Wave Modulation and Breaking

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Wave Modulation and Breaking

by

H. H. Hwung and W. S. Chiang Tainan Hydraulics Laboratory,

Department of Hydraulic and Ocean Engineering National Cheng-Kung University

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Introduction

• _Motivation

– Academic excellence project

– Wave modulation due to sideband instability – Long wave flume

• Purpose

– Long time evolution of nonlinear wave trains in deep water

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Literature review

– Sideband instability (Benjamin and Fier, 1967)

– Lake et al. (1977) : wave flume ~ 12m

– Dysthe (1979)

– Melville (1982) : wave flume ~ 28m

– Lo and Mei (1985) – Trulsen (1990)

– Tulin and Waseda(1999) : wave flume ~ 50m

– Hwung and Chiang (2005) : wave flume ~ 300m







₂ 1/ 2



exp

2

i

a



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

_

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k h

c



1.363

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Summary of previous studies

• _{Some interesting phenomena such as} _{disintegration of}

wave train, recurrence of initial status and frequency

downshift of wave spectra have been reported in

previous studies for different kind of wave group.

• However, the observed evolution of nonlinear wave train s in fetch is still limited due to the limitation on length of wave flume.

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IPC1 MNDAS 238 m 5. 0 m wave generator wave gages No. 4~17 spacing 6 m wave gages No. 17~63 spacing 3 m 0 : configuration file 1 : instruments database 2 : measured signals 3 : processing results 4 : AD/DA instruction file 5 : wave board control signals

0 3 MEASURE 6 1 6 GENERATE 4 5 PROCESS 2 6 AD/DA CED1401 5. 2 m 3. 5 m 1:7 1:10 15m 47 m 0.4 m

side view of the flume

plane view of the flume

IPC2

frame grabber

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• _{Initial uniform wave trains}

• Initial imposed sidebands wave trains

• Initial bichromatic wave trains

0

a

c

sin(

c

t

)







E

xperimental wave conditions

sin(

)

sin(

)

sin(

)

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a

c c

t

a

t

a

t

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sin(

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)

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sin(

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)

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Experimental procedures

• _{Calibrate the wave gages}

• Check the accuracy and stability of the

wave maker

• _{Conduct the experiments under still wind}

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0 0 . 5 1 1 . 5 2 2 . 5 am pl itu de (c m ) 3 0 0 3 1 0 3 2 0 3 3 0 3 4 0 3 5 0 t i m e ( s e c ) - 4 - 2 0 2 4 di sp la ce m en t ( cm )

a

_c

=2.25 cm, T

_c

=1.6sec

a

₊

/a

_c

=0.35, a

_-

/a

_c

=0.35

E

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Data analysis

• Analysis of local maximum wave

– Measured crest elevation and wave period defined by zeroup crossing method

– Stokes third order wave theory

• Analysis of wave spectrum

– Hanning window

– frequency range of 0 to 12.5 Hz – a resolution bandwidth of 0.006 Hz

• Analysis of initial growth rate

– Normalized sideband amplitudes – Exponential curve fitting

• Hilbert-Huang Transformation (HHT)

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Noise characteristics

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Noise characteristics

• _{First mode: a=0.18 cm, T=0.12sec, ka=0.254.} • Second mode: a=0.27cm, T=0.23sec, ka=0.100.

• Third mode : a=10.3cm , T=1.6sec, ka=0.162.

• The results demonstrate that the electronic noise of the wave gauges and the ripple due to light wind blowing over the water surface in the flume are much smaller than that of the wave generated by the wave maker both in length and time scales.

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Transformation of transient wave front

Uniform wave train

Modulation of transient

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Effect of wave reflection

time = 10 sec _{time = 50 sec}

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Wave trains propagation without modulation

0.102 



w

av

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Quasi-steady modulation of wave trains

0.165 



w

av

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Quasi-steady modulation of wave trains

0.165 



w

av

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Fastest growth mode and initial growth rate

0 . 0 0 . 1 0 . 2 0 . 3 0 . 4  0 . 0 0 . 1 0 . 2 0 . 3 0 . 4  / c

┼: Melville(1982); ○: Waseda&Tulin(1999); : present experiments; thick line: Tulin a◇

nd Waseda (1999) calculated based on Krasitskii’s (1994) theory; dash line: Dysthe(197 9); dot line: Longuet-Higgins(1980); thin line: Benjamin&Feir(1967).

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Spatial evolution of normalized wave amplitudes

circle: carrier wave; diamond: lower sideband; cross: upper sideband; solid line: growth curve of sidebands predicted by Benjamin and Feir(1967); dash line: growth curve of sidebands

0.149 



Benjamin and Feir (1967)

Tulin and Waseda (1999)

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Quasi-steady modulation evolves from initial

imposed sidebands wave trains

Two series of experiments were conducted with constant wave steepness but varied frequency difference between the carrier wave and imposed sidebands.

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)

sin(

)

sin(

)

1

4

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a

t

a

t

a

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c

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Quasi-steady modulation evolves from initial

imposed sidebands wave trains

wave conditions of initial imposed sidebands wave trains

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Evolution with breaking and with downshifting

The amplitude contours of initial modulated wave train versus dimensionless frequency difference and fetch.

