Download Lecture_12 - Dept of Maths, NUS

USSC3002 Oscillations and Waves
Lecture 12 Water Waves
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
http://www.math.nus/~matwml
Tel (65) 6874-2749
1
VELOCITY POTENTIAL
We assume that the water has constant density 
and that its velocity u  (u1 , u2 , u3 ) is irrotational
 u3
 u2


j
k
i
 x2   x3
  u1  u3 


 

curl u    u  det   x1  x2  x3     x3   x1   0
 u2
 u1 

 u1 u2 u3 


x
 1  x2 
Theorem 1. If u is irrotational on a domain   R
then there exists a velocity potential function  :   R
such that
p   and
Proof
Choose
any

 x 
   path  ( p, x ) Stokes Thm 
u  grad        x 

( x)    u  dy


 
y ( p , x )
x 
2
1
2
3
INCOMPRESSIBILITY
We further assume that the water is incompressible.
Theorem 2. If the flow is incompressible then
div u    u 
 u1
 x1

 u2
 x2

 u3
 x3
0
Proof Follows from the divergence theorem.
Corollary 1 If u is both irrotational and incompressible
u   ,    0.

1  x1
Definition u    u
u

2  x2
u

2  x2
(this operator can operate on real or vector valued functions)
Corollary 2 If u is both irrotational and incompressible
( 12 u  u)  u  u
3
MATERIAL DERIVATIVE
Lemma Along the flow of any particle, whose
position is x = x(t), the rate of change of any (real
or vector valued) function H(t,x) is given by the
material derivative or total derivative defined by
DH
Dt

H
t
 u  H
Proof This follows directly from the chain rule.
Theorem 3 Every fluid satisfies the equation
Du
 F    where F  F (t , x) is the

Dt


body force density and    (t , x) the stress tensor.
Proof Follows from Newton’s 2nd Law.
4
NAVIER STOKES EQUATIONS
Definition A Newtonian fluid is one satisfying
Stokes assumptions    pI   (  u ) I  2  D
where I = identity matrix, p = pressure,
 ,   viscosity coeffs, D  12 (u  (u)T ) = strain tensor.
Corollary 4. Newt. fluids satisfy the Navier-Stokes eq.

Du
Dt
  F  p  ( (  u))    (2 D)
Corollary 5. The incompressible Navier-Stokes eq. are
 F  1 p  u,   u  0

where    = kinematic viscosity coefficient.
Du
Dt
5
BERNOULLI’s EQUATION
Corollary 6. An irrotational flow of an incompressible
inviscid Newtonian fluid for which the body force is
conservative satisfies Bernoulli’s equation
p

 u  u V 
1
2

t
 C(t )
where V is the potential for the body force F  V .
Proof Corollaries 1 and 2  ( p  12 u  u  V  t ) 
1

p  u   u  F 
u
t
 u by corollary 5.


The inviscid assumption     0 hence the left
side is independent of x and therefore is a function
of time. It is customary to absorb C(t) into  .
6
TIDAL WAVES
are also called long waves in shallow water. Their
wavelengths are much longer than the water depth so
we may ignore the vertical component of acceleration.
We will assume that water is inviscid. Since F3   g
D u3
p
1
corollary 5  0  D t  F3   p   g   x3
hence p   g  x3  constant. Let x3  0 be the
undisturbed free (horizontal) surface and  (t, x1, x2 )
be the elevation of the water above the point ( x1 , x2 ,0)
Therefore p(t , x)  pair  g  (  (t , x1 , x2 )  x3 ).

 x1
Hence
and
g
independent of depth.
D u1
Dt
D u2
Dt
g

 x2
are
7
TIDAL WAVES IN A STRAIGHT CHANNEL
Fig 1 shows a wave moving in the x1 direction in a
channel whose cross-section area A and surface
breadth b vary slowly so that u1 is independent of x2
D u1
 u1
 u1
 u1
 u1
 u1

Then  g  x1  D t   t  u1  x1  u2  x2  u3  x3   t

x3
Since the net influx through
x1
planes x1  s and x1  s  s,
equals the increase of water in
s
the region between these planes
s


and | b |  A    x ( Au )  b  t


 
(
Ag
)

b
so above eqn   x
x
t
1
2
1
rectangle A=bh 
2
1
2 
 x1
2

2
1  
c2  t 2
h
, c  gh
8
SURFACE WAVES
If the velocity is small and gravity is the only body
force then u u  0 and V  g  therefore
Bernoulli’s equation   
 

t
1
g
free surface
and since a particle on the free surface stays there
0
D
Dt
hence
also
( (t , x1 , x2 )  x3 ) 

 x1

t

1
g

 
 x1  t
 u3  


1
g
 
fs
1 
g  x1 fs

t


1  x1
u
 
 t  x1
2 
t
2

g
fs

2  x2
1  u1
g
t
u


 x3
 u3
 
fs
0
in a long rectangular tank these and boundary cond 
2 h
g
2
2
2
  C cosh  ( x3  h) cos  ( x1  ct ), c  2 tanh 
9
TUTORIAL 12.
Problem 1. Locate a statement of Stokes theorem and
use it to give a detailed proof of Theorem on vufoil 2.
Problem 2. Prove corollaries 1 and 2 on vufoil 3.
Problem 3. Derive the identity   (2 D)  u  (  u )
and use it to show that cor 4 implies cor 5.
Problem 4. Compute the speed of a tsunami travelling
in a single direction for sea depths: 40m, 400m, 4km.
Problem 5. (Extra) Compute u for problem 4 if
  cos (t  k x1 ) and wavelength = 10h.
Hint Use   u  0 to compute u3 from u1 .
Problem 6. (Extra) Show that the group velocity for
surface water waves equals c/2 .
10