 23, 24,
25, 26] and [27] for a unified description. The last solution appears for both
SFB and Ma-solutions as a limiting case and has polynomial decay in time and
space to a uniform wave train. Since our interest is in the signalling problem
with a slowly modulated wave at the initial position, i.e. the situation of
the onset of the BF-instability at one specific position (the wave maker), it
leads us to select the SFB-solution for detailed consideration. Large
amplitude amplification of monochromatic wave trains is also observed to occur
in water wave models different than the NLS model. For example, in Green-
Naghdi Level III equations, the band-modulated Stokes waves show amplitude
amplifications up to a factor of $3.1$ [28]. Numerical investigations of
modulated wave trains within a fully nonlinear irrotational numerical flow
solver show that the amplitude amplification can be even larger than $3.3$ for
a wide range of initial wave steepness [16]; a similar result is demonstrated
also in [29].

We now briefly summarize the main contributions reported in this paper. We
will describe that when the modulation is long enough, at the extreme position
(where the wave heights are largest), phase singularities appear periodically
in time. These phase singularities correspond to wave dislocations of two
types, dislocations of wave splitting and of wave merging, at which energy
flow in opposite direction takes place. This leads to the understanding that
these successive splitting and merging phenomena are responsible for a
temporal separation and discrimination between ‘extreme’ waves and
intermittent waves of much smaller amplitude. At the extreme position, this
amplitude difference can be essentially larger than when the maximal amplitude
is compared to the amplitude of the far-field; this last amplification factor
is at most $3$ [30]. We will show, in fact, that the time signal at the
extreme position has very special properties, and can be described by a Newton
type of second order equation. The (classical-mechanical) potential of that
system is a symmetry-breaking unfolding of the potential that describes the
steady states of NLS, such as the standard soliton and periodic oscillations
in the amplitude around an equilibrium value of a nonlinear monochromatic
mode. In the limit for long modulation lengths, the extreme signal tends to a
soliton on finite, negative, background. The extreme waves in this signal are
almost independent of the number of intermittent waves, which number varies
from zero to infinity. This shows that the extreme waves have, practically
speaking, a generic form. Aside from clearly observable phenomena as wave
dislocations and the generic extreme waves, the SFB solutions show interesting
properties that are less easy to observe directly. This includes the spatial
evolution of the spectrum which shows large energy exchange between modes as a
consequence of different contributions from the extreme and intermittent
waves.

In this paper, we address these properties and discuss the applicability for
the sake of deterministic extreme wave generation. Actually, the findings here
have motivated and designed a series of experiments in the large wave tank
($200$ m long, $4$ m deep) of MARIN, the Maritime Research Institute
Netherlands [31]. These experiments revealed that most of the generated waves
did not ‘break’, which could not have been predicted within the NLS-model that
is used here; numerical indications that breaking will occur only in the most
extreme cases have already been reported [20].

The organization of this paper is as follows. In the next Section 2, the NLS
equation and the explicit family of SFB solutions are presented. The
phenomenon of phase singularity and wave dislocation, and the explanation of
the energy exchange, is described in Section 3. In Section 4, the extreme
signal of SFB will be investigated in detail. In Section 5 the energy exchange
during down-stream evolution is shown in spectral components of the successive
time signals. Various cases for characteristic parameters and samples of
extreme wave generation in laboratory coordinates are presented in Section 6,
together with the further explanation of the maximal temporal amplitude. In
the last section, we give a brief summary and some concluding remarks.

##  2 Spatial NLS and soliton on finite background

###  2.1 The NLS equation

In the following, we use the notation $\eta(x,t)$ for the physical wave
amplitude. The variables $\eta$, $x$, and $t$ are non-dimensionalized and are
related to laboratory variables by the scaling $\eta_{lab}=\eta\,\ast h$,
$x_{lab}=x\ast h$ and $t_{lab}=t\ast\sqrt{h/g}$ where $h$ is water depth and
$g$ is gravitational acceleration. For linear dispersive wave equations, a
simple linear monochromatic wave is described by
$\eta(x,t)=A\exp\left[i(k_{0}x-\omega_{0}t)\right]+\text{cc}$, where
$\omega_{0}$ is the frequency and $k_{0}$ is the wavenumber that are related
by the dispersion relation, say $\omega_{0}=\Omega\left(k_{0}\right)$, $A$ is
a constant amplitude and ‘cc’ denotes complex conjugation of the previous
term. For small amplitude surface waves, the linear description is governed by
the dispersion given by $\Omega\left(k\right)=\sqrt{k\tanh(k)}$.

For weakly nonlinear phenomena, the deformation of a linear monochromatic wave
is commonly described by a series expansion in the (small) amplitude. The
resulting wave group centred around the basic frequency $\omega_{0}$ then
describes the spatial deformation with a complex amplitude $A$ that is allowed
to vary in time and space, i.e. in the lowest order

$\eta(x,t)\approx
A(\xi,\tau)\exp\left[i(k_{0}x-\omega_{0}t)\right]+\text{cc}.$ (1)

Here local coordinates in a coordinate system moving with the group velocity
$V_{0}=\Omega^{\prime}(k_{0})$ are used: $\tau=t-x/V_{0}$, $\xi=x$. For weakly
nonlinear physical phenomena, the complex amplitude $A(\xi,\tau)$ satisfies a
spatial NLS-type of equation for solutions for which there is a balance
between dispersive and nonlinear effects:

$\partial_{\xi}A+i\beta\partial_{\tau}^{2}A+i\gamma|A|^{2}A=0.$

The parameters $\beta$ and $\gamma$ depend on the monochromatic frequency;
with a KdV-type of equation to model the surface waves, these parameters are
easily written down as follows (see [32])
$\beta=-\frac{1}{2}\Omega^{\prime\prime}(k_{0})/V_{0}^{3}$,
$\gamma=\frac{9}{4}k_{0}(\sigma_{0}+\sigma_{2})/V_{0}$, with
$\sigma_{0}=1/(\Omega^{\prime}(k_{0})-\Omega^{\prime}(0))$,
$\sigma_{2}=k_{0}/(2\Omega(k_{0})-\Omega(2k_{0}))$ and $V_{0}$ is the group
velocity. We will restrict to the case of interest in the ‘self-focussing’
regime for which sign$(\beta\gamma)>0$, which is a restriction on the maximum
wavelength of the carrier waves: $k_{0}>k_{\text{crit}}$ with
$k_{\text{crit}}\approx 1.23$.

