Chapter 3.1 Weakly Nonlinear Wave Theory for Periodic Waves (Stokes Expansion)

IntroductionThe solution for Stokes waves is valid in deep or intermediate water depth.

It is assumed that the wave steepness is much smaller than one.

(1)

where k is the wavenumber and h is the

water depth which is assumed constant.

kh O1ak

Nondimensional Variables

2

, , ,

, / ,

, , ,

where , , , , and are dimensional

variables and , , , , and are

corresponding nondimensional variables.

X xk Z zk Y yk

t t a

C gkC D

ag ag

x z t h C

X Z t C

•Nondimensional Governing Equation & Boundary Conditions

2 2 2

2 2 2

2

0 (3.1.1)

0 at (3.1.2)

at (3.1.3)

1 at (3.1.4

2

h h

kh ZX Y Z

Z khZ

D D ZZt

D C Zt

)

where and stand for gradient and horizontal

gradient, respectively.h

Perturbation (Stokes Expansion)

1 2 32 1

1 2 32 1

1 2 32 1

Assuming the wave train is weakly nonlinear

( 1), its potential and elevation can

be perturbed in the order of :

jj

jj

jj

ak

C C C C C

Hierachy Equations

Using the Taylor expansion, the free-surface

boundary conditions (Equations (3.1.3) and (3.1.4)

are expanded at the still water level (Z = 0).

Then we substitute perturbation forms of potential

and ele

(j)

vation into the Laplace Equation, bottom

and free-surface boundary conditions. The equations

are sorted and grouped according to the order in

wave steepness . The governing equations for -

order

j th solutions is given by:

2 ( )

11

11

0 0 (3.1.5)

, at 0 (3.1.6)

, at 0 (3.1.7)

0 at

j

jj jj j

j jjj j

j

hk Z

P Zt

D Q ZZt

Z khZ

(3.1.8)

( ) ( )where the and can be derived in terms

of the solutions for the potential and elevation of

order ( -1) or lower. Therefore, the above

hierarchy equations must be solved sequentially

from lowe

j jP Q

j

( ) ( )

r to higher order until the required

accuracy is reached. To derive the third-order

solution for a Stokes wave train, it is adequate

to truncate the equations at 3.

Up to 3, and are givej j

j

j P Q

n below.

(1)1 1

12 21 (2)2 1

121 12 12

2213 1 2 1

1 122 32 1 (3)

2

1 22 32 132

and 0

2

2

1

2

2

h h

P C Q

DP C

Z t

Q D DZ

DP D

Z t

CZ t Z t

DQ D

Z

1

3

1 22 1

21 11

+

h h h h

h h

Z

D

DZ Z

• Solving the non-dimensional Equations from lower order (j=1) to higher order (j=3) for the non-dimensional solutions (wave advances in the x-direction).

1

1 1

23

2 2

2

cosh( )sin( ),

cosh

cos( ), 0

3 cosh(2 2 )sin(2 2 )

8 sinh cosh

(3 1)cos(2 2 )4

1

2sinh 2

kh ZX t

kh

X t C

kh ZX t

kh kh

X t

Ckh

3 2 2 2

3 4 2

6 2 2

3

21 1 2 2 2

1( 1)( 3)(9 13)

64cosh(3 3 )

sin(3 3 )cosh

3( 3 3)cos( )

83

(8 ( 1) )cos(3 3 )64

0

91 1

8

where coth

kh ZX t

kh

X t

X t

C

D

kh

•The non-dimensional solutions are then transferredback to the dimensional form.

(1)

(1)

First-order:

cosh[ ( )]sin

cosh( )

cos

where

and .

k z hA

kh

a

kx t a Ag

(2) 2 2

2 2 2

2 2

Second-order:

3( 1) cosh[2 ( )]sin 2

81

(3 1) cos 24

1Bernoulli Constant: ( 1)

4o

akAk z h

a k

C a kg

(3) 2 2 2

2 2

(3) 4 2 3 2

6 2 2 3 2

22 2 2 2

Third-order:

1( 1)( 3)(9 13)

64cosh[3 ( )]

sin 3cosh 3

3( 3 3) cos

83

(8 ( 1) ) cos364

Nonlinear Dispersion Relation:

9tanh( ) 1 1

8

k z ha k A

kh

a k

a k

gk kh k a

2

Convergence

For the fast convergence of the perturbed coefficient, , must be much smaller than unity, which is consistent with weakly nonlinear assumption. However, when the ratio of depth to wave length is small, the Stokes perturbation may not be valid.

