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Chapter 6
Wave Instability and Resonant Wave Interactions in A
Narrow-Banded Wave Train
Narrow –Banded Wave Train• Definition: The frequencies of all free waves
in a wave train are close to its spectral peak frequency and almost travel in the same direction.
Sketch of a narrow-banded spectrum
Order Analysis of Narrow-Band Waves
(1)
1
(1)
1
1( , , ) exp( ) . ,
2
( , , ) exp . . 2
= ,
( ) , , 1
For a narrow-band wave train,
( 1) ( ),
N
n nn
Nn
n nn n
n nn n n nx ny
n m
n
p p
p
p
x n
p
y
p
x y t a i c c
a gix y t k z i c c
x k t k ik jk
n m m
k
k
N
k kNO
k
1, for 1, 2,..,n N
The superposed elevation & potentialin terms of a carrier wave train with slowly varying amplitude
(1)
1
(1)
( 1)
1
1( , , ) ( , , ) . ,
2
( , , ) exp ( ) ,
( ) ( ) ( ) . (slow phase)
( , , ) (
e
, , , ) exp . 2
( , , )
xpT P p p p
N
T n n pn
n p nx p ny n p n p
T
jT j
j
P
p p
x y t a x y t c c k x t
a x y t a i
x k k yk t
ix y t A x y z t
i
k c cz i
A A x y t z
1
1
( ), and ( , , )
( 1)!
exp [( ( ) ( ) ] .
jNn pn
jn n
nx p ny n p n p
k ka gA x y t
j
i x k k yk t
2nd –order solution expressed in terms of a carrier wave train with slowly varying amplitude
12
1 1 1
2
(2)
1
1 1
nd
1 exp ( ) . .
4
The first two terms are of '2 harmonic and the last is 'zero harmoni
1 1exp 2 exp ( )
c'
4 4
1( , , ) ex
4
N N N
j j j
N N
j l l j l jl j j
j l l j l jj l j j
p
a a k k i c
a k i a a k k i
a x y t
c
k
1
( )(2)
1
2
1
nd
2
exp ( ) . . 2
The 2 order potential is of zero ha
1p 2 . ( , , ) cos(2 ) ( )
2
rmonic
l j
N Nk k z
j l l l
p T p p T
jl j j
ia
i c c a x y t k
k Ae i
a
c c
O
The derivatives of ‘zero’ harmonic potential & elevation w.r.t. time,space coordinates are at most of third order.
Constraints on AT
• Laplace Equation
• Linear free-surface dynamic B.C
22 2 0 T Tp p T
A Aik k A
x z
11( , , ) ( , , ). T
p p
Ai gA x y t a x y t
t
• Based on the definition
1
1
( , , , 0) ( , , ).
( ) , ( ) , and
( ) , ( ) , for 1, 2,.., 1.
T
T T T Tp T p T
T Tp T j j
A x y t z A x y t
A A A AO k A O A
x y z t
a aO k a A O A j N
x y
Derivation of Nonlinear Schrodinger
Equation for A Narrow-Banded Wave Train • Many different methods (see notes)• MCM is used here.
21
12 2 (1)2 3 2 1
2 2
1 22 (1) 2 (2)3 2 1
2 2
211 2 2
12 2 (1)21 1
2 2
, at 0, ,
12
2
g P P z P gz z z tt t
P g gz z z zt t
gz t t zz t
2(1) (1)1.
2
Limited to the first harmonic terms (L.H.S)
22
2 20 0
2
2
2 2
2 2
exp . .2 2
From the Laplace eqn.
1fourth-order terms
2
L.H.S exp .2 2 4
T T Tp p
z z
T T T
p
T T T Tp p
p
A A Ai igg i c cz t zt t
A A Ai
z x k y
A A A Ai g igi c
t x kt y
c
The 1st & 3rd terms are of 2nd order & 2nd & 4th Terms are of 3rd order
Limited to the first harmonic terms (R.H.S)
1) is of 3rd-order but only contribute to zero-harmonic. It has no contribution to the first harmonic up to 3rd.
2) The 1st & 3rd terms in are of 4th-order at most.
3) The derivatives of 2nd-order potential with respect to the space coordinates & time are of 3rd-order the 2nd & 4th terms in are of 4th-order at most.