Remark.  In all of the following, we will use relation (1) to translate
properties of the NLS equation to the physical variables, and thereby do not
take higher order contributions into account. This is done for ease of
presentation. But it should be noted that, for instance, the characteristic
Stokes phenomenon from second-order terms is therefore missing, and that the
quantitative wave heights are not very accurate. Indeed, for laboratory
practice, second-order effects have to be added, which may increase the wave
height with up to 10%.

###  2.2 The family of SFB solutions

A so-called nonlinear monochromatic wave of constant amplitude $r_{0}$ is
described by

$A_{0}=r_{0}e^{-i\gamma r_{0}^{2}\xi},$

and corresponds to a harmonic physical wavefield with (nonlinearly) adjusted
dispersion relation:

$\eta_{nm}=2r_{0}\cos\left((k_{0}-\gamma r_{0}^{2})x-\omega_{0}t\right).$

Figure 2: Density plot (left) with horizontally the time $\tau$ and vertically
$\xi$, and a 3D plot (right) of $2a=2|A|$, i.e. twice the absolute value of
$A_{\text{SFB}}(\xi,\tau)$ as given by (2), for the set of parameter values:
$\hat{\nu}=1,\omega=2.5$ and $M=0.1$.

Another well-known solution of the NLS equation is the soliton solution which
decays to zero at $\xi=\pm\infty$. We will return to this ‘standard’ soliton
in Section 4 to relate it to a description of the extreme signal. The solution
of main interest in this paper is the soliton on finite background, SFB. This
is the solution that arises when a nonlinear monochromatic wave is perturbed
with a long-period modulation of the basic frequency, and that describes the
connection of a nonlinear monochromatic wave of amplitude $2r_{0}$ at
$\xi=-\infty$ to a similar (shifted) wave of the same amplitude at
$\xi=\infty$, while undergoing a Benjamin-Feir instability stimulated by the
modulation. This linear instability leads to exponential growth of the ‘side-
band’ perturbations. In fact, perturbations in frequency at sidebands
$\omega_{0}\pm\nu$ will grow (or decay) with the Benjamin-Feir growth rate
$\sigma(\nu)=\beta\nu\sqrt{2\nu_{\ast}^{2}-\nu^{2}}=\gamma
r_{0}^{2}\hat{\nu}\sqrt{2-\hat{\nu}^{2}}$ provided this is real, i.e. provided
$\hat{\nu}<\sqrt{2}$ where $\hat{\nu}=\nu/\nu_{\ast}$ with
$\nu_{\ast}=r_{0}\sqrt{\gamma/\beta}$ a normalized frequency; the maximal
growth rate is attained for $\hat{\nu}=1$ and has value $\gamma r_{0}^{2}$.
The linear modulational instability is described precisely by the asymptotic
behaviour of the SFB, which solution can, therefore, be seen as a continuation
of the Benjamin-Feir instability into the nonlinear regime (see e.g. [1]).
This SFB is explicitly given by

$A_{\text{SFB}}(\xi,\tau;\omega_{0},r_{0},\nu)=A_{0}\frac{(\hat{\nu}^{2}-1)\cosh(\sigma\xi)-i\hat{\nu}\sqrt{2-\hat{\nu}^{2}}\sinh(\sigma\xi)+\sqrt{1-\hat{\nu}^{2}/2}\cos(\nu\tau)}{\cosh(\sigma\xi)-\sqrt{1-\hat{\nu}^{2}/2}\cos\nu\tau}.$
(2)

Figure 3: Density plot (left), with horizontally the physical time shifted
with the group velocity and vertically the physical space, and 3D plot (right)
of the real wave field $\eta_{\text{SFB}}(x,t)$ corresponding to the case
depicted in Fig. 2.

Note the periodicity in $\tau$ with period $T_{\text{mod}}=2\pi/\nu$ and the
soliton-type of shape in $\xi$ on top of the level at value $r_{0}$ with a
phase shift $2\phi$ from $\xi=-\infty$ to $\xi=\infty$ with
$\tan\phi=\hat{\nu}\sqrt{2-\hat{\nu}^{2}}/(\hat{\nu}^{2}-1)$.

Through formula (1) this SFB solution gives an approximate solution of the
physical system; we will denote this solution in the original space-time
variables by $\eta_{\text{SFB}}$, depending on three parameters,
$\omega_{0},r_{0}$ and $\hat{\nu}$. By writing the complex amplitude $A$ in
‘polar’ coordinates by introducing a real amplitude $a$ and phase $\phi$
according to $A=ae^{i\phi}$, the real wavefield is given by

$\eta_{\text{SFB}}=2a\cos\Phi,\qquad\text{with}\quad\Phi=k_{0}x-\omega_{0}t+\phi(x,t).$
(3)

In order to give a graphical illustration of the contents of the explicit
formula (2), we show in figure Fig. 2 a density plot and a 3D plot of the
amplitude $2a=2|A|$; the corresponding real wave field $\eta_{\text{SFB}}$ is
shown in Fig. 3. Parameters used for these plots are $\hat{\nu}=1$,
$\omega=2.5$ and $M=0.1$.

From the explicit formula (2) it is easily seen that the maximal amplitude of
$\eta_{\text{SFB}}$, which will be denoted by $M$, is attained for
$\xi=\tau=0$, and has value given by

$M=2r_{0}\left(1+2\sqrt{1-\hat{\nu}^{2}/2}\right).$ (4)

This expression shows that the maximum amplitude amplification, defined as the
ratio of the maximal amplitude and the amplitude of the linear monochromatic
wave at infinity $\alpha=M/(2r_{0})$, is monotonically increasing for
decreasing $\hat{\nu}$ with limiting value $3$ attained for
$\hat{\nu}\rightarrow 0$, see Fig. 4.