(2)

(1)

Convergence rate:

,

is the ratio of the potential magnitude of

second-order to that of first order solution at

0.

mag

mag

R

R

z

2 2

1 3

3 ( 1).

8For fast convergence, should be << 1. This is

true when ~ (1). When 1, we have :

3~ ( ) , hence ~ ( )

8

may be much greater than unity

R

R

kh O kh

kh R O kh

R

Ursell number

2 31

=( ) ( )

8For 1, then .

3

r

r

a

h kh kh

R U

U

A few striking features of a nonlinear wave train can be described for the above equation:

• The crests are steeper and troughs are flatter; (see applet (Nonlinear Wave Surface)).

• Phase velocity increases with the increase in wave steepness.

• Non-closed trajectories of particles movement. (see applet (N-Trajectory)).

• Nonlinear wave characteristics (up to 2nd order).

2 2

3

2 2

3

cosh[ ( )] 3 cosh[2 ( )]cos cos 2

cosh 4 sinh cosh

sinh[ ( )] 3 sinh[2 ( )]sin sin 2

cosh 4 sinh cosh

Particle velocity

Acceleration x

V iu kw

akg k z h a k g k z hu

kh kh kh

akg k z h a k g k z hw

kh kh kh

aa i

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(1) (1)

2 23

(1) (1)

2 23

cosh[ ( )]sin

cosh3 cosh[2 ( )] 1

sin 22 sinh cosh sinh 2

sinh[ ( )]cos

cosh3 sinh[2 ( )]

2 sinh cosh

z

x

z

a

u k z ha V u akg

t khk z h

a k gkh kh kh

w k z ha V w akg

t khk z h

a k gkh k

k

��������������

��������������

��������������

1cos 2

sinh 2h kh

Wave advancing in the x-direction

Particle Trajectory

Denoting the mean position of a particle by ( , ) , and its

instantaneous displacement from the mean position by ( , ),

the Lagrangian velocities of the particle are hence

( , ) and ( , ),

x z

u x z w x z

(1) (1)(1) (2) (1) (1) 2 (1)

(1) (1)(1) (2) (1) (1) 2 (1)

they are related to the

Eulurian velocities through a Taylor Expansion:

( , ) ( , ) ( , ) ( )

( , ) ( , ) ( , ) ( )

wh

u uu x z u x z u x z O u

x z

w ww x z w x z w x z O w

x z

(1) (2) (1) (2)ere ( , ), ( , ), ( , ) and ( , ) are first- and second-

order horizontal and vertical velocities.

u x z u x z w x z w x z

0 00 0

(1)

( , ) are calculated by integrating the related Lagrangian velocities.

( ) ( , , ) ; ( ) ( , , )

We intend to compute ( , ) up to second order in wave steepness

( ) ( ,

t tt u x z d t w x z d

t x

(2)(2) 2 (1)

(1) (2) 2 (1)

, ) ( , . ) ( , ) ( )

( ) ( , , ) ( , , ) ( )

where superscripts stand for orders and overbar denotes a secular term.

At leading-order, the solution is the same as that in L

z t x z t x z O

t x z t x z t O

(1) (1) (1)00

(1) (1)00

WT,

cosh[ ( )]( , , ) sin

sinhsinh[ ( )]

( , , ) cossinh

t

t

k z hu x z d a

khk z h

w x z d akh

(1) (1) 2 2 2 2(1) (1) 2 2

2 2

(1) (1)(1) (1)

cosh [ ( )] sinh [ ( )]sin cos

sinh cosh sinh cosh

cosh 2 ( )1 cos 2

sinh 2 sinh 2

0

u u a k g k z h k z h

x z kh kh kh kh

k z ha k g

kh kh

w w

x z

u

(1) (1) 2 2(2) (1) (1)

3

2 2

(1) (1)(2) (1) (1)

