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PHY 711 Fall 2014 -- Lecture 34 111/14/2014
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 34:
Chapter 10 in F & W: Surface waves
-- Non-linear contributions and soliton solutions
PHY 711 Fall 2014 -- Lecture 34 211/14/2014
PHY 711 Fall 2014 -- Lecture 3411/14/2014 3
p0
hz
x
z
y
212
2
Within fluid: 0
constant
0
( , , , )
( , , , ( , , , )
At surface: with
)
z h
v g z ht
x y z
x y z t
xt
z h
y z t
v v
, ,
, ,
where , , ,x y x y x y
x y t
dv v v v x y h t
dt t x y
General problem including non-linearities
Surface waves in an incompressible fluid
PHY 711 Fall 2014 -- Lecture 3411/14/2014 4
p0
hz
x
z
y
dt
d
zthzxv
hz
txtzx
y
z
,, :surfaceAt
0 :fluidWithin
, ,,
dependence- trivialassume tions;simplificaFurther
PHY 711 Fall 2014 -- Lecture 34 511/14/2014
p0
hz
x
z
yz=0
2 00 0
0
0
Dominant non-linear effects soliton solutions
3( , ) sech
2
where 11 / 2
constantx ct
x th h
ghc gh
h h
Non-linear effects in surface waves:
PHY 711 Fall 2014 -- Lecture 34 611/14/2014
Detailed analysis of non-linear surface waves[Note that these derivations follow Alexander L. Fetter and John Dirk Walecka, Theoretical Mechanics of Particles and Continua (McGraw Hill, 1980), Chapt. 10.]
We assume that we have an incompressible fluid: constant
Velocity potential: ( , , ); ( , , ) ( , , )x z t x z t x z t
v
The surface of the fluid is described by z=h+z(x,t). It is assumed that the fluid is contained in a structure (lake, river, swimming pool, etc.) with a structureless bottom defined by the z = 0 plane and filled to an equilibrium height of z = h.
PHY 711 Fall 2014 -- Lecture 34 711/14/2014
Defining equations for F(x,z,t) and z(x,t)
where 0 ( , )z h x t
2 2
2 2
( , , ) ( , , )0 0
Continuity equation:
x z t x z t
x z
v
Bernoulli equation (assuming irrotational flow) and gravitation potential energy
2 2( , , ) 1 ( , , ) ( , , )
( ) 0.2
x z t x z t x z tg z h
t x z
PHY 711 Fall 2014 -- Lecture 34 811/14/2014
Boundary conditions on functions –
Zero velocity at bottom of tank:
( ,0, )0.
x t
z
Consistent vertical velocity at water surface
( , , ) . z z h
dv x z t
dt t
vt
( , , ) ( , , ) ( , ) ( , )0z h
x z t x z t x t x t
z x x t
t
PHY 711 Fall 2014 -- Lecture 34 911/14/2014
Analysis assuming water height z is small relative tovariations in the direction of wave motion (x)Taylor’s expansion about z = 0:
2 2 3 3 4 4
2 3 4( , , ) ( ,0, ) ( ,0, ) ( ,0, ) ( ,0, ) ( ,0, )
2 3! 4!
z z zx z t x t z x t x t x t x t
z z z z
Note that the zero vertical velocity at the bottom ensures that all odd derivatives vanish from the
Taylor expansion . In addition, the Laplace equation allows us to convert all even derivatives with respect to zto derivatives with respect to x.
( ,0, )n
nx t
z
2 2 4 4
2 4Modified Taylor s expansion: ( , , ) ( ,0, ) ( ,0, ) ( ,0, )
2 4!
z zx z t x t x t x t
x x
PHY 711 Fall 2014 -- Lecture 34 1011/14/2014
Check linearized equations and their solutions: Bernoulli equations --
Bernoulli equation evaluated at ( , )
( , , )( , ) 0
Consistent vertical velocity at ( , )
( , , ) ( , )0z h
z h x t
x h tg x t
tz h x t
x z t x t
z t
t
Using Taylor's expansion results to lowest order
2
2
( , , ) ( ,0, ) ( , , ) ( ,0, )x z t x t x h t x th
z x t t
2 2
2 2
( ,0Decoupl
, )ed equations:
( ,0, ).
x t x tgh
t x
PHY 711 Fall 2014 -- Lecture 34 1111/14/2014
Analysis of non-linear equations -- keeping the lowest order nonlinear terms and include up to 4th order derivatives in the linear terms. Let ( , ) ( ,0, )x t x t
222 3 2
2 2
22 3
2
Approximate form of Bernoulli equation evaluated at surface:
( ) 1( ) 0
2 2
10.
