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).