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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
������������������������������������������
����������������������������
(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