4) The 5th term in is calculated below.
(2)P
(3)P
(3)P
2 21 2 rd
2 2(1) (1) 4
0
1exp(2 ) 3 -order terms.
21
exp( ) . . 2 2
p T p
p T T pz
k A k z
ik A A i c c
(3)P
Nonlinear Schrodinger Equation
2 224
2 2
rd
2 2 22th th
2 2 2
2 21 1 1
2 2 2
, at 0.2 2 4 2
3 -order terms.2
4 order terms 4 order terms2 4
8 4
T T T Tp p T T
p
T T
p
T T T
p p
p pg
p p
A A A Ag i ig ik A A z
t x kt y
A Ag
t x
A A Ag g
t xt x
i iA A A AC
t x k x k
4
22 22
2 2 2 2
211 12
, 2
28 4
p p p pT T T T
g T Tp p
p
p
i i i ka a a aC a a
t
i
x k x k
kA
y
y
A
Steady Solution Of NSE
22 22
2 2 2 2
In the moving coordinates, and - .
. 8 4 2
g
p p p pT T TT T
p p
t x C t
i i i ka a aa a
k k y
•A Periodic Wave Train
2 2
22 2 2 2 4
exp( ), constant
1,
2
1 1 0( ) ,
T p p p
p p
p p p p
a a i a
a k
gk k a
Nonlinear dispersion relation
• Solution for Envelope Soliton (uni-directional wave)
222
2 2
2 2 2
2 2 2
(1) 2 2 2
. (independent of ) 28
sech 2 exp( )4
sech 2 ( ) exp( ). 4
1( , , ) sech 2 ( ) cos
4
p p pT TT T
p
T p p p p p p
p p p g p p p
p p p g p p p p
i i ka aa a y
k
ia a a k a k
ia a k x C t a k t
x y t a a k x C t a k t
Snapshot of the elevation of an envelope soliton in deep water at t = 0. The carrier wave’s period and amplitude are 2s and 0.1m , respectively
Elevation of the same envelope soliton at x = 410 m as a function of time
• Solution for Conoidal Envelope
•Envelope soliton is a special case of Conoidal Envelope
•See hand-written notes
0 5000-2
-1
0
1
2
x
A snapshot of a wave train with a Cnoidal envelope in deep water (Emax=1.0 m2, Emin=0.1 m2, Tp=10sec).
Side-Band Instability• Initial InstabilitySuperposing infinitesimal disturbances on a steady periodic
wave train.
2 21exp( ) 1 ,
2( ),
where , and are the amplitude, freq. and wavenumber
of a periodic wave train and constant. and are the
normalized amp
i iT p p p p
x y
p p p
a a ik a a e a e
i K K y
a k
a a
litudes of imposed disturbances. Their
wavenumbers are , , and freq. .
/ , / , / , and ( ).
p x y p
x p y p p
k K K
K k K k a a O
(1) ex1
( , , ) ( , , ) . , p2 P pPT p pix y t a x y t c c k x t
2 2 2 22 2
2 22 2
112 0 ,
1 28 42
p p p p x p yp p p
p pp p p
P k a a K KP k a
a k kk a P
To have a non-trivia solution for the system, the determinant of the matrix must be zero, leading to the solution for
2 22 2 22 2 2 2 2
2 2 2 2
222 2
2 2
2 2max
1,
2 8 4 8 4
and 2 , is imaginary.8 4
1For =0, 2 Maximum growth rate (Im ) .
2
y yx xp p p p p p
p p p p
yxp p x y
p p
xy p p p p p
p
K KK KP k a k a
k k k k
KKk a K K
k k
KK k a a k
k
Side-band (Benjamin-Fier) Instability Growth Rate (Yuen & Lake 1982)
Ky = 0 K = Kxao = ap , ko = kp
Long-Term evolution
•ZEM
•MCM (see notes for the details)
•Numerical simulation
Measured wave spectrum of a wave train experienced the side-band instability.
a) at x = 5 ftb) at x = 10 ftc) at x = 15 ftd) at x = 20 fte) at x = 25 ftf) at x = 30 ft
Time aeries at different locations
Coupling Eq.s derived using MCM
• Identify the forcing terms which may force or nonlinear interact related ‘free’ waves. (Resonance conditions)
• Quartet Resonance Interaction
- Coupled Equations - Phase governing equations - Long-term evolutions (energy conservation)
3 3 3(3)
1 1& 1
3 2 1 3 2 1
3 1 2 3 1 2
1 2 3 0 1 2 3 0
1 2 3 1 2 3
sin sin(2 )
sin( )
sin( )
, ,
sin
( )
i i i i i j i ji i i j j
P A P P
PP
P
k k k k
MCM
Then we consider four- (quartet)-wave interaction among free waves ‘0’, ‘1’,’2’ and ‘3’. The P(3) for 4 free waves can be extended from the general solution of above P(3). How?
