Dec 7, 2000

PHY 711 -- Notes on Hydrodynamics PHY 711 - Notes on Hydrodynamics - ("Solitary Waves"[])

Basic assumptions

We assume that we have in incompressible fluid (r = constant) a velocity potential of the form F(x,z,t). The surface of the fluid is described by h+z(x,t) = z. The fluid is contained in a tank with a structureless bottom (defined by the plane z = 0) and is filled to a vertical height h at equilibrium. These functions satisfy the following conditions.

Poisson equation:

2 F(x,z,t)
x2
 + 2F(x,z,t)
z2
 = 0
(1)

Zero vertical velocity at bottom of the tank:

F(x,0,t)
z
 = 0
(2)
Bernoulli's equation:
- F(x,z,t)
t
 + 1
2
 é
ê
ë 		æ
ç
è 		F(x,z,t)
x
 ö
÷
ø 		2

 
 + æ
ç
è 		F(x,z,t)
z
 ö
÷
ø 		2

 
 ù
ú
û 		+gz(x,t) ú
ú
û

z = h+z 
 = 0
(3)
Surface equation:
- F(x,z,t)
z
 + F(x,z,t)
x
 z(x,t)
x
 - z(x,t)
t
 ú
ú
û

z = h+z 
 = 0
(4)

In this treatment, we assume seek the form of surface waves traveling along the x- direction and assume that the effective wavelength is much larger than the height of the surface h. This allows us to approximate the z- dependence of F(x,z,t) by means of a Taylor series expansion:

F(x,z,t) » F(x,0,t) + z F
z
 (x,0,t) + z2
2
 2 F
z2
 (x,0,t) + z3
3!
 3 F
z3
 (x,0,t) + z4
4!
 4 F
z4
 (x,0,t)¼
(5)

This expansion can be simplified because of the bottom boundary condition (2) which ensures that all odd derivatives [(n F)/( zn)](x,0,t) vanish from the Taylor expansion (5). In addition, the Poisson equation (1) allows us to convert all even derivatives with respect to z to derivatives with respect to x. Therefore, the expansion (5) becomes:

F(x,z,t) » F(x,0,t) - z2
2
 2F
x2
 (x,0,t) + z4
4!
 4F
x4
 (x,0,t)¼
(6)

For convenience we define f(x,t) º F(x,0,t). Using Eq. (6), the Bernoulli equation (3) then becomes:

- f
t
 + (h+z)2
2
 3 f
t x2
 + 1
2
 é
ê
ë 		æ
ç
è 		f
x
 ö
÷
ø 		2

 
 + æ
ç
è 		(h + z) 2 f
x2
 ö
÷
ø 		2

 
 ù
ú
û 		+gz = 0,
(7)
where we have discarded some of the higher order terms. Keeping all terms up to leading order in non-linearity and up to fourth order derivatives in the linear terms, the Bernoulli equation becomes:

- f
t
 + h2
2
 3f
t x2
 + 1
2
 æ
ç
è 		f
x
 ö
÷
ø 		2

 
 +gz = 0.
(8)

Using a similar analysis and approximation, the surface definition equation (4) becomes:

x
 æ
ç
è 		(h+z(x,t)) f
x
 ö
÷
ø 		- h3
3!
 4 f
x4
 - z
t
 = 0,
(9)

We would like to solve Eqs. (8-9) for a traveling wave of the form:

f(x,t) = c(x-ct) and z(x,t) = h(x-ct),
(10)
where the speed of the wave c will be determined. Letting u º x-ct, Eqs. (8 and 9) become:
d
d u
 æ
ç
è 		(h + h(u)) d c(u)
d u
 ö
÷
ø 		- h3
6
 d4 c(u)
d u4
 +c d h(u)
d u
 = 0,
(11)
and
c d c(u)
d u
 - ch2
2
 d3 c(u)
d u3
 + 1
2
 æ
ç
è 		d c(u)
d u
 ö
÷
ø 		2

 
 + g h(u) = 0.
(12)

The modified surface equation (11) can be integrated once with respect to u, choosing the constant of integration to be zero and giving the new form for the surface condition:

(h+h) c¢- h3
6
 c¢¢¢ +ch = 0,
(13)

where we have abreviated derivatives with respect to u with the ``¢" symbol. This equation, and the modified Bernoulli equation (8) are now two coupled non-linear equations. In order to solve them, we use, the modified Bernoulli equation to approximate c¢(u) and its higher derivatives in terms of the surface function h(u). Equation (8) becomes approximately:

c¢ = - g
c
 h+ h2
2
 c¢¢¢ - 1
2c
 (c¢)2 » - g
c
 h- h2g
2c
 h¢¢ - g2
2c3
 h2.
(14)
Using similar approximations, we can eliminate c¢(u) and its higher derivatives from the surface equation (13):
(h+h) æ
ç
è 		- g
c
 h- h2g
2c
 h¢¢ - g2
2c3
 h2 ö
÷
ø 		+ h3 g
6c
 h¢¢ + c h = 0,
(15)

where some terms involving non-linearity of higher than 2 or involving higher order derivatives have been discarded. Collecting the leading terms, we obtain:

æ
ç
è 		1 - gh
c2
 ö
÷
ø 		h- g h3
3 c2
 h¢¢ - g
c2
 æ
ç
è 		1 + gh
2c2
 ö
÷
ø 		h2 = 0.
(16)

For the second two terms, Fetter and Walecka argue that it is consistent to approximate gh » c2, which reduces (16) to

æ
ç
è 		1 - hg
c2
 ö
÷
ø 		h(u) - h2
3
 h¢¢(u) - 3
2h
 [ h(u) ]2 = 0.
(17)

Your text shows that a solution to Eq. (17) (corresponding to Eq. 56.30 of the text), with the initial condition h(0) = h0 and h¢(0) = 0, is the solitary wave form:

z(x,t) = h(x-ct) = h0   sech2 æ
ç
ç
ç
è 		  æ
 ú
Ö
3h0
h
 
    x-ct
2h
 ö
÷
÷
÷
ø 		,
(18)
with
c =   æ
 ú
Ö
gh
1-h0/h
 
 »   __
Ögh
 
 æ
ç
è 		1+ h0
2h
 ö
÷
ø 		.
(19)

The ``standard'' form of the related Korteweg-de Vries equation[] is given in terms of the scaled variables [`t] and [`x] in terms of the function h([`x],[`t]) by

h
_
t
 
+ 6 h h
_
x
 
+ 3 h
_
x
 
 3
 = 0 ,
(20)
which has a solution
h( _
x
 
 , _
t
 
 ) = b
2
   sech2 é
ê
ê
ê
ë
Ö
b
 
2
 ( _
x
 
 - b _
t
 
 ) ù
ú
ú
ú
û 		.
(21)
This form is related to our results in the following way.
b = 2 h0,            _
x
 
 =   æ
 ú
Ö
3
2h
 
 x
h
 ,             and             _
t
 
 =   æ
 ú
Ö
3
2h
 
 ct
2 h0 h
 .
(22)
To show how the reduced equation (17) is related to the Korteweg-de Vries equation, we first take the u derivative to find:
h0
h
 h¢ - h2
3
 h¢¢¢ - 3
h
 hh¢ = 0,
(23)
where we have used the relation
h0
h
 = 1 - gh
c2
 .
(24)
Then we notice that
h
t
 = -c d h
d u
           and          h
x
 = d h
d u
 ,
(25)
so that Eq. (23) can be written:
- h0
ch
 h
t
 - h2
3
 3 h
x3
 - 3
h
 h h
x
 = 0.
(26)
Substituting the transformation (22) into this partial differential equation yields the Korteweg-de Vries equation (20).

References

[]
Alexander L. Fetter and John Dirk Walecka, Theoretical Mechanics of Particles and Continua, (McGraw Hill, 1980), Chapt. 10.
[]
Websites concerning solitons:

UK,
K. Brauer,
Japan

File translated from TEX by TTH, version 2.20.
On 7 Dec 2000, 14:51.