Figure 4: The plot of the maximum amplification factor (ratio of the maximal
wave amplitude and the amplitude of the linear monochromatic wave at infinity)
as a function of $\hat{\nu}$, with limiting value $3$ for infinitely long
perturbations. Values larger than $1$ are only found within the Benjamin-Feir
instability interval $\hat{\nu}\in(0,\sqrt{2})$.

In the next sections, we will further investigate several physical phenomena
that are hidden in the explicit expressions given above, and exploit the
formulas to predict and prescribe extreme wave generation in hydrodynamic
laboratories.

##  3 Phase singularity

###  3.1 Phase jumps for indefinite envelopes

Investigating the complex amplitude $A_{\text{SFB}}$, it is noted that this
amplitude is real-valued (only) at $\xi=0$, which is the extreme position
where the maximal amplitude is obtained for
$\tau=0{\,\text{mod}}\left(T_{\text{mod}}\right)$, where
$T_{\text{mod}}=2\pi/\nu$ is the modulation period. The signal of the physical
solution $\eta_{\text{SFB}}$ at the extreme position $\xi=x=0$ will be called
the extreme signal. This extreme signal is given by
$\eta_{\text{SFB}}(x=0,\tau;\omega_{0},r_{0},\nu)=s_{0}\left(\tau\right)$,
with

$s_{0}(\tau)=2S(\tau)\cdot\cos(\omega_{0}\tau)\qquad\text{with}\quad
S(\tau)=r_{0}\cdot\frac{\hat{\nu}^{2}-\left[1-\sqrt{1-\hat{\nu}^{2}/2}\cos(\nu\tau)\right]}{1-\sqrt{1-\hat{\nu}^{2}/2}\cos\nu\tau},$
(5)

(Note that at $\xi=0$, also $\tau=t$). The function $S(\tau)$ is periodic with
the modulation period, and is strictly positive for $3/2<\hat{\nu}^{2}<2$,
while for $\hat{\nu}^{2}<3/2$ it is partly positive and partly negative,
vanishing at the points $\tau=\zeta$ for which

$\cos(\nu\zeta)=\frac{1-\hat{\nu}^{2}}{\sqrt{1-\hat{\nu}^{2}/2}}.$ (6)

In the following, we will call the possibly non-sign definite function $S$ the
‘envelope’, to distinguish it from (half) the ‘amplitude’ $|S|$ which is
nonnegative. In fact, in polar coordinates, the signal can be written like

$s_{0}(t)=2|S(t)|\cos\left(\omega_{0}t+\pi\cdot\left(1-\text{sign}\,S\right)/2\right).$

This makes it explicit that, since $2|S|$ is the amplitude, the signal has a
phase jump $\pi$ at the points where $S$ vanishes. As stated above, this
happens for parameter values $\hat{\nu}^{2}<3/2$, which is for part of the B-F
instability interval. This is illustrated in Fig. 5 which shows plots of
$S(\tau)$ for three different values of $\hat{\nu}$ that all lie inside B-F
instability interval $(0,\sqrt{2})$, namely $\hat{\nu}=1$, $\sqrt{3/2}$,
$1.3$. Observe the limiting case $\hat{\nu}=\sqrt{3/2}$ which is the value for
the signal that separates the strictly positive functions $S$ from the
indefinite functions, and the limiting case $\hat{\nu}\rightarrow 0$ for which
the function $S$ becomes like a soliton on finite (negative) background. In
the next section, we show that it is, in fact, a real soliton; the negative
value of the finite background is precisely the value of the wave field at
infinity, $2r_{0}$.

Figure 5: Plot of (twice) the extreme envelope, the function $2S$ (the factor
of $2$ for ease of relating the plots to those of the physical waves:
$\eta=2S\cos\left(\omega_{0}t\right)$) for three different values of
$\hat{\nu}$: (a) $\hat{\nu}=1.3$, (b) $\hat{\nu}=\sqrt{3/2}$, (c)
$\hat{\nu}=1$. In case (a) there are no phase jumps, while in case (c) there
are, namely the zero’s of $S$, and (b) is the separating case. Also shown
(repeated in the inset on a larger horizontal scale) is the graph (d) of the
limiting function $2S$ for $\hat{\nu}\rightarrow 0$ which is a soliton on
negative finite background.

In the next section, we will further analyse the extreme signal. For here it
is sufficient to use the appearance of a phase jump in the extreme signal as a
motivation to investigate what the related phenomena are for the total
wavefield. This leads to the finding of wave dislocations, as we will detail
further on.

###  3.2 Wave dislocation

The phenomenon of wave dislocation is known to appear in surface waves [33],
but little attention seems to have been given to it. It is shown to illustrate
the performance of numerical codes in [16, 34], and was explained as a result
of vertical vorticity in [35]. In optics literature, see e.g. [36, 37, 38, 39,
40, 41], the phenomenon of wave dislocation is often called phase-singularity,
and has been measured experimentally and simulated as reported in [36]; in the
linear case, the intensity amplification is of course lacking.

A wave dislocation, a phenomenon of merging of two waves into one, or the
inverse process of splitting of one wave into two, is a consequence of the
phase singularities in SFB. Fig. 6 is an amplification of part of the
wavefield shown in Fig. 3 (left) to show the wave splitting and merging in
more detail.

Figure 6: Density plot of the wavefield showing in more detail the wave
splitting at the left of the origin and wave merging at the right.
Horizontally is the physical time $t$ shifted with the group velocity, and
vertically the physical space $x$.

In Fig. 7 this wave splitting and merging is shown to be related to the
vanishing of the amplitude of the extreme wave signal $S(\tau)$ at $t=\zeta$
for values $\hat{\nu}<\sqrt{3/2}$. In Fig. 7 are shown plots of the time
signal of $\eta_{\text{SFB}}$ and its envelope (dashed) at various positions
near the extreme position $x=0$. At $x=0$ the amplitude vanishes at $t=\zeta$,
described as ‘disapearance of waves’ in [33]; the relation to the singular
behaviour will be described further on.

