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Proc. R. Soc. A (2007) 463, 85–112
doi:10.1098/rspa.2006.1759
Published online 1 August 2006
Observations on the evolution
of wave modulation
B Y H WUNG -H WENG H WUNG 1, * , W EN -S ON C HIANG 2
1
AND S HIH -C HUN H SIAO
1
Department of Hydraulics and Ocean Engineering, and 2Tainan Hydraulics
Laboratory, National Cheng Kung University, No. 1, Ta-Hsueh Road,
Tainan, 701 Taiwan, Republic of China
A series of laboratory experiments on the long-time evolution of nonlinear wave trains in
deep water was carried out in a super wave flume (300!5.0!5.2 m) at Tainan
Hydraulics Laboratory of National Cheng Kung University. Two typical wave trains,
namely uniform wave and imposed sideband wave, were generated by a piston-type
wavemaker. Detailed discussions on the evolution of modulated wave trains, such as
transient wavefront, fastest growth mode and initial wave steepness effect, are given and
the results are compared with existing experimental data and theoretical predictions.
Present results on the evolution of initial uniform wave trains cover a wide range of initial
wave steepness (kc ac Z 0:12–0:30) and thus, greatly extend earlier studies that are confined
only to the larger initial wave steepness region (kc ac O 0:2). The amplitudes of the fastest
growth sidebands exhibit a symmetric exponential growth until the onset of wave breaking.
At a further stage, the amplitude of lower sideband becomes larger than the carrier wave
and upper sideband after wave breaking, which is known as the frequency downshift.
The investigations on the evolution of initial imposed sideband wave trains for fixed initial
wave steepness but different sideband space indicate that the most unstable mode of initial
wave train will manifest itself during evolution through a multiple downshift of wave
spectrum for the wave train with the smaller sideband space. It reveals that the spectrum
energy tends to shift to a lower frequency as the wave train propagates downstream due
to the sideband instability.
Experiments on initial imposed sideband wave trains with varied initial wave steepness
illustrate that the evolution of the wave train is a periodic modulation and demodulation
at post-breaking stages, in which most of the energy of the wave train is transferred
cyclically between the carrier wave and two imposed sidebands. Meanwhile, the wave
spectra show both temporal and permanent frequency downshift for different initial wave
steepness, suggesting that the permanent frequency downshift induced by wave breaking
observed by earlier researchers is not permanent. Additionally, the local wave steepness
and the ratio of horizontal particle velocity to linear phase velocity at wave breaking in
modulated wave group are very different from those of Stokes theory.
Keywords: modulation; recurrence; sideband instability; frequency downshift
* Author for correspondence ([email protected]).
Received 13 January 2006
Accepted 29 June 2006
85
q 2006 The Royal Society
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H.-H. Hwung and others
1. Introduction
The evolution of nonlinear wave trains in deep water exhibits many subtle
phenomena, such as chasing, integration and disintegration among waves.
Pioneering research was done by Benjamin & Feir (1967), who showed that weakly
nonlinear deep-water wave trains are theoretically unstable to modulational
perturbations. Lake et al. (1977) further confirmed the validity of Benjamin and
Feir’s theory experimentally and discussions on how the energy interchanges
between different modes are also given. Later, Melville (1982, 1983), Su (1982),
Tulin & Waseda (1999) and Hwung & Chiang (2005a) all explored the wave
modulation experimentally in detail.
Theoretical and numerical progresses to the evolution of nonlinear wave trains are
further advanced by Zakharov (1968), Dysthe (1979), Lo & Mei (1985) and Krasitskii
(1994) based on different assumptions and numerical schemes. Specifically, Zakharov
showed that the leading order complex amplitude of the modulated surface wave
satisfies the nonlinear Schrödinger (NLS) equation for weakly nonlinear and narrowbanded wave trains. On the basis of the NLS equation, the evolution of a wave train is
expected to have a recurrence of initial state and the envelope of surface elevation is
symmetric with respect to the peak of wave profile. On the other hand, previous
studies on the numerical simulation of the NLS equation had revealed that a wave
field with initial narrow bands could breakdown due to the energy leakage to highfrequency modes, resulting from the violation of the narrow bandwidth constraint of
the NLS equation. Therefore, Dysthe (1979) derived the fourth-order NLS equation,
which includes the small mean current induced by the radiation stresses of the
modulated wave trains, whereby the asymmetric envelope of surface elevation with
respect to the peak of the wave profile was found. Crawford et al. (1981) investigated
the evolution of nonlinear wave trains using an improved approximation from
Zakharov and showed that the predictions of Benjamin and Feir’s theory regarding
the most unstable wave mode and the maximum growth rate could differ as large as
30% from their numerical results for an initial wave steepness of 0.2. Stiassnie &
Kroszynski (1982) analytically examined the long-time evolution of unstable wave
trains, which combines with a carrier wave and two small sideband components;
especially, the long waves induced by wave groups and the recurrence period of
modulation–demodulation were discussed. Lo & Mei numerically solved the fourthorder NLS equation and demonstrated the asymmetric evolution of initial symmetric
wave trains. Besides, the long-time evolution of wave trains reveals a periodic
modulation and demodulation process, in which the recurrence of initial state of wave
trains was found. The corresponding wave spectrum discloses a periodic frequency
downshift and upshift. Using the same numerical scheme proposed by Lo & Mei,
Trulsen & Dysthe (1990) included an artificial function of energy dissipation into the
fourth-order NLS equation and produced permanent frequency downshift in their
numerical simulation on the evolution of initial modulated wave trains. Later, they
further proposed the band-modified nonlinear Schrödinger equation (BMNLS), in
which the restriction of spectrum bandwidth was relaxed while retaining the same
accuracy in nonlinearity. Numerical investigations with the BMNLS by Trulsen &
Dysthe (1996, 1997) for the evolution of weakly nonlinear narrow band wave trains in
deep water indicate that permanent frequency downshift is prohibited in twodimension. It is worth noting that Trulsen & Dysthe’s studies imply that energy
dissipation is the dominant factor leading to permanent frequency downshift.
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Observations on wave modulation evolution
87
In contrast, there are only a few experiments available on the long-time
evolution of nonlinear wave trains in deep water due to the limitation of
appropriate experimental facilities. Lake et al. (1977) conducted experiments on
the evolution of nonlinear wave trains and confirmed Benjamin & Feir’s finding
on the initial growth of sideband amplitudes. They reported that the evolution of
wave trains occurs near recurrence of initial state and the corresponding wave
spectrum reveals a frequency downshift. Later, Melville (1982) underwent a
transformation analysis on initial regular waves in deep water with high
nonlinearity, where the wave steepness ranges between 0.20 and 0.28. Melville
found that the development of sideband energy mainly resulted from the
repeated reflection of wavefronts in wave flumes. Following an initial exponential
growth, two sidebands would grow asymmetrically and the location would mirror
to the approximate point of wave breaking. The frequency downshift of wave
spectrum was investigated experimentally after wave breaking and Melville
concluded that the spectrum evolution of nonlinear wave trains due to the
sideband instability is not only restricted to a few discrete frequencies, but also
involves a growing continuous spectrum. On the other hand, Su et al. (1982)
provided experimental findings on the evolution of a finite length wave packet.