0.173 





ˆ



0.7

(corresponds to the most unstable mode)

w

av

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0.173 



_

ˆ

_

_0.5

Evolution with breaking and with multiple downshifting

(smaller than that of the most unstable mode)

w

av

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Effect of wave steepness on the evolution of

nonlinear wave trains

Experiments were conducted with varied wave steepness and the imposed sidebands roughly correspond to the most unstable mode

.

sin(

)

sin(

)

sin(

)

1

4

o c c c

a

t

a

t

a

t

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









 

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       

















k a

_{c c}

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Evolution of wave trains without breaking

0.109   0.109





w

av

e

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Evolution of wave trains without breaking

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 k _cx T 1 7 2 F 1 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 a / a ₀

0.109 



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Evolution of wave trains with breaking

0.130

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(a) _(b) (c) (d)

0.11 







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





0.17

0.23  

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Evolution of bichromatic wave trains

Fourth-order nonlinear Schrödinger Equation

• Spatial domain model of extended nonlinear Schrödinger e

quation : Trulsen and Stansberg (2000, 2001)

• Numerical scheme: Lo and Mei (1985)

• Breaking dissipation: Trulsen (1990) ; Kato and Oikawa (19

95)

Multi-Layer Boussinesq Equation

• The multi-layer model is derived through piecewise

integration of the primitive equations of motion. The complete derivation, variable definitions and final equations

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Spatial evolution of normalized wave amplitudes

circle: carrier wave; diamond: lower sideband; cross: upper sideband; solid line: growth curve of sidebands predicted by Benjamin and Feir(1967); dash line: growth curve of sidebands

predicted by Tulin and Waseda(1999).

0.149

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Results of NLS Results of NLS Non-breaking case Non-breaking case a1=5 cm, a2=5 cm, T1=1.5 s , T2=1.7 s

Evolution of wave groups for non-breaking type

w

av

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Results of Boussinesq Non-breaking case Non-breaking case a1=5 cm, a2=5 cm, T1=1.5 s , T2=1.7 s

w

av

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w

av

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NLS

Boussinesq

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symbol : experimental data

line: results of fourth-order Schrödinger eq. with Trulsen & Dysthe’s (1990) breaking model

a

_₂

/ a

_o

a

₂_₁_-_₂

/ a

_o

a

₂__2-_₁

/ a

_o

a

_₁

/ a

_o

Evolution of wave groups for breaking type

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Characteristics of large transient wave

• Wave trains propagate with energy focusing on the front of strong modulated wave train due to sideband instability was shown in previous chapters. Whereby, the transient large wave on the front of wave trains was observed. Maximum wave crest in the transient wave front was investigated up to the onset of incipient wave breaking.

large transient wave

w

av

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Transient large wave on the wave front

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Transient large wave at quasi-steady state

0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 i n i t i a l w a v e s t e e p n e s s , a _ok _o 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 lo ca l w av e st ee pn es s c r i t e r i o n o f S t o k e s w a v e

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Transient large wave at quasi-steady state

0 0 . 1 0 . 2 0 . 3 k _ca _c 0 0 . 2 0 . 4 0 . 6 0 . 8 1 u m / c

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Transient large wave at quasi-steady state

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Time-frequency features of large transient wave

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Time-frequency features of large transient wave

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Time-frequency features of large

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Conclusions

Initial imposed sidebands wave trains

Effect of sideband space on the evolution of wave trains • The most unstable mode of initial wave train will manifest

itself with propagation. Especially, the most unstable mode of initial wave train develops through a multiple-downshift of wave spectrum for the wave train with smaller sideband space, which is observed at first time in literature. It reveals that the wave spectrum tends to be lower frequency as the wave train propagates downstream due to the sidebands instability

.

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Conclusions

Initial imposed sidebands wave trains

Effect of wave steepness on the evolution of wave trains • The evolution of wave train is a periodic modulation and

demodulation at post breaking stages. Meanwhile, the wave spectrum shows temporal and permanent frequency downshift respectively for different initial wave steepness, which suggests that the permanent frequency downshift induced by wave breaking observed

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Conclusions

Initial bichromatic wave trains

• Near recurrence of initial status of bichromatic wave train

is demonstrated for non-breaking case. However, for

breaking case, the amplitude of lower sideband is

selectively amplified through the breaking process. The

results confirm the findings on the evolution of initial uniform and imposed sidebands wave trains.

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Conclusions

Large transient wave

• The local wave steepness and the ratio of horizontal particle velocity to linear phase velocity at wave breaking in modulated wave group indicate that the breaking criterion of Stokes wave, which is derived based on the uniform wave, is not consistent with the results of modulated wave trains.

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Conclusions

Large transient wave

explanation of frequency downshift

• In deep water, wave breaking is observed near the large wave on the front of modulated wave group, where the energy peaks of the upper sideband and carrier wave appear. So, the upper sideband and carrier wave are most affected by the wave breaking. Whereby, the lower sideband is selectively amplified through the breaking process.

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Future works

• _{The present experimental results largely extended our}

knowledge on the long time evolution of nonlinear wave trains. The valuable data provides an useful information for the calibration of numerical models which should be improved in the near future.

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Future works

• _{Large transient wave was observed during the propagati}

on of nonlinear wave trains through Benjamin-Feir instabi lity which may provide a mechanism of freak wave gener ation in deep water and further study is needed.

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Future works

• _{Based on linear theory, the wave height distribution sh}

ould be Rayleigh distribution. However, the observed w ave height distribution in the ocean is much close to th e Weibull distribution. The relation between sideband i nstability and transformation of wave height distribution is an important topic for ocean engineering.

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Future works

• _{The evolution of modulated wave trains from deep}

water to shallow water plays an important role for coastal engineering. In which, the effects of sideband instability and shoaling of wave trains coexist and contribute to the transformation of nonlinear wave trains. More attention should be paid to the interesting topic.

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Wave Modulation and Breaking