Figure 7: (Above) Slices at various positions near the wave dislocation at
$(x,t)=(0,\zeta)$ (with $\zeta>0$) were wave merging takes place of the
solution depicted in Fig. 6. Shown are plots at different positions $x$,
increasing from negative values below to positive values upwards, of the wave
elevation (solid lines) and amplitude (dashed lines) as a function of
(shifted) time. The vanishing of the envelope at $(0,\zeta)$ leads to the
flattening of the surface in the focusing region, resulting in the
disappearance of one wave when crossing $x=0$ to positive time, as shown by
the curve connecting two wave crests. When the sign of the vertical axis is
reversed, wave splitting is seen that happens at the point $(0,-\zeta)$.
(Below) The manifestation of the phase singularity shown in the time signal at
the extreme position; the box indicates the waves in the signal which are
visible in the middle of the picture above.

###  3.3 Dispersion plots

We will now describe the phenomenon of wave dislocation in a more analytical
way by showing that it is related to large changes in the local wavelength and
frequency. This is best illustrated using the phase-amplitude description for
the wave field (3) and introducing local wavenumber and local frequency as
usually:

$\eta(x,t)\approx 2a(x,t)\cos\Phi\left(x,t\right),\qquad
k(x,t)=\partial_{x}\Phi(x,t);\qquad\omega(x,t)=-\partial_{t}\Phi(x,t).$

With these definitions, the evolution of the local wavenumber and local
frequency can first be investigated in the dispersion plane of frequency
versus wavenumber. In figure Fig. 8 (left), for various positions $x$ the
trajectories $t\rightarrow(k(x,t),\omega(x,t))$ are shown parameterized by
$t$. Observe that for the value $x=0$ a straight line appears connecting at
$\omega=2.5$ the $k$-values $\infty$ and $-\infty$, which describes the phase
singularity at the position and time at which the wave number $k$ becomes
unbounded. This phenomena can further be seen from Fig. 8 (right) where
contour plots are drawn for the wave number $k(x,t)$ near the location
$\xi=x=0$. Large and abrupt changes of the local wave number are visible near
the wave dislocations at $t=\zeta>0$ (wave merging) and at $t=-\zeta$ (wave
splitting), showing the singular behaviour at the point of the dislocation and
justifying the use of the word ‘phase-singularity’. For later reference we
note that the contour lines show that $\partial_{x}k<0$ for merging, and
$\partial_{x}k>0$ for splitting.

Figure 8: (Left) In the dispersion plane of local frequency (vertical) vs
local wavenumber (horizontal), trajectories parameterized by $t$ are drawn of
$\eta_{\text{SFB}}$ for various positions, $x=0<x_{1}<x_{2}<\cdots$. The phase
singularity is observed for $x=0$ where the line $\omega=2.5$ connects
$\infty$ and $-\infty.$ The dashed line is the linear dispersion relation
$\omega=\Omega(k)$. (Right) Contour plots of the wave number as function of
$\tau=t-x/V_{0}$ (horizontal) and $\xi=x$ (vertical). Observe the abrupt
change of the wave number for $\xi=0$ at the wave dislocations: splitting at
the left ($t=-\zeta$) and merging ($t=\zeta$) at the right.

###  3.4 Energy exchange and wave discrimination

We will now present an explanation of the large amplitude amplification
between phase singularities, or better to say, to the separation process of
successive intermittent and extreme waves. Therefore we return to the
governing NLS equation and rewrite this equation in physical coordinates in
the real amplitude $a$ and local wave number and local frequency. The
governing equations, obtained by looking at the real and imaginary parts, are
the so-called phase-amplitude equations. The phase equation is given by

$\omega-\Omega(k)=V_{0}\beta\frac{\partial_{t}^{2}a}{a}+V_{0}\gamma a^{2},$

where $\Omega(k)$ is the linear dispersion relation, and the amplitude
equation is effectively the energy equation

$\partial_{t}E+\partial_{x}\left[V\left(k\right)E\right]=0,\qquad E=a^{2}.$

The phase equation contains the so-called Fornberg-Whitham term
$\beta\frac{\partial_{t}^{2}a}{a}$ which becomes singular when the amplitude
vanishes; this leads to the singular behaviour in the dispersion plane as
described above.

The resulting large amplitudes in between wave-splitting and wave-merging are
caused by nonlinear effects, although the wave dislocation itself is actually
a linear phenomenon. The amplitude amplification is caused by the interplay
between the envelope and the changes in phases of the carrier wave, and can be
seen as follows. The energy equation describes conservation of integrated
energy density between group lines (see [2]). Stated differently, this
equation shows that the change of energy transported with the local group
velocity is forced by local changes in wavelength:

$\left[\partial_{t}+V\left(k\right)\partial_{x}\right]E+E\partial_{x}V\left(k\right)=0.$

These changes are largest near the dislocations. Since
$\partial_{x}V\left(k\right)=\Omega^{\prime\prime}\left(k\right)\partial_{x}k$,
and $\Omega^{\prime\prime}\left(k\right)<0$ for surface wave dispersion, there
is a decrease or increase of energy near the dislocations depending on the
character of the dislocation. At wave splitting, it holds $\partial_{x}k>0$ as
shown above, and hence the energy increases while travelling with the group
velocity. The group velocity is for surface wave dispersion less than the
phase velocity, and hence there is a nett energy flow crossing the wave at the
merging position from left to right. Reversely, at a position of wave merging
$\partial_{x}k<0$ and there is a nett energy flow from right to left. These
two effects enhance each other and lead to energy increase between a splitting
singularity and a merging singularity. This nett transport of energy causes
that the waves are divided into intermittent waves and extreme waves. This
energy flow leads to changes of the amplitudes of the waves that can become as
large as illustrated in the plots (see also Section 5). Hence, the process of
wave discrimination is caused by energy exchange through successive wave
dislocations.

##  4 Extreme signals

We now return for a further study to the extreme signal, in particular to the
(possibly non sign-definite) envelope function $S$ in (5) as plotted in Fig.
5. Of special interest are the cases for which $S$ is partly negative: the
cases of phase singularities and wave discrimination, i.e. values
$\hat{\nu}<\sqrt{3/2}$.