The frequency downshift during the evolution of a short wave packet is
significant for initial wave steepness as low as 0.1. The evolution of longer wave
packets is more complex than that of shorter wave packets. Thereafter, Su et al.
conducted a series of experiments on the nonlinear instabilities of continuous
wave trains in deep water, both in a wave tank and a wide basin. It was found
that two-dimensional wave trains evolved into a series of three-dimensional
spilling breakers for initial large wave steepness, followed by a series of twodimensional wave groups. Melville (1983) further analysed the instantaneous
amplitude, frequency, wavenumber and phase speed in deep-water wave trains
using the Hilbert transform technique. The increasing modulation that evolves
from sinusoidal perturbations finally results in a very rapid frequency jump or phase
reversal near the location, where the local amplitude of wave trains approaches zero.
Recently, Waseda & Tulin (1999) demonstrated experimentally that the
growth rate of sideband amplitudes decreases as wind speed increases, but
the sideband amplitude is still larger for stronger wind conditions. Moreover, the
near recurrence of the initial state of wave trains without frequency downshift
was observed in a non-breaking case. For a wave-breaking case, the spectrum
evolution at post-breaking stage indicates that the amplitude of the lower
sideband almost coincides with that of the carrier wave in their experiment.
Further evolution was not investigated due to the limitation on the length of the
wave flume. More recently, some of the remarkable results on long-time evolution
of nonlinear wave trains in deep water were reported by the research team of
Tainan Hydraulics Laboratory. Particularly, Hwung & Chiang (2005a) studied
experimentally the evolution of wave trains from the initial onset of instability on
initial modulated wave trains to the long-fetch evolution of the wave trains. The
modulation and demodulation of wave trains occur cyclically at post-breaking
stages. The phenomenon was reconfirmed experimentally on the evolution of
initial uniform wave trains by Hwung & Chiang (2005b).
Although successful results and interesting phenomena, such as disintegration of
wave trains, recurrence of initial status of wave trains and frequency downshift of a
wave spectrum, have been reported in earlier studies for different kinds of wave
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88
H.-H. Hwung and others
grouping, the observed evolutions of nonlinear wave trains in fetch are only a few
(e.g. Stansberg 1998; Trulsen et al. 1999; Onorato et al. 2004) due to the limitation
on the length of wave flume. To further extend the earlier studies, the present paper
considers the evolution characteristics of nonlinear modulated wave trains for a
wide range of initial wave steepness. In particular, the transient wavefront, the
evolution of wave trains and related wave spectra, the fastest growth mode and
time-frequency features of wave breaking are discussed in detail. A brief description
of the experimental set-up, data acquisition system and the measurement technique
is given in §2. The experimental data are analysed and discussed in §3. Section 4
outlines the conclusions obtained from this study.
2. Experimental apparatus and data analysis
The success of carrying out experiments on the modulation of nonlinear wave
trains highly depends on the following terms (Hwung & Chiang 2005b): (i) a
carefully constructed long wave tank, (ii) a high accuracy and programmable
wavemaker, (iii) the experimental instrumentation, and (iv) a suitable analysis
technique. The experimental facilities, instruments and the method of data
analysis used in this study are described in the following sections.
(a ) Experimental facilities and set-up
The experiments were performed in a super long-wave flume at Tainan
Hydraulics Laboratory. The flume is 300 m long, 5.0 m wide and 5.2 m deep.
Figure 1 shows the schematic diagram of the experimental set-up. A programmable,
high-resolution wavemaker is located at one end of the tank and an effective waveabsorbing structure at the opposite end. The wave-absorbent structure comprises
two slopes covered by pebbles with a mean diameter of 10 cm. The first slope (1 : 7)
of 14.9 m in length is located at 240 m and the second slope (1 : 10) of 25.8 m in
length is located at 260.3 m from the position of the waveboard. Between the two
slopes, there is a berm of 5.4 m in length in order to stabilize the armour pebbles. In
general, the reflection coefficients are between 3 and 5% and the maximum
reflection coefficient is about 7% in the experimental tests. The wavemaker is a
piston-type paddle activated by a hydraulic cylinder. The motion of the hydraulic
cylinder is prescribed by a programmable controller, which takes an external input
signal for the motion. The input signal at 25 Hz can be either generated in real time
or read from a data file stored on the hard disk. The wavemaker is designed to have
optimal performance for wave periods between 1 and 3 s. The evolution of surface
wave trains is recorded by using 66 capacitance-type wave gauges, which are
located between 15 and 240 m downstream of the wavemaker. Each wave gauge has
a dynamic range of 4.5 m at 12 bits digitization, with noise typically less than 4
counts (approx. 0.5 mm). The wave gauges are calibrated before and after the
experiments. The linearity of wave-gauge response is indicated by a correlation
coefficient of 0.999. The data acquisition is PC-based multi-node data-acquisition
system, which is developed by Tainan Hydraulics Laboratory specially to cope with
two problems. One is synchronization of signals from a large number of parallel
inputs and the other is the signal decay due to long-distant transmission of data in
the wave flume. The time delay of the measured data between first and last gauge in
the data acquisition system is 0.01 s, which is relatively small compared to 0.04 s
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Observations on wave modulation evolution
3.5 m
1:7
1:10
wave
generator
5.2 m
video camera
video camera
side view of the flume
wave gauges
No. 17–63 spacing 3 m
5.0 m
wave gauges
No. 4 –17 spacing 6 m
0.4 m
238 m
plane view of the flume
15 m
6
1
0
MEASURE
6
2
6
PROCESS
3
IPC1 MNDAS
GENERATE
4
5
AD/DA
CED1401
47m
0 : configuration file
1 : instruments database
2 : measured signals
3 : processing results
4 : AD/DA instruction file
5 : wave board control signals
6 : command file
IPC1
fram grabber
Figure 1. Schematic diagram of the experimental set-up.
sampling interval and 1.6 s wave period. So, the time-series of water surface
elevations are acquired almost simultaneously at 25 Hz and stored for further
processing. For initial uniform wave trains, the experimental run is recorded for
40 min of real time in order to provide data samples sufficiently long for accurate
calculation of the power spectra. Each experimental run is recorded for 10 min of
real time for initial imposed sideband wave trains in order to provide data samples
sufficiently long for accurate calculation of the power spectra. The time-interval
between two successive measurements is 35 min for any long wave fluctuations to be
damped out in the wave flume.
