nonlinear long wave in shallow water. contents two important parameters for waves in shallow water ...

Nonlinear Long Wave in Shallow Water

Contents

Two Important Parameters For Waves In Shallow Water

Nondimensional Variables

Nondimensional Governing Equation & Boundary Conditions

Perturbation Series of Potential

Depth Average Horizontal Velocity & Wave Elevation

Ariy's Approximation

Boussenesq's Approximation

Steady Kdv Equation

Solitary Wave

Cnoidal Wave

2

2 1 and

~ (1) : Ariy's approximation;

1: Boussinesq's approximation;

where is amplitude, wavenumber,

water depth (constant).

Urseell number = , when >> 1,

t

r r

h akh

L hO

a k

h

U U

he Stokes expansion may not be valid anymore,

a shallow water perturbation should be used.

Two Important Nonlinear Parameters for Waves in Shallow Water

Non-dimensional Variables

, / , ,

/ , ,

where , , and are dimensional

variables and , , , and are the

non-dimensional variables.

X xk Z z h t t ghk

aa gh

khx z t

X Z t

Nondimensional Governing Equation & Boundary Conditions

2 22

2 2

2

2 22 2

0, 1 (1)

0, at 1 (2)

, at (3)

1

2

ZX Z

ZZ

ZX X Zt

X Zt

0,

at (4)Z

Perturbation Series for Potential :

0

22

22

Let ( , , ) ( 1) ( , ).

To satisfy the Laplace Equation (1), we have

the following recursive relation:

, 0,1,2,3,.. (5)( 1)( 2)

Noticing 1, the series, , decays w

nn

n

n

n

n

X Z t Z X t

X nn n

1

2 1

ith the

increase in . To satisfy the bottom boundary

condition (2), 0. Based on the recursive

relation (5), 0, thus the odd-number

terms in the series are equal to zero.n

n

222 0

0 2

444 0

4

222 0

2

Hence, the perturbation series for reduces to,

( , , ) ( 1)2!

( 1) ...4!

( 1) ( 1) ... (6)

(2 )!

nn nn

n

X Z t ZX

ZX

Zn X

Depth Average Horizontal Velocity &

Wave Elevation

2 220 0

0 2

32 3 403

0 00

2

Substituting (6) into the free-surface boundary

conditions (3) and (4), we obtain,

1

2

( ) (7)6

1

U UH H HU H

X Xt X

UH o

X

U UHU

X Xt

H

X

2 2 2 22 20 0 0

0 2

4

2 2 2

( ) (8)

U U UH HU

Xt X X

o

00

0

0

where , 1 ,

and .

The truncated Equations (7) and (8) can be used

to solve the two unknowns and , which are

related to and . Equations (7) and (8) can

al

U HX

H H

X Xt t

H U

22 2 400 21

02 22 4

0 2

so be expressed in terms of , the average

horizontal velocity,

1( )

6

Inversely, can be expressed in terms of .

( )6

U

UU dz U H o

XH X

U U

UU U H o

X

0

4

2 22

2

2 22

Substituting by and after algebraic manipulation,

Equations (7) and (8) reduce to,

( ), (9)

1

6

2

U U

HHU o

Xt

U H U UU H

X Xt t X

H U

X t

22 22

2

4

3 2

( ). (10)

(9) & (10) are the depth-average horizontal

velocity and wave elevation equations.

H U H UU

XX X

o

Ariy's Theory For Very Long & Very Steep Waves

2

2

2

Very long wave 1, very steep ~ (1),

, it applies to waves in the surf zone.

Neglecting the terms of ( ) in Equations (9) &

(10), we have,

0

r

O

U

O

HHU

Xt

(11)

10 (12)

U H UU

X Xt

The corresponding dimensional equations are,

( ) 0 (13)

0 (14)

where is the dimensional counterpart of

d

d dd

d

h ut x

u uu

t x x

u U

.

Equation (13) describes the mass conservation of

an incompressible flow. The solution for the Ariy's

wave can be found in Mei (1983)

Boussinesq's Approximation

2 2

2

4 2

2

Very long wave 1, but ~ 1,

~ (1), which is applied to waves outside the

surf zone. Neglecting the terms of ( ), ( ) &

O( ) in Equations (9) and (10), we have,

0

rU o

O O

HHU

Xt

2 3

2

(15)

10 (16)

3

U H U UU

X Xt X t

2 3

2

In comparison with the corresponding Equations (10)

& (11) based on the Ariy's theory, Equation (16)

involves an additional term , which 3

accounts for the wave dispersion due to water dep

U

X t

th.

The solutions for Boussinesq approximation may lead

to two distinct shallow water waves:

(1) Solitary waves;

and (2) Cnoidal waves.

Steady KdV (Korteweg-de Vries) Equation

2 2 3

2

Letting and noticing 1 , we integrate

(16) with respect to X,

0. (17)2 3

Equation (15) relates to through the operator ,

( )

U HX

Xt X t

t

2 2

2 2

22 2 2 4 2

2 2 2 22

.

Applying the operator on (17) and noticing the above

equation, we obtain,

(18)3

X X X X

Xt tX Xt X t

The coordinates ( - ) move at the non-dimensional

phase velocity (which is defined as ) of the

shallow water wave train, where is related to by,

. Assuming the shallow water wave tr

Z

CC

gh

X

X Ct

222 (4) 2

ain

is steady in the moving coordinates, then

and .