###  4.1 Phase plane representation

We will start to draw the ‘phase portrait’ of $S$, that is to say,
parameterized by $t$, the function values are plotted against the values of
the derivative in figure Fig. 9 (left), leading to a trajectory in the
phaseplane of $\partial_{\tau}\,2S$ versus $2S$. The different trajectories
correspond to different values of the normalized modulation frequencies
$\hat{\nu}=\sqrt{1/2}$, $1$, $\sqrt{3/2}$. The corresponding signals
$s_{0}(\tau)$ for those values are plotted in Fig. 9 (right).

Figure 9: (Left) In the phase plane of $2S$ (horizontally) vs
$\partial_{t}\,2S$ (vertically) the trajectories are plotted of $S$ as
function of time for three values of the normalized modulation frequencies
$\hat{\nu}$: $\sqrt{1/2}$, $1$, $\sqrt{3/2}$ (from outside to inside). The
‘limiting’ case $\hat{\nu}\rightarrow 0$ is approximated by $2S$ for very
small value of $\hat{\nu}$ and leads to the most outward curve. (Right) Plot
of the signals $s_{0}(\tau)=2S(\tau)\cos\left(\omega_{0}\tau\right)$ for the
same values of $\hat{\nu}$ as in the left picture. The resemblance of the
extreme waves for all these different values shows once more the generic form
of these extreme profiles. In the inset the limiting case
$\hat{\nu}\rightarrow 0$ is shown on a much larger time interval.

Plots as these are often found in systems of one degree of freedom in
classical mechanics. Indeed, it can be verified that the extreme signal
satisfies Newton’s equation

$\beta\partial_{\tau}^{2}S+\gamma S^{3}-\kappa S-\lambda=0,$ (7)

with parameters $\lambda,\kappa$ given by

$\lambda=\gamma r_{0}^{3}(2-\hat{\nu}^{2}),\qquad\kappa=\gamma
r_{0}^{2}(3-\hat{\nu}^{2}).$ (8)

When written like $\beta\partial_{\tau}^{2}S+\partial V/\partial S=0$, the
equation can be interpreted as the equation of motion of a particle with mass
$\beta$ subject to a conservative force with potential
$V_{\lambda}(S)=\frac{\gamma}{4}S^{4}-\frac{1}{2}\kappa S^{2}-\lambda S$
(where we made, for later reference, the dependence of the potential on
$\lambda$ explicit in the notation). The trajectories in the phase plane are
therefore level lines with $\mathcal{E}>0$ of the total (conserved)
‘energy’-like quantity:

$\mathcal{E}=\frac{\beta}{2}\left(\partial_{t}S\right)^{2}+V_{\lambda}(S),\qquad\partial_{\tau}\mathcal{E}=0.$

The potential and the corresponding phase plane are shown in Fig. 10 (left
column).

Figure 10: (Left column) For normalized variables $\gamma=\beta=1$, and
$\kappa=2.5$ and $\lambda=0.5$, the phase plane is shown with the potential
$V_{\lambda}$ above it. The interesting extreme signals, of amplitude $M/2$,
are obtained for $\mathcal{E}=V_{\lambda}\left(M/2\right)>0$, and correspond
to the periodic motions that are mainly in the positive half plane but have
some negative values. Observe the homoclinic orbit through the equilibrium
(the local maximum) which is $-2r_{0}$. These trajectories correspond to those
shown in Fig. 9 (left). (Right column) For the same variables, the symmetric
potential $V_{0}$ with $\lambda=0$ is shown, with below it the phase plane.
The steady-state profiles of NLS, nonlinear normal mode, periodic amplitude
oscillations and the soliton correspond to the (positive) equilibrium, the
periodic oscillations around it, and the homoclinic orbit through $(0,0)$.

The extreme signal can be written down in the form of an elliptic integral

$t=\sqrt{\beta}\int^{S}\frac{ds}{\sqrt{2(\mathcal{E}-V_{\lambda}(s))}}$

where the total energy can be expressed in the maximal amplitude $M$ as
$\mathcal{E}=V_{\lambda}\left(M/2\right)$.

###  4.2 Signal potential as unfolded NLS-steady state potential

It is illustrative to compare the above phase plane analysis with ‘standard’
soliton solution of NLS. This solution and periodic modulations of the
nonlinear monochromatic mode are found from the NLS equation by looking for
solutions of the form in separated variables

$A=e^{-i\kappa\xi}f\left(\tau\right)$

with $f$ a real function and $\kappa$ a real number to be determined together
with $f$. In physical variables, the corresponding wave field looks like

$\eta_{ss}(x,t)=2f(t-x/V_{0})\cos\left((k_{0}-\kappa)x-\omega_{0}t\right),$

i.e. a monochromatic carrier wave with adapted dispersion, modulated with a
profile function $f$ that steadily progresses with the group velocity. For
this reason, solutions of this kind are usually called ‘steady state’
solutions. Substitution of the above Ansatz into the NLS equation leads to the
ODE for $f$

$\beta\partial_{\tau}^{2}f+\gamma f^{3}-\kappa f=0.$

This is recognised as Newton’s equation with a potential
$V_{0}(f)=\frac{\gamma}{4}f^{4}-\frac{\kappa}{2}f^{2}$. The potential and the
corresponding phase plane trajectories, satisfying the ‘energy’ conservation
$\beta\left(\partial_{\tau}f\right)^{2}+V_{0}\left(f\right)=$ constant, are
shown in Fig. 10 (right column) for the (only) interesting case of $\kappa>0$.
In the phase plane, the nonlinear monochromatic mode, for which the potential
is minimal, and periodic oscillations around this monochromatic mode, and the
soliton solution as the homoclinic orbit through $f=0$, are to be noted.
Explicit formulæ in terms of elliptic integrals can be written down as before.