The accurate performance of the wave generator is obviously a key factor in
successfully carrying out experiments on wave modulations and wave breaking.
In order to ensure that the experimental data are of high quality and
uncontaminated by background noise, the wave generator system was checked
before carrying out the experiments. The detailed description can be found in
Hwung & Chiang (2005b). The results suggest that the noise energy level
generated by wave generator is very small compared to the given wave
conditions both in time- and length-scale. Hence, the noise effect on the evolution
of wave modulation can be ignored in the study.
(b ) Test conditions
The experiments were performed using computer-generated wave forms. Two
different types of wave generation mechanisms are investigated, which includes
uniform wave trains and imposed sideband wave trains, given by equations (2.1)
and (2.2), respectively, as input to the wavemaker servo-system.
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H.-H. Hwung and others
(i) Uniform wave trains
ð2:1Þ
hðtÞ Z ac sinðuc tÞ;
where h is the surface displacement, ac and uc are the given amplitudes and
angular frequency of carrier wave, and t is the time.
(ii) Imposed sideband wave trains
hðtÞ Z ac sinðuc tÞ C aCsinðuCt C fCÞ C aKsinðuKt C fKÞ;
uGZ uc GDu;
fGZKp=4;
ð2:2Þ
ð2:3Þ
ð2:4Þ
where ac and aG are the amplitudes of the carrier wave and imposed
sidebands, uc and uG are the angular frequency of the carrier wave and
imposed sidebands, Du is the frequency difference between the carrier wave and
imposed sidebands, fG is the initial phase difference between the carrier wave
and imposed sidebands. The stroke of the wave paddle is related to the given
wave amplitude by the linear wavemaker theory.
According to Benjamin & Feir’s (1967) theory, the growth rates of the
sidebands depend on the dimensionless frequency difference between the carrier
^ Du=ðuc kc ac Þ, where kc is the wavenumber of
wave and the sidebands, dZ
pffiffiffi
^
the carrier wave. When d is smaller than 2 and the depth parameter ðkc hÞ
is larger than 1.363, the Stokes wave is unstable to small sideband perturbations.
^
The most unstable mode of sidebands corresponds to dZG1:0.
Tulin & Waseda
(1999) demonstrated that the initial growth rates of sidebands decrease as the
initial wave steepness increases, indicating that initial wave steepness is also an
important factor in the modulation of wave trains through sideband instability.
Therefore, the effects of three important parameters, namely dimensionless
^ depth parameter ðkc hÞ and initial wave steepness
frequency difference (d),
parameter ðeZ kc ac Þ, are investigated in this study. For initial uniform wave
trains, the wave period and amplitude are prescribed and the wave conditions are
listed in table 1. In particular, the evolution of wave trains is examined for varied
initial wave steepness parameter and water depth parameter.
The initial imposed sideband wave trains are composed of a carrier wave of
prescribed angular frequency and a pair of sideband components of prescribed
frequency difference between the carrier wave and imposed sidebands. The initial
magnitudes of the sideband amplitudes relative to that of the carrier waves are
also prescribed. A wide range of initial wave steepness and the frequency
difference between the carrier wave and sideband components are generated and
details are listed in table 2.
We noted that the existence of background noise might mislead our
understanding of the wave evolution. In this study, the measured noise data in
the wave flume are composed of the electronic noise and wind ripple, which are
obtained by the analysis of the Hilbert-Huang transform (Huang et al. 1998).
Generally, the mean amplitude of noise is less than 0.2 cm depending on the
environmental condition, which is not substantial in comparison with the given
wave amplitudes in the experiments. Additionally, the peak frequency is about
4.0 Hz according to the results of the Fast-Fourier transform analysis, which is far
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Observations on wave modulation evolution
Table 1. Experimental wave conditions of initial uniform wave trains.
water depth (hZ3.5 m)
water depth (hZ2.5 m)
eZ kc ac
kc h
eZ kc ac
kc h
0.072–0.240
0.081–0.252
0.063–0.354
0.063–0.312
0.107–0.357
0.111–0.349
1.68
3.53
5.50
7.29
8.33
9.78
0.056–0.444
0.088–0.446
0.133–0.433
0.117–0.430
—
—
3.53
5.50
8.33
9.78
—
—
Table 2. Experimental wave conditions of initial weakly modulated wave trains.
wave type
ac (cm)
Tc (s)
eZ kc ac
^ Du=ðuc eÞ
dZ
aG=ac
initial modulated
wave trains
0.04–0.22
1.1–2.0
0.081–0.230
0.10–1.4
0.05–0.35
beyond the carrier frequency used in the experiments and the corresponding
instability region. The results demonstrate that the electronic noise of wave
gauges and the ripple due to light wind blowing over the water surface in the flume
are much smaller than the wave generated by wavemaker both in length- and
time-scale. More detailed analysis can be referred to Chiang (2005).
(c ) Data analysis techniques
(i) Analysis of the local maximum wave
In order to quantitatively describe the evolution of a wavefront, the wave data
before wave reflection are extracted from the experiments. According to the
measured surface elevation at the station near the end of the tank, the duration
before the wave approaching the wave absorber is identified. The measured wave
data before wave reflection from the wave absorber at all gauges are used in the
following analysis.
The normalized local wave parameters provide a key to understanding the
transient large wave. Following Grue et al. (2003), the local wavenumber is
estimated from Stokes third-order wave theory in conjunction with the measured
surface elevation and the dispersion relation. A set of local wave parameters in
equations (2.5)–(2.7) are defined according to the Stokes third-order wave theory
and the measured surface elevation data. By solving the coupled equations (2.6)
and (2.7), the local wave steepness em and wavenumber k m that corresponds to
the measured maximum crest elevation are calculated. Then, the local wave
steepness based on crest elevation, k m hM , and maximum wave height, k m a m , can
be obtained.
2p
uZ
;
ð2:5Þ
Tzc
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H.-H. Hwung and others
u2
Z 1 C e2m C Oðe4m Þ;
gk m
1
1
k m hM Z em C e2m C e3m C Oðe4m Þ;
2
2
k H
k ma m Z m m ;
2
rffiffiffi
g
um Z
em ekhM ;
k
ð2:6Þ
ð2:7Þ
ð2:8Þ
ð2:9Þ
where Tzc is the wave period of measured surface elevation defined by zero up
crossing method, hM is the measured maximum crest elevation, Hm is the wave
height corresponding to the crest elevation hM and u m is the water particle
velocity at crest elevation.