Using above equalities, Equation (18) reduces to,

31 (19)

3 2

dC C

X d t

CC C

2

4 2 2

22 (4) 2

Equation (19) indicates 1 ( , ), and hence (19)

canbe further simplified by neglecting the high-order term

of ( , , ),

31 . (20)

3 2

Inte

C O

C

22 (3) 21

''

22 2 2

grating (20) with respect to once, we have

31 . (21)

3 2

Multiplying (21) by and then integrating it one more

time, we have,

1 11

2 6

C A

C

31 2

1 2

, (22)2

where & are arbitrary constants to be determined

using lateral boundary condition later.

A A

A A

2 2

22 2 2 3

1 2

Noting ( , ) ( , ) ,

Equation (22) can be rewritten in terms of :

11 , (23)

2 6 2which is known as the . Based on

different

C O Ot

C A A

steady Kdv Equation

lateral boundary conditions, the solution for

the Kdv equation leads to two different wave trains,

(1)

(2) .

Solitary Wave;

Cnoidal Wave

Solitary Waves

'

1 2

22' 2

2

Far away from the crest of a solitary wave (at crest 0), we

have 0 and 0, as . These two lateral boundary

conditions lead to 0, and Equation (23) reduces to,

13

A A

C

2

2'

. (24)

To obtain a non-trivial solution, we must have 1, and

1. At the crest, 0 and reaches the maximum

magnitude which is one (due to non-

C

C

2

max

2

dimensionalization).

11 (25)

1 (26)

C

C

22 2

22

Equation (26) indicates that the previous assumption

11 ( , ) is valid. Noting that =1,

we solve Equation (24) and have,

1 3sec .

2

CC O

h

23

(27)

The corresponding dimensional solutions for the solitary

wave are, 1 (28)

3sec

2

aC gh

h

aa h x Ct

h

(29)

The elevation of a solitary wave train is sketched in Fig. 1

Figure 1. The sketch of a non-dimensional solitary wave train.

Cnoidal Wave

2 2

23 2

1 2

3 2 1

1 2

The steady Kdv equation (23) can be written as,

' ( ) (30)3

1where ( )

( )( )( ),

,

f

Cf A A

3

1 2 3

, are the three roots of the third polynomial

( ( )). By default, .f

Figure 2. The sketch of ( ) vs. .f

2 3 max 3 min 2

For a Non-trivial and meaningful solution, ( ) must be

positive and is limited. Then the solution for must be

in the range of , i.e., and ,

which correspond to the crest and tr

f

2 23 2

3 2

ough of the cnoidal

wave, respectively.

Let cos sin , (31)

where is a function of , = ( ).

Differentiating (31), sin 2 ( ) . (32)d

d

2

23 12

3 2

3 1

Substituting (31) and (32) into (30), after some algebraic

manupulations, we obtain,

3( ) 1 sin (33)

4

where

dm

d

m

1 2 3

3 1 020

(34)

Noticing , it is known that 0 1. Solving

Equation (33), we have

3( , ) ( ). (35)

21 sin

m

dF m

m

0The arbitrary constant is related to the initial position

of the Cnoidal wave train with respect to the original of

the moving coordinates -Z. ( , ) is known as the

Elliptic integral of the first k

F m

3 1 0

3 1 0

ind with modulus . It can

also be expressed as ,

3cos ( ) (36)

2

3sin ( ) (37)

2

m

Cn

Sn

1

22 3 1

2 3 2 3

2

The corresponding dimensional solution is

( )3( ) ( ) (39)

2

where is the initial phase & .

In order to compute , we need to know the relations

among

i i

Cn x ct

h

a

1 2 3, , , and the wave characteristics, such as

non-dimensional wave length and wave height . They

are given below.

L H

2

3 1 20

Noticing that the periods of cos and cos are 2 and , respectively.

Based on Equation (35) the wave length of the Cnoidal wave is equal to,

32 ( ) (4

2 1 sin

dL K m

m

2

20

3 1

2 3

0)

where ( )1 sin

4 ( ) (41)

3 ( )

· Noticing that and corresponds to the elevation at the trough and

crest, resp

dK m

mK m

L

3 2

ectively, the nondimensional wave height is equal to,

. (42)H

2

2 31

0

Comparing the coefficient of the polynomial ( ) and

its three roots,

1 . (43)

Because 0 is defined at the still water level,

0 . L

f

C

d

21 3 12

20

(44)

· Making use of Equations (31) and (33), Equation (44)

can be reduced to,

( )(1 sin )0 . (45)

1 sin

md

m

22

0

1 3 1

3 1

Noticing ( ) 1 sin , ( ) is the elliptical

integral of the second kind with modulus . Equation (45)

is reduced to,

( ) ( ) ( ) 0 (46)

Noticing , we sol

E m m d E m

m

K m E m

Hm

1 2 3

1

2

3

ve Equations (42), (43), and (46)

for , , and ,

( ) , (47)

( )

( )1 , (48)

( )

( )1

H E m

m K m

H E mm

m K m

H E m

m K

. (49)( )m

1 2 3

2

Having known , , and , the dimensional wave characteristics

can be expressed in terms of , wave height and wave depth .

34 ( ) or ( ) (50)

3 2

( )1 2 3

( )

m H h

mhL K m h K m m

H

H E mC gh m

hm K m

22

(51)

2 ( )( ) (52)

4 ( ) 3 (53)

( )1 2 3

( )

K mH Cn x ct

L

mhK m h

L HTC H E m

gh mhm K m

Computation Procedure:

•When water depth h, wave length L (or T) and wave height H are given, we may use a try and error (or iterative) method to determine m based on Equation (50) (or (53)).

•Once m is obtained, the characteristics of a Cnoidal wave train can be computed using Equations (47), (48), (49), (51) & (52).