3 cosh[2 ( )] 1( , ) cos 2

4 sinh cosh sinh 2

cosh 2 ( ) + ,

sinh 2

( , )

u u a k g k z hx z

x z kh kh kh

k z ha k g

kh

w ww x z

x z

2 2

3

3 sinh[2 ( )]sin 2

4 sinh cosh

a k g k z h

kh kh

(1) 2 (1) 2

2

The leading-order trajectory of a particle is an ellipse of the center at ( , )

cosh[ ( )] sinh[ ( )]and a major-axis and minor-axis .

sinh sinh

( ) ( )

cosh[ ( )] sinh[sinh

x z

k z h k z ha a

kh kh

k z h ka a

kh

2

(2) 24 2

(2

1.( )]

sinh

The secon-order solutions for the displacement are calculated by integrating

the related second-order lagrangian velocities.

3 cosh[2 ( )] 1sin 2

8 sinh 4sinh

z hkh

k z ha k

kh kh

) 2

2

(2) 24

cosh[2 ( )]

2sinh3 sinh[2 ( )]

cos 28 sinh

k z ha k t

khk z h

a kkh

(2)The secular term ( ) in the horizontal displacement indicates the

particles will continuously move in the wave direction. Hence, the

trajectory of a particle is no longer an ellipse. Becasue the hor

22

iztonal

mean position of a particle is not fixed at but change with time, we

re-define the horizontal mean position by

cosh[2 ( )]' and ' ' .

2sinhCorrespondingly, the displacemen

x

k z hx x a k t kx t

kh

(1) (1)

(2) 24 2

(2) 2

t with respect to the instantaneous

mean position ( ', ) is given by,

cosh[ ( )] sinh[ ( )]sin ', cos ',

sinh sinh3 cosh[2 ( )] 1

sin 2 ',8 sinh 4sinh

3 si

8

x z

k z h k z ha a

kh khk z h

a kkh kh

a k

4

nh[2 ( )]cos 2 '.

sinh

k z h

kh

(2)

The trajectory of a particle based on the solution is plotted in Applet

(N-trajctory). The time average Lagragian velocity of a particle is

equal to the derivative of the secular term ( ) with respe

22

ct to time.

cosh[2 ( )]

2sinhThe integral of the average Lagragian velocity with respect to water

depth renders the average mass flux induced by a periodic wave

train over a unit width.

Mass

lk z h

u a kkh

0 2 21 1 flux = / / ,

2 2which is consistent with the result derived using Eulurian approach.

l phu dz a a kg E C

Dynamic Pressure

(1) (2)2(1)

0

(1)

Using the Bernoulli equation, dynmaic pressure head induced by a

periodic wave train can be calculated up to second-order,

1 1,

2

cosh[ ( )]co

cosh( )

pC

g g t t g

p k z ha

g kh

(2) 2 22 2 2

(2)

2 2

s ,

3( 1) cosh[2 ( )]cos 2 ( 1)cos 2 ,

4 4

( 1) 1 cosh[2 ( )] .4

p a k a kk z h

g

pa k k z h

g

Radiation Stress

Radiation stress: defined as the time average of excess quasi momentum flux due to the presence of a periodic wave train.

0 0 02 20 0 0

20

20 (1)2 (1)2 (1)

0

Up to second order, a wave train advancing in the - axis,

Noticing that ,

( )

xx xy

yx yy

xx h h h h

xx h

S Sx S

S S

S p u dz p dz u dz p p dz pdz

p w gz p

gaS u w dz p dz

2

20 02

0 0 0

1.

sinh 2 4

2sinh 20

yy h h h

xy yx

khga

kh

ga khS p v dz p dz p p dz pdz

khS S

2

2 10

1sinh 2 2 , where , the energy density.2

0sinh 2

In deep water In shallow water

1/ 2 0 3/ 2 0 .

0 0 0

kh

khS E E gakh

kh

S E S E

.1/ 2

In the case of a wave train having an angle, , with respect to the -axis,

3 cos 21 1 2 sin 21

sinh 2 2 2 2 sinh 2 4

3 cos 22 sin 2 1 11

sinh 2 4 sinh 2 2 2 2

x

kh kh

kh khS E

kh kh

kh kh