(ii) Analysis of wave spectrum
Spectrum analysis is used to analyse the phenomena of wave evolution, such
as the sideband energy level, the initial growth of sidebands and the evolution
of the wave spectrum. The spectrum is obtained by using a discrete Fourier
transformation with the Hanning window. In the present study, the spectrum
is calculated within the frequency range of 0–12.5 Hz and a resolution
bandwidth of 0.006 Hz, which is smaller than the frequency difference between
the carrier wave and the naturally evolved and initial imposed sidebands in all
the experiments.
(iii) Analysis of initial growth rate
From the analysed wave spectrum, the amplitudes of wave modes ðac ; aGÞ can
be expressed as the square root of the total energy of the corresponding wave
spectral peaks. The normalized amplitudes of sidebands, bG, are defined in
equation (2.10). The first-order effect of dissipation is removed by using the
normalized amplitudes due to the small frequency difference between the carrier
wave and sidebands. The dissipation rates are almost identical for the sidebands
and carrier wave owing to the small frequency difference between them. Next, the
normalized amplitudes are plotted as a function of normalized fetch, x~, which is
defined in equation (2.11). Then, the exponential curve, equation (2.12), is fitted
by the least square method to the plotted data for normalized sideband
amplitudes between 0.05 and 0.2, whereby the initial growth rate, b, of the
sideband amplitudes is determined.
a
bG Z G ;
ð2:10Þ
ac
x~ Z kc x;
ð2:11Þ
bGð~
x Þ Z bGðx 0 Þexpðb~
x Þ:
ð2:12Þ
Note that x 0 is the initial position of the sideband growth downstream of the
wavemaker, kc is the wavenumber of the carrier wave and, x is the distance
measured downstream of the initial position, x 0.
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Observations on wave modulation evolution
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3. Experimental results
(a ) Modulation of wavefront and quasi-steady state
The evolution of wave trains from initial transient wave generation up to quasisteady state is investigated in this section. The modulation of initial uniform
wave trains is initiated from transient wavefront accompanied by the increase of
sideband energy. The sideband energy further enhances from the effect of wave
reflection until a quasi-steady state is achieved. Details are given as follows.
(i) Transient wave transformation
The unsteady wavefront, where the local wave amplitude increases gradually
from zero to a steady state, is observed on the surface elevation due to the
variation of initial transient motion of the wavemaker. Transformation of
the unsteady wavefront develops with the propagation downstream of the wave
flume. The combining effect of nonlinearity and dispersion results in the
interesting and complicated evolution of the unsteady wavefront. Owing to
the nonlinearity of wave trains, the wave crest with large amplitude travels fast.
So, the evolution of wave trains following the initiation of modulation shows that
the wave energy focuses into a short group of steeper waves. As the wave trains
evolve further, the modulation becomes stronger, which either initiates wave
breaking or reaches a maximum modulation. Roughly, at this stage, the
modulated wave trains form a pulse train, where the minimum level of individual
wave amplitude approaches zero and a crest pairing is observed. However, the
wave trains demodulate at a later stage and crest splitting is exhibited during the
demodulation process. A typical evolution of initial uniform wave trains without
breaking is shown in figure 2. The initial wave steepness ðkc ac Þ of the wave trains
is 0.110 and the relative water depth ðkc hÞ is 5.5. The top time-series of surface
elevations measured at 15 m fetch is about four wavelengths away from the
wavemaker. It shows that the wave trains just off the paddle are almost uniform
except the initial transient wavefront. The following time-series illustrates the
developing modulation and the formation of leading group in front. Owing to
small initial wave steepness, the evolution of the wave trains is relatively slow.
To understand the effect of water depth, the evolution of wave trains with eZ 0:112
and kc hZ 9:8 is shown in figure 3. Under the same wave steepness, it is evident that
the evolution of wave trains evolves faster as the relative water depth increases.
Moreover, two wave groups form at the head of the following uniform wave trains
shown at fetch kc x Z 456. No breaking is observed in this case according to the video
recording of the experiment. For sufficiently large wave steepness, the initial
transient wavefront evolves due to the Benjamin–Feir instability leading to wave
breaking. Then, the wave front goes through a region of breaking and becomes a
region of unbroken gentle wave as shown in figure 4 with eZ 0:149 and kc hZ 5:5.
The above-mentioned experimental data suggest that the modulation of wave
trains induces a large transient wave group, characterized by an extreme wave
height in front of the wave trains, which is significantly higher than all the
proceeding and the following waves. Besides, these experimental results
demonstrate that the source of modulational disturbances lies in a disturbance
travelling with the wavefront, and the nonlinear instability of wavefronts in initial
uniform wave trains does not lead to permanent disintegration of wave trains. It is
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H.-H. Hwung and others
20
0
–20
kcx= 30
kcx= 49
kcx= 68
kcx= 86
kcx=105
kcx=124
surface elevation (cm)
kcx=143
kcx=162
kcx=181
kcx=200
kcx=219
kcx=237
kcx=256
kcx=275
kcx=294
kcx=313
kcx=332
kcx=351
kcx=369
0
20
40
60
80
100
120 140
time (s)
160
180
200
220
240
Figure 2. The measured surface elevations of the wave front of initial uniform wave trains,
kc ac Z 0:110, kc hZ 5:5 (non-breaking).
worth noting that Yue (1980) solved the NLS equation numerically for
the evolution of a transient wavefront. His calculated results indicated that the
undulation developed behind the wavefront is qualitatively consistent with the
present experimental results.
On the other hand, the evolution of transient wavefronts of initial imposed
sideband wave trains is also examined as shown in figures 5 and 6 for eZ 0:109 and
0.173, respectively. Generally, the same qualitative features as the evolution of
transient wavefront of initial uniform wave trains are found. However, the
modulation of wave trains occurs simultaneously on the transient wavefront and
the following wave trains due to the nonlinear interaction between the carrier wave
and initial imposed small perturbations through the mechanism of sideband
instability. In figure 6, wave breaking was observed during the propagation of the
wavefront. First breaking region locates between kc x Z 105–143 and second
breaking region locates between kcxZ303 and 360. It is observed that these
breaking regions of transient wavefront may not correspond to those of the quasisteady state.