20 0
0 0 0 1 2 3 2 1 32
32 2 2 0
0 0 0 0 01
0
0 0 0
200
0
0
0
sin sin( + ),
( 2 )
Conbsidering that is modulating
exp( ) . 2
10 (from the Laplace Eq.)
2
, ,
( )
i ii i
g A P Pt z
P gk a k k a k
A
ik z i cA x z c
A Ai
t
Ax z k
th0 0 +4 -order error A A
ix z
The related forcing terms are now applied to free wave ‘0’.
The free-surface B.C. for free wave ‘0’ are split to
0 0 00
0 0
22 0 1 2 30 0 02
0 0
2 1 3 0
1 1 1
1 1
1
3 2
1
3
0
0
2
sin2
1,
2 2
1cos ,
Similarly, we have equations for free waves 1,2 and
sin2
3.
1
2 2
A AgA
t x t
A Pgk P
A t A
A A
P
g P
t x t
1
22 3 2 011 1 12
1 1
,
1cos ,
A
PAgk P
A t A
2
2 2 22
2 2
22 3 1 022 2 22
2
3
2
3 3 33
3 3
22 3 2 1 03 3 32
3
2
1 0
1 0
3
3
1,
2 2
1cos ,
si1
n22 2
1co
n
s
si2
A AgA
t x t
PAgk P
A t A
A AgA
t x
P
t
A Pgk P
P
A t A
1 2 0 3 1 2 0 3
1 2 3 0
and ,
,
k k k k
t
0 31 21 2 0 3
1 2 0 3
2 22 20 31 2
2 2 2 21 2 0 31 2 0 3
3 2 0 3 1 0 1 2 3 2 1 0
1 2 0 31 2 0 3
1
2
1 1 1 1 1
2
1cos , .
2i i
P PP P
t
A AA A
t t t tA A A A
P P P Pgk
A A A A
3 30 1 2 3 0 3 2 01 1
2 23 30 0 1 10 1
333 1 0 3 2 1 0 32 2
2 23 32 2 3 32 3
1 2 0 3
(1 )sin , (1 )sin , 2 2 2 24 4
(1 )sin , (1 )sin ,2 2 2 24 4
A P A PA A
t tt t
P A P AA A
t tt t
Five Coupling Eq.
Special case: Side-Band Instability
2 3 1 2 3 1
2 2 2 2 2 22 1 32 2 2 2 1 1 1 3 3
22 231 1
21 3 1 2 2 2 123 2 1
2 21 23 2 2 3 2 1 3 1 2 2
2 , 2 .
1( / 2 ) ( )
2
1 + 2 ( ) cos
2
1sin , sin ,
2
k k k
k k a k k a k a kt
aa ka a k k a k
a k a
A AA a k A a a k k
t t
2
2 23 11 2 1
3
1 3 2 1 2 3
1sin .
2
Considering the case , , . p p
AAa k
t
a a a a k k k k
Resonant Wave Interactions in Ocean Waves
• Quartet wave interaction
•WAM----- Wave Energy Budget
4 1 2 3 4 1 2 3 44
1 2 31 2 3 4 1 2 3 4 3 4 1 2
62
1 2 3 4
* ,
( ) , , ,
, , , 4 / ,
where is the mean wavenumber ve
g in dis nl
nl
EC U E S S S
t
S k G k k k k k k k k
n n n n n n n n dk dk dk
G k k k k k
k
����������������������������
ctor
6 22 3
1 2 3 4 1 22 2 22 3 2 3
21 3
2 21 3 1 3
4, , , 1 3
,
where is defined in the same way as the previous equation, and , ,ii i
kG k k k k
k kk
k