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