Furthermore, based on the visual investigation of experiments, the observed
wave breaking during the evolution of the wavefront shows both two- and threedimensional features, which depend not only on the initial wave steepness, but
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Observations on wave modulation evolution
20
0
kcx=53
–20
kcx=87
kcx=120
kcx=154
kcx=187
kcx=221
surface elevation (cm)
kcx=254
kcx=288
kcx=321
kcx=355
kcx=388
kcx=422
kcx=456
kcx=489
kcx=523
kcx=556
kcx=590
kcx=623
kcx=657
0
20
40
60
80
100
120
140
160
180
200
220
240
time (s)
Figure 3. The measured surface elevations of the wave front of initial uniform wave trains,
kc ac Z 0:112, kc hZ 9:8 (non-breaking).
also on the frequency difference between the carrier wave and sideband
components. Figure 7 displays two typical images of two- and three-dimensional
wave breaking. The breakers were observed intermittently at the centre and the
sidewall of the wave flume with the propagation of the wavefront.
In order to reveal the evolution of amplitude spectra of wave trains in space
domain, the amplitude contours of wave trains are plotted versus frequency and
space at different time instants for initial uniform wave trains eZ 0:220 as shown
in figure 8. It indicates the energy distribution of wave trains in the frequency
and fetch at the initial stage of wave generation, tZ10 s, in which the wave
energy focuses in the fundamental and its first higher harmonic frequencies. At
tZ50 s, the wave trains propagates a little further and the corresponding energy
spreads to the sidebands of fundamental frequency, which are enhanced after the
reflection of wavefront from the absorbing structure as shown at tZ120 s.
However, the peak frequencies of naturally evolved sidebands are obscure at this
stage. At tZ2000 s, the contours of wave amplitudes reveal evident energy peaks
at the carrier wave and naturally evolved frequencies of sidebands, especially a
pair of fastest growth sidebands. These results demonstrate that the sideband
energies initiate from the transient wavefront and increase as the wavefront
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20
0
–20
kcx = 30
kcx = 49
kcx = 68
kcx = 86
kcx =105
kcx =124
surface elevation (cm)
kcx =143
kcx =162
kcx =181
kcx =200
kcx =219
kc x =237
kcx =256
kcx =275
kcx =294
kcx =313
kcx =332
kcx =351
kcx =369
0
20
40
60
80
100
120 140
time (s)
160
180
200
220
240
Figure 4. The measured surface elevations of the wave front of initial uniform wave train,
kc ac Z 0:149, kc hZ 5:5 (breaking).
propagates downstream of the wave flume. Ideally, the wave trains will evolve
further in an infinite domain. However, the wave trains reflect at the end wall of
wave flume, which further magnifies the strength of the sidebands through
nonlinear interaction between the carrier wave and the sidebands. Eventually, a
quasi-steady state is reached after 40 min of wave generation for initial uniform
wave trains. For initial imposed sideband wave trains, the quasi-steady state of
wave modulation is achieved earlier than the initial uniform wave trains. Since
the characteristics of evolution are similar to that of initial uniform wave trains,
the detailed patterns are omitted here.
(ii) Quasi-steady modulation of wave trains
Figure 9 shows the records of water surface elevations and the related amplitude
spectra, when the quasi-steady state is approached at several locations downstream
of the wave flume for the initial uniform wave train with kc ac Z 0:165 and kc hZ 5:5.
Initially, the wave profile is almost uniform with the wave spectrum comprising a
carrier wave and its related weak higher harmonics. Modulational instability causes
an almost uniform wave train to gradually develop a well-defined wave group at the
fetch kc x Z 221 accompanied with the growth of a pair of sideband amplitudes,
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Observations on wave modulation evolution
20
0
–20
kcx=45
kcx=74
kcx=102
kcx=131
kcx=160
kcx=188
surface elevation (cm)
kcx=217
kcx=245
kcx=274
kcx=302
kcx=331
kcx=360
kcx=388
kcx=417
kcx=445
kcx=474
kcx=502
kcx=531
kcx=560
0
20
40
60
80
100
120 140
time (s)
160
180
200
220
240
Figure 5. The measured surface elevations of the wave front of the initial imposed sidebands wave
^ 1:0 and kc hZ 8:3 (fully developed without breaking).
train, kc ac Z 0:109, dZ
which can be seen in the evolution of wave spectra. Further evolution shows that the
large wave amplitude occurs in front of the wave group and then initiates wave
breaking. Around the initiation of the breaking stage, the wave spectrum indicates
an asymmetrical development of amplitudes of the lower and upper sideband
components. Meanwhile, the energy of the wave train spreads over frequencies that
are higher than the upper sideband. During the breaking process, the energy of the
lower sideband is selectively amplified compared to those of the carrier wave and
upper sideband. Therefore, the frequency downshift of wave spectrum is observed
after wave breaking. The wave train then undergoes a demodulation process until
the end of the wave flume, in which the carrier wave gradually recovers part of its
initial energy through nonlinear wave interaction due to wave breaking, an earlier
observed irreversible process. Finally, the amplitudes of the carrier wave and lower
sideband are almost identical in this case.
(iii) Spatial evolution of normalized wave amplitudes
To describe the modulation of wave trains quantitatively, the spatial
evolution of wave mode amplitudes is discussed in this section. The spectrum
discussed in §3a(ii) showed that most of the energy of the wave trains resides
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20
0
–20
kcx =30
kcx =49
kcx =68
kcx =86
kcx =105
kcx =124
surface elevation (cm)
kcx =143
kcx =162
kcx =181
kcx =200
kcx =219
kcx =237
kcx =256
kcx =275
kcx =294
kcx =313
kcx =332
kcx =351
kcx =369
0
20
40
60
80
100
120
140
160
180
200
220
240
time (s)
Figure 6. The measured surface elevations of the wave front of the initial imposed sidebands wave
^ 0:5 and kc hZ 5:5 (fully developed with breaking).
train, kc ac Z 0:173, dZ
Figure 7. Photographs of the breaking events of initial imposed sidebands wave train with
^ 0:50.
uc Z 0:625 Hz, eZ 0:173 and dZ
in the carrier wave and some discrete sidebands. The amplitudes of wave mode
components are calculated from the spectrum of each corresponding wave
mode. The spatial evolution of the normalized amplitudes of the carrier wave
and a pair of fastest growth sidebands with four different wave steepness,
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Observations on wave modulation evolution
frequency
(a)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
t =10 s
frequency
(b)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
t =50 s
frequency
(c)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
t =120 s
frequency
(d )
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
t =2000 s
70
120
170
220
270
320
kc x
– 0.2
0
0.2
0.4
amplitude in log scale (cm)
0.6
0.8
Figure 8. The contours of amplitude spectra of initial uniform wave train, kc ac Z 0:220, kc hZ 5:5
versus frequency and non-dimensional fetch at four different times.
kc ac Z 0:149, 0.165, 0.220 and 0.252, are discussed and the results are shown in
figure 10. For the sake of comparisons, the theoretical predictions based on
Benjamin & Feir (1967) and Tulin & Waseda (1999) are also given. As a
result, the initial evolution of sideband amplitudes displays an exponential
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10
20
0
–20
kcx =23
0
kcx =30
–10
kcx =58
kcx =87
kcx =116
kcx =144
kcx =163
kcx =178
kcx =206
kcx =221
kcx =235
kcx =249
amplitude (cm)
surface elevation (cm)
kcx =192
kcx =264
kcx =278
kcx =292
kcx =306
kcx =321
kcx =335
kcx =349
kcx =364
kcx =375
0
10
20
time (s)
30
40
0
0.5
1.0
1.5
2.0
frequency (Hz)
Figure 9. The measured surface elevations and related wave spectra at quasi-steady state for initial
uniform wave train, kc ac Z 0:165, kc hZ 5:5.
growth tendency that is consistent with the prediction of Benjamin & Feir.
