Transcript Propagation of waves - Dalhousie University

Propagation of
waves
Friday October 18, 2002
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Propagation of waves in 3D

Imagine a disturbane that results in waves
propagating equally in all directions
 E.g.
sound wave source in air or water, light
source in a dielectric medium etc..
The generalization of the wave equation to
3-dimensions is straight forward if the
medium is homogeneous
 Let  = amplitude of disturbance (could be
amplitude of E-field also)

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Propagation of waves in 3D
 depends on x, y and z such that it satisfies the wave equation
 2  2  2
1  2


 2
0
2
2
2
2
x
y
x
v t
or,
2
1


2
 2
0
2
v t
where in cartesian co-ordinates,



  xˆ  yˆ  zˆ
x
y
z
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1. Special Case: Plane Waves along x
Suppose (x, y, z, t)=(x, t) (depends only
on x)
 Then  = f(kx-ωt) + g(kx+ωt)
 Then for a given position xo,  has the
same value for all y, z at any time to.
 i.e. the disturbance has the same value in
the y-z plane that intersects the x-axis at
x o.
 This is a surface of constant phase

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Plane waves along x
kxˆ
Planes perpendicular to the x-axis are wave fronts – by definition
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2. Plane waves along an arbitrary
direction (n) of propagation


Now  will be
constant in plane
perpendicular to n – if
wave is plane
For all points P’ in
plane

r  nˆ  d
z
nˆ
P
P’
d
O

r
y
x
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2. Plane waves along an arbitrary
direction (n) of propagation
For all points P’ in plane


 f kd  t 

 f k r  nˆ   t 
or, for the disturbance at P

 f kd  t 
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2. Plane waves along an arbitrary
direction (n) of propagation
If wave is plane,  must be the
same everywhere in plane  to n
z
nˆ
This plane is defined by
P

r  OP nˆ  0
or,

r  nˆ  nˆ  OP  d  const
d
O
P’

r
y
is equation of a plane  to n,
a distance d from the origin
x
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2. Plane waves along an arbitrary
direction (n) of propagation



 f k r  n   t 

 f knˆ   r  t 
 
 f k  r  t


is the equation of a plane wave propagating in k-direction
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3. Spherical Waves



 r , t 
Assume
has spherical symmetry about
origin (where source is located)
In spherical polar co-ordinates
1   2  
1
 
 
1
2
  2
r
 2
 sin 
 2
r r  r  r sin   
  r sin 2   2
z
2
θ r
y
x
φ
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3. Spherical Waves


Given spherical symmetry,  depends only on r, not φ or
θ
Consequently, the wave equation can be written,
1   2   1  
r
 2 2  0
2
r r  r  v t
or,
2
2   2 1  2
 2  2 2 0
r r r
v t
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3. Spherical Waves
Now note that,
 2 r   
 
   r
2
r
r 
r 

 2
2
r 2
r
r
 2   2 
 r
 2
 r r r 
r  2
 2 2
v t
 2 r 
r 2
1  2 r 
 2
v
t 2
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3. Spherical Waves
But,
 2 r  1  2 r 
 2
0
2
2
r
v
t
is just the wave equation, whose solution is,
r  f kr  t   g kr  t 
1
  f kr  t 
r
i.e. amplitude decreases as 1/ r !!
Wave fronts are spheres
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4. Cylindrical Waves (e.g. line source)
The corresponding expression is,

A

cosk  t 
for a cylindrical wave traveling along positive 
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Electromagnetic waves

Consider propagation in a homogeneous
medium (no absorption) characterized by a
dielectric constant
  
o
o = permittivity of free space
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Electromagnetic waves
Maxwell’s equations are, in a region of no free charges,

  E  4  0

B  0


B
 E  
t



E 
E
   o
  B   o  4j 


t
t


Gauss’ law – electric field
from a charge distribution
No magnetic monopoles
Electromagnetic induction
(time varying magnetic field
producing an electric field)
Magnetic fields being induced
By currents and a time-varying
electric fields
µo = permeability of free space (medium is diamagnetic)
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Electromagnetic waves
For the electric field E,








2
2
    E     E   E   E

2


 E

  B    o
2
t
t


or,


 E
2
 E   o 2  0
t
2
i.e. wave equation with v2 = 1/µo
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Electromagnetic waves


 B
2
 B   o 2  0
t
2
Similarly for the magnetic field
i.e. wave equation with v2 = 1/µo
In free space,
c
 =  o = o
( = 1)
1
 o o
c = 3.0 X 108 m/s
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Electromagnetic waves
In a dielectric medium,
1
 = n2
and
 =   o = n2  o
1
c
v


 o
n o o n
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Electromagnetic waves: Phase relations
The solutions to the wave equations,


 E
2
 E   o
0
2
t
2


 B
2
 B   o
0
2
t
2
can be plane waves,

 i k r t 
E  Eo e

 i k r t  
B  Bo e
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