Compared to the potentials $V_{\lambda}$ defined above that describe the
extreme signals for $\lambda>0$, the potential $V_{0}$ considered here for the
standard NLS is precisely the potential $V_{\lambda}$ for the special case
$\lambda=0$. Hence $V_{\lambda}$ can be seen as an unfolding of the potential
$V_{0}$. For $\lambda>0$ the symmetry that is present in $V_{0}$ is broken, as
is seen in the plot of the potential and in the phase plane. Although, from
this point of view, the unfolding is a rather simple phenomenon, on the level
of solutions of the NLS equation, this unfolding has large consequences for
the governing dynamics in $\xi$, which shows itself in the very different NLS-
solutions: when $\kappa$ and $\lambda$ are related to the quantities $r_{0}$
and $\hat{\nu}$ according to (8), the corresponding SFB-solutions are
obtained.

###  4.3 Phase singularity related to extreme signal

From the equation (7) satisfied by $S$, we can extract more information about
the solution in a neighbourhood of the extreme position $\xi=0$. To that end
first note that at $\xi=0$, $A=S(\tau)$. Then since $A$ satisfies the NLS
equation the restriction to the point $\xi=0$ implies that

$\left.\partial_{\xi}A\right|_{\xi=0}=-i\gamma\left(\frac{\beta}{\gamma}\partial_{t}^{2}S+S^{3}\right)=-i\gamma\left(\lambda_{2}S+\lambda_{1}\right)$

leading to

$A=e^{-i\gamma\lambda_{2}\xi}S(\tau)-i\gamma\lambda_{1}\xi\qquad\text{near}\qquad\xi=0.$

Writing $\Phi$ for the phase of $A$, $A=|A|e^{i\Phi}$, it holds that

$\Phi=\tan\left(\sigma\xi\right)+\frac{\xi}{\cos\left(\sigma\xi\right)}\frac{\lambda}{S(\tau)},$

confirming that $\Phi=0$ at $\xi=0$, while a simple calculation leads to

$\left.\partial_{\xi}\Phi\right|_{\xi=0}=\frac{\lambda}{S(\tau)}.$

This shows once again the appearance of the phase singularity, now explicitly
and directly relating the zero’s of $S$ to the singularity in the change of
phase of $A$, and hence in the phase of $\eta_{\text{SFB}}$, while crossing
the extreme position at times near the events of wave splitting/merging.

##  5 Down-stream spectral evolution

In Subsection 3.4 we have used the energy equation to investigate the flow of
energy in the physical time-space near the phase singularities. Another common
way to study the deformations is to investigate the evolution of the spectral
components. In the present case of a signalling problem, this means that we
Fourier analyse the signal at each position and follow the individual Fourier
coefficients for increasing position. Although this information is not easily
related to actual changes of the physical signal, it contains the same
information.

To find the spectral evolution, we write the solution as a superposition of
contributions at the carrier frequency $\omega_{0}$ and at sidebands of all
orders: $\omega_{0}\pm m\nu$ for $m=1,2,\dots$:

$\eta_{\text{SFB}}\left(x,t\right)=\sum_{-\infty}^{\infty}c_{m}(x)\exp\left(i\left(\omega_{0}+m\nu\right)t\right)+\text{cc}=\sum_{-\infty}^{\infty}a_{m}(x)\cos\left((\omega_{0}+m\nu)t+\phi_{m}(x)\right)$

with real coefficients $a_{m}$ and phase factors $\phi_{m}$ or with complex-
valued coefficients $c_{m}$. The energy contents of the central frequency and
the sidebands are defined by

$\displaystyle e_{0}(x)$
$\displaystyle=|c_{0}(x)|^{2}=\frac{1}{4}a_{0}^{2}(x),$ $\displaystyle
e_{m}(x)$
$\displaystyle=|c_{-m}(x)|^{2}+|c_{+m}(x)|^{2}=\frac{1}{4}[a_{-m}^{2}(x)+a_{+m}^{2}(x)]\qquad\text{for}\quad
m\geq 1.$

For SFB, the asymptotic behaviour is known to consist of only the carrier
wave, and hence

$\textnormal{for}\quad x\rightarrow\pm\infty:\quad a_{0}(x)\rightarrow
2r_{0},\qquad a_{m}\left(x\right)\rightarrow 0\quad\textnormal{for}\quad
m=1,2,\dots.$

The Fourier coefficients for the sidebands are given by

$c_{m}\left(x\right)=\frac{1}{T}\int_{-T/2}^{T/2}\eta_{\text{SFB}}(x,t)\exp\left(-i(\omega_{0}+m\nu)t\right)\,\text{d}t,$

where $T=T_{\text{mod}}$. For the spatial evolution, the total ‘energy’,
$\int\eta^{2}/2\,\text{d}t$, is conserved which implies that the sum of the
energy contents of the modes is conserved:

$\frac{\text{d}}{\text{d}x}\sum_{m=0,1,2,\dots}e_{m}(x)=0,$

and hence equal to the asymptotic value $r_{0}^{2}$.

In Fig. 11 the Fourier spectrum of the extreme signal is plotted for three
cases. The spatial dynamics of the energy in sidebands during the downstream
evolution is shown in figure Fig. 12. Plotted are the energy content of the
carrier wave and of the first few sidebands as a function of position. The
extreme position is now at $x=0$, and since the plot is symmetric, only the
negative part is shown. As is seen, the major part of the total energy remains
in the first few sidebands. To make this explicit, also is plotted the
remainder energy, which is the energy in the sidebands not shown in this
picture.

Figure 11: Fourier spectrum of the extreme signal of $\eta_{\text{SFB}}$ for
$\hat{\nu}=\sqrt{3/2}$, $1$ and $\sqrt{1/2}$ from left to right. It is to be
noted that the energy content that is asymptotically concentrated in the
carrier frequency $\omega_{0}=2.5$ has been redistributed over many sidebands.
In the special case $\hat{\nu}=\sqrt{1/2}$ the energy at the carrier frequency
vanishes precisely.

Figure 12: For $\hat{\nu}=\sqrt{3/2}$, $1$, $\sqrt{1/2}$ from left to right
are shown the evolution of the energy of the carrier frequency $e_{0}$ (a), of
the three lowest sidebands $e_{m}$, $m=1,2,3$ (b), (c) and (d) respectively,
and of the remaining energy denoted and defined by
$e_{\ast}\left(x\right)=\Sigma_{m>3}e_{m}(x)$, (e).