However, the growth rate of Benjamin & Feir is overestimated, especially for
initial larger wave steepness.
Detailed inspection further shows that the initial growth rate of the sidebands
matches well with Tulin & Waseda’s calculation up to the breaking area.
Moreover, the symmetric growth of the lower and upper sidebands develops until
it reaches a level, where the energy is defined by the square of the normalized
sideband amplitudes, approximately Oð10K1 Þ. Subsequently, the amplitudes of
the lower and upper sidebands diverge, which roughly corresponds with the onset
of wave breaking. While the lower sideband retains tendency of exponential
growth during the breaking stage, the upper sideband declines slowly after
approaching a certain smaller maximum value. When the wave breaking ceases,
the energy of the lower sideband reaches its local maximum. Meanwhile, the peak
of the wave spectrum shifts to the lower sideband. At post-breaking stages, the
evolution of wave trains discloses a periodic demodulation and modulation
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Observations on wave modulation evolution
(a) 1.2
1.0
ai / ao
0.8
0.6
0.4
0.2
0
(b) 1.2
1.0
ai / ao
0.8
0.6
0.4
0.2
0
(c) 1.2
1.0
ai / ao
0.8
0.6
0.4
0.2
0
(d ) 1.2
1.0
ai / ao
0.8
0.6
0.4
0.2
0
50
100
150
200
kc x
250
300
350
400
Figure 10. Evolution of normalized amplitudes of one carrier wave and a pair of fastest growth
sidebands versus dimensionless fetch. (a) eZ 0:149, (b) eZ 0:165, (c) eZ 0:220 and (d) eZ 0:252.
Circle, carrier wave; diamond, lower sideband; cross, upper sideband; solid line, growth curve of
sidebands predicted by Benjamin & Feir (1967); dashed line, growth curve of sidebands predicted
by Tulin & Waseda (1999). Dotted region indicates the occurrence of a wave breaking event.
process. The amplitude of the lower sideband increases while that of the carrier
wave decreases with the increase of the modulation. However, a contrary trend of
the evolution of the carrier and lower sideband is found during the demodulation
process. For initial larger wave steepness, the energy transfer back to the carrier
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H.-H. Hwung and others
(a) 1.2
ai / ao
1.0
0.8
0.6
0.4
0.2
0
(b) 1.2
ai / ao
1.0
0.8
0.6
0.4
0.2
0
(c) 1.2
ai / ao
1.0
0.8
0.6
0.4
0.2
0
ai / ao
(d ) 1.2
1.0
0.8
0.6
0.4
0.2
0
50
100
150
200
kc x
250
300
350
400
Figure 11. The evolutions of dimensionless amplitudes including one carrier wave and two pairs of
sidebands for wave trains with the same wave steepness (eZ 0:173), but different normalized
^ 0:30, (b) dZ
^ 0:50, (c) dZ
^ 0:70 and (d) dZ
^ 1:0. Circle, uc ; diamond, uc KDu;
sideband space. (a) dZ
cross, uc C Du; square, uc C 2Du; triangle, uc K2Du. Dotted region indicates the occurrence of a
wave breaking event.
wave during the demodulation process is lower than that for smaller wave
steepness after wave breaking. More specifically, a permanent frequency
downshift is observed for initial larger wave steepness, while the temporary
frequency downshift is found for initial smaller wave steepness. The
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Observations on wave modulation evolution
(a)
h (cm)
20
10
0
–10
frequency (Hz)
(b)
1.5
1.0
0.5
0
125
130
135
140
145
150
155
time (s)
1
2
3
4
5
6
amplitude (cm)
7
8
9
10
Figure 12. Wavelet spectrum at kc x Z 253 for initial uniform wave trains (kc ac Z 0:29). (a) Free
surface displacement and (b) spectrum contour. The black line indicates the frequency of maximum
wave energy and white line is the initial fundamental frequency.
(a)
h (cm)
20
10
0
–10
frequency (Hz)
(b) 1.5
1.0
0.5
0
135
140
145
150
155
160
time (s)
1
2
3
4
5
6
amplitude (cm)
7
8
9
10
Figure 13. Wavelet spectrum at kc x Z 278 for initial uniform wave trains (kc ac Z 0:29). (a) Free
surface displacement and (b) spectrum contour. The black line indicates the frequency of maximum
wave energy and white line is the initial fundamental frequency.
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H.-H. Hwung and others
present experimental results provide much longer spatial evolution of wave
trains, compared to previous researchers’ results, and reveal that the frequency
downshift induced by wave breaking is not permanent but partial recurrence of
the initial state of wave trains due to the wave breaking is an irreversible process.
(iv) Effect of sideband space on evolution of normalized wave amplitudes
For the wave trains with constant wave steepness but different imposed sideband
space, the evolution of normalized amplitudes of one carrier wave and two pairs of
sidebands ðuc ; uc GDu; uc G2DuÞ versus dimensionless fetch is shown in figure 11
for eZ 0:173. Note that the hatched area indicates the breaking region identified
according to the visual inspection during the experiments. Consistently, the
sidebands exhibit a symmetrical growth at the initial stage until the dimensionless
energy of the imposed sidebands approaches Oð10K1 Þ, where the onset of
asymmetric evolution of the sidebands is observed.
As the wave trains evolve further, the modulation is strong enough to initiate
wave breaking in front of the wave trains. Several different types of evolution are
shown at the post-breaking stages. It is observed that the multiple downshift occurs
for smaller imposed sideband space as shown in figure 11a, suggesting that the most
unstable mode eventually manifests itself instead of the imposed sideband
components during the evolution of the wave trains. When the imposed sideband
components coincide with the most unstable mode, wave breaking occurs earlier, as
shown in figure 11b, due to the larger initial growth rate of the imposed sidebands.