2 2
z h
hh g
t t x x x
hg
t t x x
3 4
4
Approximate form of surface velocity expressio
( ( , )) 0.3!
n :
hh x t
x x x t
The expressions keep the lowest order nonlinear terms and include up to 4th order derivatives in the linear terms.
PHY 711 Fall 2014 -- Lecture 34 1211/14/2014
22 3
2
3 4
4
10.
2 2
Cou
( ( ,
pled
)) 0.
equation
!
s:
3
hg
t t x x
hh x t
x x x t
Traveling wave solutions with new nota
t
( , ) ( ) and ( , ) (
io
)
n:
u x ct x t u x t u
22 3
3
3 4
4
( ) ( ) 1 ( )( ) 0.
2 2
( ) ( ) ( )( ( )) 0.
6
d u ch d u d uc g udu du du
d d u h d u d uh u c
du du du du
Note that the wave “speed” c will be consistently determined
PHY 711 Fall 2014 -- Lecture 34 1311/14/2014
Integrating and re-arranging coupled equations
22 3
3
2 2 22 2
3
3 4
4
3
( ) ( ) 1 ( )( ) 0.
2 2
1' ''' ( ') ''
2 2 2 2
( ) ( ) ( )( ( )) 0.
6
( ) ' ''' 06
d u ch d u d uc g udu du du
g h g h g g
c c c c c
d d u h d u d uh u c
du du du du
hh c
PHY 711 Fall 2014 -- Lecture 34 1411/14/2014
Integrating and re-arranging coupled equations – continued --Expressing modified surface velocity equation in terms of h(u):
2Note: c gh
2 2 32
3
32
2 2 2 2
22
2
( ) '' '' 02 2 6
1 '' 1 03 2
31 ( ) ''( ) ( ) 0.
3 2
g h g g h gh c
c c c c
gh gh g gh
c c c c
hg hu u u
c h
PHY 711 Fall 2014 -- Lecture 34 1511/14/2014
Solution of the famous Korteweg-de Vries equation
Modified surface amplitude equation in terms of h
2
2
2
31 ( ) ''( ) ( ) 0.
3 2
hg hu u u
c h
Soliton solution
2 00
00
0
is a constan
3( , ) ( ) sech
2
1 where 1 / 2
t
x ctx t x ct
h h
ghc gh
h h
PHY 711 Fall 2014 -- Lecture 34 1611/14/2014
Relationship to “standard” form of Korteweg-de Vries equation
00
3 32 , , and .
2 2 2
New variables:
x ctx t
h h h h
3
3
Standard Korteweg-de Vries eq
6 0
i n
.
uat o
t x x
2( , ) sech ( ) .
Soliton soluti
2
on:
2x t x t
PHY 711 Fall 2014 -- Lecture 34 1711/14/2014
More details
2
2
2
02
20
Modified surface amplitude equation in terms of :
31 ( ) ( ) ( ) 0.
3 2
Some identities: 1 ; ; .
Derivative of surface amplitude equation:
3''
3' ' '
hg hu u u
c h
gh d dc
h c t du x du
h
h h
2 30
3
3
3
0.
Expression in terms of and :
30.
3Expression in terms of and :
6 0.
x t
h
ch t x h xx t
t x x
PHY 711 Fall 2014 -- Lecture 34 1811/14/2014
Soliton solution
2 00
00
0
is a constan
3( , ) ( ) sech
2
1 where 1 / 2
t
x ctx t x ct
h h
ghc gh
h h
Summary
PHY 711 Fall 2014 -- Lecture 34 1911/14/2014
Some links: Website – http://www.ma.hw.ac.uk/solitons/
Photo of canal soliton http://www.ma.hw.ac.uk/solitons/
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