Maximal Temporal Amplitude
a new tool for deterministic extreme wave generation
Andonowati & E. van Groesen
Institut Teknologi Bandung, Indonesia & University of Twente, The Netherlands
Summary
This research is motivated by the requirement of hydrodynamics laboratories to
generate extreme waves for testing ships in steep, large amplitude wave fields. To
facilitate a convenient way to characterise suitable wave fields we use the concept of the
Maximal Temporal Amplitude (MTA). This quantity measures at each downstream
position the maximal amplitude of the time signal of the nonlinearly evolving wave.
The MTA gives a direct view of the position where the maximal waves occur and which
signal should be generated at the wavemaker.
As illustration of the use of the MTA and to prepare for actual measurements, we aim to
generate extreme waves that are nonlinear extension of the modulational Benjamin-Feir
instability. For a simplified model this is described by an exact solution of the spatial
NLS known as the (spatial) Soliton on Finite Background (SFB). With the modulation
length and the prescribed maximum amplitude as parameters of SFB, the MTA’s of the
wavefields can be calculated explicitly. These show which parameter values are feasible
to be generated in a tank of given length and with given restrictions on the wave maker.
Key-words: Extreme waves, non-linear propagation, maximum temporal
amplitude, soliton on finite background
Background
The direct motivation of this research stems from the need of hydrodynamic laboratories to generate
`extreme' (`freak' or `rogue') waves to test ships and marine constructions in wave tanks under extreme
sea-conditions. The wavemaker at one end of the tank will generate a wave of initially moderate wave
height. Traveling downstream along the tank, non-linear effects and wave focusing may lead to large and
steep waves within the wave tank that are much larger than can be produced by the wavemaker directly.
The problem of ‘deterministic extreme wave generation’ is to find wave fields and precise wavemaker
signals such that at a specified position in the tank extreme waves are being produced.
Maximum Temporal Amplitude
The Maximal Temporal Amplitude (MTA) of a wave measures at each position the maximal amplitude of
the waves at that position, so the maximum over time of the time signal at that position:
α ( x) = max t η ( x, t ) . To facilitate the understanding of this MTA, see the figure below.
wave elevation
envelope
maximum temporal amplitude
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Shown is a snapshot of a wave field traveling to the right; the physical unit along the axes is meter, and
the wavelength is approximately equal to the depth of the layer. The dashed line denotes the wave
envelope, and shows that the field is composed of wavegroups which consist of about 5 waves. The plot
clearly shows large deformations of the wave groups while traveling downstream. The dotted line through
the maxima of the wave envelopes is the maximum temporal amplitude MTA.
Use of MTA for wave generation
From a practical point of view, knowledge of the MTA is very useful in the design of a generation
strategy. Indeed, let
x max be the ‘extreme position’, i.e. the point of maximal amplitude:
M = max x α ( x) = α ( x max ) . If L is in practice the distance from the wave maker where the largest
waves are desired to occur, then the signal at the position x max − L , i.e. η ( x max − L, t ) should be the
initial signal to be generated by the wave maker. This initial signal will be feasible if the technicalphysical constraints of the wave maker can be satisfied for this signal, in particular if the maximal wave
height at the wavemaker is feasible or not. The MTA also shows what the practical amplification factor is
from the wave height at the wavemaker to the largest wave is that is achieved, namely M / α ( x max − L) .
Example: Modulation instability described by SFB
The plot above corresponds to a wavefield that is described by a member from a family of Solitons on
Finite Background (SFB), a family of solutions of the spatial NLS that describe the continuation into the
nonlinear regime of the Benjamin-Feir instability. The basic parameters of the family are the maximal
waveheight and the modulation length which is a measure of the number of waves in one wavegroup.
Characteristic of SFB’s with sufficiently long modulation length is that wave focusing takes place which leads
to large local wave amplification in between phase singularities. This is shown in the plots below, where we
also show the wavemaker signal that leads to the extreme waves after a distance of 150 m downstream. In a
tank of smaller length, the MTA shows the wave height that has to be generated depending on the distance to
the extreme position.
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Density plot of the SFB, horizontally the
This is the time signal at the extreme
This is the time signal at a wave maker
time and vertically the spatial distance.
position of one SFB. Observe the large
that generates the wave field with the
Observe that the mono-chromatic time
differences in amplitudes between the
above plotted characteristic properties.
signals for large negative and positive
intermittent waves and the extreme
The extreme position will be at a
spatial distances are connected by large
waves which are separated by phase
position 150 m downstream of the wave
deformations and wave dislocations.
singularities
maker where the wave signal is given by
vanishes.
where
the
envelope
the plot at the left.