Furthermore, the breaking event is seen to cover a larger area and last for a longer
time. While the lower sideband continues to grow during this period, the upper
sideband almost remains the same and the carrier wave is further dissipated
through the breaking process. The permanent downshift of peak of wave spectrum is
observed at the post-breaking stage in this case. In figure 11c, the growth rate of the
imposed sideband components is smaller than that shown in figure 11b, as the
imposed sideband space increases. The amplitudes of the carrier wave and lower
sideband almost coincide at post-breaking stage in this case. For the wave trains
^ 1:0, no frequency downshift is observed after
with imposed sideband space, dZ
wave breaking due to the decrease of the growth rate of imposed sidebands as shown
in figure 11d. Interestingly, these wide sideband spaces lead to less modulation and
downshift shown in figure 11c,d confirms earlier findings from irregular wave
spectra (Stansberg 1995).
(b ) Wavelet analysis of wave breaking
In this section, the wavelet analysis (Torrence & Compo 1998) is employed to
investigate the energy transport mechanism of modulated wave trains. Figures
12 and 13 show the free surface variation and related wavelet spectra before and
after wave breaking, respectively. Specifically, on the leading wavefront, both the
frequency and the amplitude modulate to high frequency before wave breaking as
shown in figure 12. This phenomenon is very similar to the findings from wave
group experiments presented by Stansberg (1998), where the Hilbert envelope
technique was used. As wave trains propagate further downstream, the wave
highly deforms due to nonlinearity and eventually a breaking wave occurs in
front. Meanwhile, the peak frequency shifts to lower frequency after wave
breaking as depicted in figure 13. The above-mentioned phenomena can also be
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Observations on wave modulation evolution
105
verified by the growth rate of evolving sidebands as shown in figure 10. For the
cases of seeded wave trains, similar results are obtained with faster growth of
wave amplitude under the same initial wave steepness.
(c ) Extreme wave
Wave trains propagating with the focus of energy on the strongly modulated
wavefront due to sideband instability are summarized in earlier sections,
whereby the transient large wave in front of the wave trains is observed during
their evolution. The maximum wave crest in the transient wavefront is
investigated up to the onset of incipient wave breaking and the quantitative
features of transient large waves are presented in this section.
(i) Large transient wave on the wavefront
Based on the background analysis described in §2, the wave period of typical
background noise existing in the experiment is less than 0.25 s. Herein, the
measured data are low-pass filtered at a cut-off frequency of 4 Hz. Then, the local
parameters of large transient wave on the unsteady wavefront are defined by the
method proposed by Grue et al. (2003).
For initial uniform wave trains with kc ac Z 0:157 and kc hZ 5:5, the spatial
evolution of local maximum wave parameters is shown in figure 14. In this case,
the wave modulation increases with fetch until it is strong enough to initiate
wave breaking at kc x Z 146, according to the visual experiment. The wave trains
then demodulate before they increase modulation again. At the fetch, kc x Z 266,
second breaking region is identified in the experiment. Although we did not show
it here, it is found that the evolution of wave trains in shallow water with about
the same initial wave steepness is slower than that in deep water.
The measured maximum transient waves for all the experimental runs are
exhibited in figures 15 and 16 for local maximum wave steepness based on the wave
amplitude and crest elevation, respectively. Note that the experimental data
include initial uniform and imposed sidebands wave trains, in which the data cover
a wide range of initial wave steepness. Clearly, the maximum local wave steepness
increases rapidly with the increase of the initial wave steepness, and levels off at an
initial wave steepness roughly equal to 0.16, despite the fact that the data exhibits
little scattering. Practically, it is difficult to exactly identify the initiation of wave
breaking in a physical experiment, especially, in a large wave flume. On the other
hand, the wave gauge stations in the wave flume are not close enough to identify the
breaking point accurately. Theoretically, even for the same initial wave steepness,
each breaking wave could have different maximum wave steepness due to the
different sideband frequencies, which either naturally evolves from initial uniform
wave trains or initially imposed in the wave trains. The above-mentioned factors
introduce scattering in the local maximum wave steepness. However, the tendency
of maximum wave steepness distribution versus initial wave steepness is consistent,
which provides useful information for the determination of the onset of wave
breaking in deep water. It is interesting to point out that the wave steepness at wave
breaking is lower than the corresponding criterion of Stokes wave, suggesting that
the breaking criteria of modulated wave trains in deep water may be quite different
from that of the uniform wave trains.
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normalized elevation
(a)
H.-H. Hwung and others
4
2
hM /a0
0
2
hm/a0
–4
(b) 1.8
Hm / H0
1.6
1.4
1.2
1.0
0.8
local wave steepness
(c) 0.6
0.5 criterion of Stokes wave
khM
0.4
0.3
kam
0.2
0.1
0
(d) 1.0
um /c0
0.8
0.6
0.4
0.2
0
50
100
150
200
250
dimensionless fetch, kc x
300
350
400
Figure 14. Evolution of local maximum wave parameters for initial uniform wave train, kc ac Z
0:157 and kc hZ 5:5. (a) Normalized crest and trough elevations, (b) normalized wave height, (c)
local maximum wave steepness, and (d) ratio of horizontal particle velocity at maximum wave
crest to linear phase velocity.
(ii) Large transient wave at quasi-steady state
Su & Green (1985) found that the ratio of the maximum wave amplitude, a m ,
to the initial wave amplitude, ac, is not a monotonic increasing function of initial
wave steepness, but has a maximum value around kc ac Z 0:14. Their experiments
were conducted with initial uniform wave trains. Tulin & Waseda (1999)
performed a series of experiments on the evolution of initial imposed sidebands
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Observations on wave modulation evolution
0.5
0.4
kHm / 2
0.3
0.2
0.1
kam = 0.451–0.0024 (tanh kcac)–2
kh>p
0.0
0.1
0.2
0.3
0.4
kcac
Figure 15. The calculated local maximum wave steepness (kHm =2) versus initial wave steepness for
initial uniform wave trains. Symbols indicate experimental data and solid line indicates the fitted curve.
0.7
0.6
khM
0.5
0.4
0.3
0.2
0.1
0.0
khM = 0.588– 0.003 (tanh kcac)–2
kh>p
0.1
0.2
0.3
0.4
kcac
Figure 16. The calculated local maximum wave steepness (khM ) versus initial wave steepness for initial
uniform wave trains. Symbols indicate experimental data and solid line indicates the fitted curve.
wave trains and analysed both the local wave steepness and wave height at wave
breaking. Generally, Tulin & Waseda’s experimental data on the amplification
factor of wave height are slightly larger than Su & Green’s results, especially for
the lower wave steepness.