##  6 Down-stream evolution and generation

As mentioned in the introduction, part of the motivation for this research is
the practical problem to generate large waves in hydrodynamic laboratories. In
this section, we will show how the previous information can be exploited for
this practical purpose. At the same time, it gives a framework to look at the
plots, amplification aspects, and the concept of the MTA.

###  6.1 Characteristic down-stream evolutions

In interpreting the graphical illustrations of spatial evolution of
$\eta_{\text{SFB}}(x,t)$ it is helpful to use the maximal amplitude $M$ and
the maximal amplitude amplification $\alpha=M/\left(2r_{0}\right)$. Actually,
instead of the original basic variables
$\left(\omega,\,r_{0},\,\hat{\nu}\right)$, one could also use
$\left(\omega,\,M,\,\hat{\nu}\right)$ or $\left(\omega,\,M,\,\alpha\right)$ as
the parameters of the family.

For the purpose of this section, we will use laboratory variables and consider
a wave tank of $5$ m depth. The waves are generated by a wave maker at the
left at $x=0$, and travel to the right. We assume the tank is at least $150$ m
long and that reflections at an artificial beach can be neglected. The
gravitational acceleration is taken to be $9.8$ m/s2.

For the three parameters of SFB, the following choices are made. The
monochromatic frequency is taken to be $\omega=3.5$ s-1; this will lead to
waves with the wavelength of approximately the depth of the layer, so
relatively short waves. For the maximum amplitude at the extreme position, we
take $M=0.5$ m. This would correspond to waves of maximal wave height
approximately $0.8$–$0.9$ m, which can be considered to be ‘extreme’ waves.
For the remaining variable the normalized frequency $\hat{\nu}$ is taken, and
we will consider three cases: $\hat{\nu}=\sqrt{3/2}$, $1$ and $\sqrt{1/2}$.
For illustrative purposes, the chosen values are to some extent arbitrary, but
the chosen ones have some characteristic property: $\hat{\nu}=\sqrt{3/2}$ is
the largest value till where phase singularities are present, for
$\hat{\nu}=1$ the B-F growth rate is maximal, while for $\hat{\nu}=\sqrt{1/2}$
the original center frequency of the monochromatic in the B-F signal in the
far field vanishes at the extreme position.

For the three cases to be considered, the following table summarizes the main
values of the physical quantities.

Table 1: Physical values of relevant quantities (in m, sec) for three cases of
$\hat{\nu}$.

$\quad\hat{\nu}$ |  $\nu\quad$ |  $\alpha\quad$ |  $2r_{0}\quad$ |  $T_{\text{mod}}\;\;$ |  $N_{\text{temp}}$ |  $\zeta\;\;$ |  $\lambda_{\text{mod}}\quad$ |  $N_{\text{spat}}$
---|---|---|---|---|---|---|---|---
$\sqrt{3/2}$ |  $0.6238$ |  $2.0000$ |  $0.2500$ |  $10.0723$ |  $\>\>5.61$ |  $5.04$ |  $14.1024$ |  $2.81$
$1$ |  $0.4220$ |  $2.4142$ |  $0.2071$ |  $14.8906$ |  $\>\>8.29$ |  $3.72$ |  $20.8466$ |  $4.15$
$\sqrt{1/2}$ |  $0.2636$ |  $2.7321$ |  $0.1830$ |  $23.8337$ |  $13.28$ |  $3.62$ |  $33.3700$ |  $6.64$

The maximal amplitude is fixed to be $M=0.5$ m, and the frequency is fixed to
be $\omega=3.5$ s-1. The frequency corresponds to a temporal period of
$T=1.7952$ s, wavenumber $k=1.25$ m-1 and wavelength $\lambda=5.0265$ m. In
the successive columns are listed the value of: the normalized modulation
frequency $\hat{\nu}$, the non-normalized modulation frequency $\nu$, the
amplitude amplification factor $\alpha$, the asymptotic level of the
background $2r_{0}$, the temporal modulation period $T_{\text{mod}}=2\pi/\nu$,
the number of waves in one temporal modulation period defined as
$N_{\text{temp}}:=T_{\text{mod}}/T$, the time $\zeta$ of the phase singularity
(when the largest wave is at $t=0$), the modulation wavelength
$\lambda_{\text{mod}}$, and the number of spatial waves in one modulation:
$N_{\text{spat}}:=\lambda_{\text{mod}}/\lambda$.

Note that a given perturbation $\nu$ in frequency corresponds to a
perturbation $\kappa$ in wavenumber according to the dispersion relation:
$\omega+\nu=\Omega(k+\kappa)\approx\Omega(k)+\Omega^{\prime}(k)\kappa$, hence
approximately $\nu=V(k)\kappa$. The spatial modulation length
$\lambda_{\text{mod}}$ can then be approximated by $2\pi/\kappa$ giving
$\lambda_{\text{mod}}=V(k)\ast T_{\text{mod}}$. Furthermore
$N_{\text{temp}}=\omega/\nu\approx\Omega(k)/\Omega^{\prime}(k)\kappa\approx
2k/\kappa$ and so the number of waves in one temporal envelope is
approximately twice the number of waves in one spatial envelope.

In Fig. 13 samples of the signals at the extreme position as well as at a
distance $150$ m are presented for the three cases. The signal at the distance
$150$ m can be interpreted as the signal at the wave maker at $x=0,$ that
evolves into the extreme signal at $x=x_{\max}=150$ m.

Figure 13: Plots of time signals of the wave elevation vs time (note the
different scaling in time in different rows) at two positions: in the second
column at the point of maximal elevation (the extremal signal), and in the
first column at a position $150$ m in front of the maximal position. The three
rows correspond to different choices for the modulation: from top down
$\hat{\nu}=\sqrt{3/2}$, $1$ and $\sqrt{1/2}$.