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H.-H. Hwung and others
1.0
um /c
0.8
0.6
0.4
0.2
0
0.05
0.10
0.15
kcac
0.20
0.25
0.30
Figure 17. The ratio of horizontal particle velocity at maximum crest elevation to the phase velocity
versus initial wave steepness. Diamond, Tc Z 1:3 s; aG=ac Z 0:3; plus, Tc Z 1:6 s; aG=ac Z 0:3; circle,
Tc Z 1:3 s; aG=ac Z 0:1, and triangle, Tc Z 1:6 s; aG=ac Z 0:1.
Normalized horizontal particle velocity at the crest of wave breaking u m =c0 is
plotted versus initial wave steepness in figure 17. Note that this particle velocity
is calculated based on equation (2.9). In general, the ratio of horizontal particle
velocity to phase velocity for a wave breaking in strongly modulated wave trains
is approximately equal to 0.5, which is much less than the kinematic criterion of
wave breaking (u m =c0 R 1) based on Stokes wave theory.
(d ) Initial growth rate and fastest growth mode
The amplitude spectra have been used as a measure of the growth of amplitude
modulation on nonlinear wave trains. The initial growth rates of the naturally
evolved sidebands from initial uniform wave trains are analysed by the method
described in §2. A typical comparison of the initial growth rate of the sideband
amplitudes between theoretical predictions and experimental data for initial
wave trains, kc ac Z 0:220, is shown in figure 18. This result indicates that the
sideband components, which meet the instability criteria, develop during the
wave propagation. The initial growth rate of sideband components increases with
normalized frequency difference between the carrier wave and sidebands until a
maximum value is reached. The initial growth rate then decreases; the initial
growth rates of the sideband amplitudes in the experiment consist of the
prediction of Crawford et al. (1981) before the peak growth rates are attained.
However, the theory over predicts the growth rate for an even larger normalized
frequency difference.
From the investigation on the evolution of initial uniform wave trains, it is
evident that a pair of specific sideband frequencies grows faster due to the
nonlinear wave interaction through sideband instability. The naturally evolved
fastest growth modes of the naturally evolved sidebands from our experiments
together with previous experimental results are compared with those by several
theoretical predictions as shown in figure 19. We note that the existing
experimental results are confined in initial larger wave steepness due to the
limitation of their experimental facility. Extended results on smaller wave
steepness are provided by the present experiments. From figure 19, it is clear
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Observations on wave modulation evolution
1.0
0.01
0.8
0.10
0.6
bx /
2
0.20
0.30
0.4
0.35
0.39
0.2
0.40
0.0
0.2
0.42
0.4
0.44 0.46 0.48
0.6
0.8
dw / wc
1.0
1.2
1.4
Figure 18. The comparison of initial growth rate between the experimental results for initial
uniform wave trains eZ 0:220 and Crawford et al.’s (1981) prediction based on Krasitskii’s (1994)
theory. The error bars denote the s.d.
0.4
dw /wc
0.3
0.2
0.1
0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Figure 19. The comparisons of normalized frequency difference between the carrier wave and the
fastest growth sideband versus initial wave steepness between present experiments, previous
experiments and theoretical predictions. Triangle, Melville (1982); circle, Waseda & Tulin (1999);
square, present experiments; solid line, Tulin & Waseda (1999) calculated based on Krasitskii’s
(1994) theory; dashed line, Dysthe (1979); dotted line, Longuet-Higgins (1980); dashed-dotted line,
Benjamin & Feir (1967).
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H.-H. Hwung and others
that Benjamin and Feir’s prediction consistently overestimated the frequency
difference between the carrier wave and the fastest growth mode for almost an
entire range of wave steepness due to the assumption of infinitesimal wave
steepness made in their theory. Besides, Dysthe’s result, which is derived based
on a narrow band assumption, is valid only for a wave steepness smaller than
0.15 and significantly underestimates the growth rate for larger wave steepness.
Overall, the experimental results agree reasonably well with the prediction of
Krasistkii’s theory derived from the Zakharov equation and that of LonguetHiggins solving the fully nonlinear equations.
4. Conclusions
Without a stable, sophisticated and programmable wave generator or a long
enough flume, it is difficult to investigate the long-time evolution of nonlinear
wave trains. Owing to these restrictions, only limited experimental studies are
available since Benjamin & Feir (1967) and most of them did not show the
complete story. To further picture the long-time evolution of nonlinear wave
trains, a series of systematic experiments were conducted in a long-wave flume
(300!5!5.2 m) at Tainan Hydraulics Laboratory. Several types of wave trains
were initially generated including uniform, imposed small sidebands and detailed
analyses of their evolution are given in the present study. The following main
findings can be drawn from this study.
(i) The evolution of a transient wavefront exhibits complicated and highly
nonlinear wave trains, in particular, a strongly modulated wave group is
observed as propagation of the wavefront, in which larger waves appear in
front. Later, a series of intermittent wave breaking occurs at the large
crest elevation. The wave group demodulates at post-breaking stage.
Meanwhile, the second modulated wave group forms at the following wave
trains. The mentioned modulation and demodulation processes appear
periodically as the wave trains propagate downstream along the flume.
Our results also indicate that the sideband energies are initiated by the
initial transient wave generation and enhanced with time and fetch until the
critical value initiates wave breaking or reaches a maximum modulation.
(ii) The present experimental results provide much longer spatial evolution of
wave trains compared to earlier researchers’ results and reveal that the
frequency downshift induced by wave breaking is not permanent, but
partial recurrence of initial state of wave trains due to the wave breaking
is an irreversible process.
(iii) The wavelet spectrum of a modulated wave train shows that both the
frequency and amplitude modulate to high frequency on leading
wavefront before wave breaking. However, the peak frequency shifts to
lower frequency after wave breaking.
(iv) According to the experimental results, the breaking wave steepness in a
modulated deep water wave is very different from that derived by Stokes,
ðkaÞmax Z 0:44. Besides, the ratio of horizontal particle velocity to phase
velocity at wave breaking in strongly modulated wave trains is
approximately 0.5, which is much less than the kinematic criterion of
wave breaking (u m =c0 R 1) based on Stokes wave theory.
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Observations on wave modulation evolution
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(v) From the investigation on the evolution of initial uniform wave trains, it
is evident that a pair of specific sideband frequencies grows faster due to
nonlinear wave interactions through sideband instability. In addition, our
results show that the fastest growth modes of the naturally evolved
sideband components and related initial growth rates of initial uniform
wave trains match reasonably well with the predictions of Tulin &
Waseda’s calculation based on Krasiskii’s theory.
The authors gratefully acknowledge the financial support provided by the Ministry of Education,
Taiwan for the Programme for Promoting University Academic Excellence under grant no.
A-91-E-FA09-7-3. They also thank the research staff of Tainan Hydraulics Laboratory for their
assistance to conduct elaborate experiments.
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