Some general observations can be made about the plots. Of course, the large
deformations in the signal profiles are clearly visible. Apart from the
amplitude amplification $\alpha$ listed in the table for these three cases, a
more practical amplification factor would now be the quotient of the maximal
wave amplitudes at the two positions. So in this set up
$\alpha_{\text{pract}}=\alpha(x_{\max})/\alpha(0)$ will give the amplification
in the interval of $150$ m; the numerical values are
$\alpha_{\text{pract}}=1.91$, $2.20$, and $2.34$ for $\hat{\nu}=\sqrt{3/2}$,
$1$ and $\sqrt{1/2}$, respectively. Concerning the wave profiles, there is a
clear difference between the three cases. For $\hat{\nu}=\sqrt{3/2}$ the
complete signal seems to consist of (relatively ‘mild’) extreme waves, and
intermittent waves are missing. That is understandable since this is the
limiting case at which the phase singularity start to develop. In the other
cases, the number of intermittent waves increases for smaller modulation.

###  6.2 MTA

The maximum temporal amplitude, MTA, has already been defined as

$\mu(x)=\max_{t}\eta(x,t)$ (9)

(see [18]), where $\eta(x,t)$ is the wave elevation. This quantity is of much
theoretical and practical importance. By its definition, it describes the
largest wave amplitude that can appear at a certain position. On the other
hand, it can be seen as a kind of stationary ‘envelope’ of the wave group
envelopes: it describes the boundary where the wall of the wave tank is wet
after a long time of waves running downstream.111We thank Gert Klopman for
this illustrative description. For the special SFB-case under consideration,
the MTA can be calculated using the explicit expressions. However, in general,
it is not clear how to obtain theoretically an expression for this function
from the nonlinear dynamical equations itself; the dependence on the initial
signal will be crucial. Some preliminary investigations will be reported
elsewhere.

Figure 14: From knowledge of the MTA of a certain initial signal, the extreme
position can be taken to be positioned at a desired place in the wave tank.
The distance to the wavemaker then determines which signal has to be taken as
the generating initial signal at the wavemaker. The plots are made with the
SFB solution for $\hat{\nu}=1$.

From a practical point of view, the MTA is very useful in the design of a
generation strategy. For an initial signal to be generated by the wave maker,
the MTA will depict the point, say $x_{\max}$, of maximal possible amplitude:
$M=\max_{x}\mu\left(x\right)=\mu\left(x_{\max}\right)$. If this position is
outside the wave tank, or too close to the beach, the inverse problem could
(theoretically) be used: if $L$ is the distance from the wavemaker where the
maximal waves are desired to occur, the signal at the position $x_{\max}-L$,
i.e. $t\rightarrow\eta\left(x_{\max}-L,t\right)$, should be taken as initial
signal at the wavemaker (provided the technical-physical constraints of the
wave maker – such as too high amplitudes, or too large frequencies – do not
prevent this). The practical amplification factor that will be achieved is
then $M/\mu(x_{\max}-L)$. In general, a direct calculation of the inverse
problem will be difficult, but in the case of SFB evolution, this is explicit.
For SFB evolution we illustrate the above-described methodology in Fig. 14,
using arbitrarily the value $\hat{\nu}=1$. For the cases considered in Table
1, the MTA curves are plotted in Fig. 15. From this figure the practical
amplification factors given earlier can be read off:
$\alpha_{\text{pract}}=\mu(150)/\mu\left(0\right)$. An expression for the MTA
can be found from the explicit expression (2.2); after some simplifications it
is found to be given by

$\left(\frac{\mu(x)}{2r_{0}}\right)^{2}=1+\frac{2\hat{\nu}^{2}\sqrt{1-\hat{\nu}^{2}/2}}{\cosh(\sigma
x)-\sqrt{1-\hat{\nu}^{2}/2}}.$ (10)

Observe in this expression the dependence on $\hat{\nu}$, and hence on the
initial signal.

Figure 15: Plots of the maximal temporal amplitude MTA for three values
$\hat{\nu}=\sqrt{3/2}$, $1$ and $\sqrt{1/2}$.

##  7 Summary and concluding remarks

In order to understand various aspects of the nonlinear distortion of waves
running downstream, we have investigated a family of specific explicit
solutions based on the approximate model described by the NLS equation. This
family of solitons on finite background, SFB, describe the full nonlinear
continuation of the linear modulational, Benjamin-Feir, instability. The
exponential growth of the linear instability of a sufficiently long modulated
monochromatic wave, is bounded by the nonlinearity, and returns to its
original monochromatic character after a long distance. During the downstream
evolution, however, large amplitude amplification and wave distortions take
place. Such properties are often observed, but difficult to describe from the
full nonlinear governing equations. The explicit expressions that are
available for SFB have been used to study in detail various aspects that are
related to the distortions. These explicit descriptions may contribute to
understand ‘extreme’ waves also in more general situations.

Depending on the origin of the physical equation, the practical interpretation
of the described qualitative phenomena and quantitative results may differ. In
this paper we have concentrated on the behaviour of gravity driven surface
waves, and its applicability to generate in a deterministic way extreme waves
in a hydrodynamic laboratory. A ship positioned at the extreme position will
experience waves described by the extreme signal described above: the
intermittent waves, which are numerous when $\hat{\nu}$ is small, resemble a
very calm ‘sea’ and are then followed by one or a few large waves (of height
that can be larger than three times the wave height in the calm sea), and a
quick return to the calm sea, which is repeated periodically in time. This
resembles somewhat as if a ‘freak’, or ‘rogue’ wave suddenly appears, as
suggested in [13].

## Acknowledgement

The authors are very grateful to Dr Pearu Peterson, Ir Gert Klopman and Dr
René Huijsmans for fruitful discussions about this research. This research is
conducted partly at Institut Teknologi Bandung, Indonesia and partly at the
University of Twente, The Netherlands, and is supported by Riset Unggulan
Terpadu Internasional (RUTI 2002/2005) of the Indonesian Ministry of Research
and Technological Development, project ‘On the onset of breaking waves’, and
by the Netherlands Organisation of Scientific Research NWO, subdivision
Applied Sciences STW, project ‘Extreme Waves’ TWI.5374. Partial support from
Hibah Bersaing 2002/2004 is also acknowledged.

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