1
Lecture 7: Helmholtz Wave Equations
and Plane Waves
Instructor:
Dr. Gleb V.
Tcheslavski
Contact:
gleb@ee.lamar.edu
Office Hours:
Room 2030
Class web site:
www.ee.lamar.edu/g
leb/em/Index.htm
“The ninth wave” by Ivan Aivazovsky (1817-1900)
ELEN 3371 Electromagnetics
Fall 2008
2
Wave Equation
The propagation of EM energy can be described by a wave equation.
We assume that the media is homogeneous and may have losses.
We also assume there are no free charges in the region of interest;
therefore, fields are studied outside the “source region”:
v = 0.
(7.2.1)
Finally, we assume no external currents.
Recall that the constitutive
relations are:
ELEN 3371 Electromagnetics
D E
B  H
J E
Fall 2008
(7.2.2)
(7.2.3)
(7.2.4)
3
Wave Equation
For the stated assumptions, the Maxwell’s equations can be rewritten as
H
  E  
t
E
 H  
 E
t
(7.3.2)
 E  0
(7.3.3)
 H  0
(7.3.4)
Equations (7.3.1) and (7.3.2) contain both electric and magnetic field terms;
therefore, they are coupled equations. When we change either electric or
magnetic field, we automatically affect the other field also.
ELEN 3371 Electromagnetics
(7.3.1)
Fall 2008
4
Wave Equation
Furthermore, the equations (7.3.1) and (7.3.2) are two first-order PDEs in the
two dependent variables E and H. We can combine them into a single secondorder PDE in terms of one of the variables.
For the E field, we take curl of (7.3.1) and substitute (7.3.2) into RHS of the result…


E 
E
2 E
   E      H      E  
  2
  
t
t 
t 
t
t
(7.4.1)
Using the vector identity:
 E    E  2 E  2 E
(7.4.2)
We obtain the wave equation:
2

E

E
2 E  
  2  0
t
t
ELEN 3371 Electromagnetics
Fall 2008
(7.4.3)
5
Wave Equation
The equation in (7.4.3) is a general homogeneous 3D vector wave
equation. It is valid for cases where there are no external sources.
We also note that the equation itself does not depend on the
coordinate system.
Solution of (7.4.3) in general case may be quite complicated…
Therefore, we assume that the wave is propagating in free space with
no currents and only y component of the electric field exists: i.e. the
wave is linearly polarized in the y-direction.
Therefore, in the CCS:
 2 Ey
z
Note:
ELEN 3371 Electromagnetics
2
 0 0
c 1
 2 Ey
t
2
0 0
Fall 2008
0
(7.5.1)
(7.5.2)
6
Wave Equation
Example 7.1: Show that the wave equation for the magnetic field intensity H can
be derived in a similar way as one for the electric field.
By taking curl of the Ampere’s law (7.3.2) and substituting the Faraday’s law
(7.3.1) to the result, we arrive at
   H      E   

H

H
 E           
t
t 
t 
t 

(7.6.1)
Using the vector identity:
 H    H  2 H  2 H
(7.6.2)
We obtain the new wave equation:
2

H

H
2
 H  
  2  0
t
t
(7.6.3)
Note: this wave equation has exactly the same form as one for the electric field.
ELEN 3371 Electromagnetics
Fall 2008
7
Wave Equation
Example 7.2: Compute an approximate numerical value for the velocity
of light c.
Using the numerical values for the electric and magnetic constants in free
space, we write:
c
1
0 0

1
1


 4 10  
109 
 36

 3 108 m / s
7
Note: the more accurate value for the dielectric constant will slightly reduce
this estimate.
ELEN 3371 Electromagnetics
Fall 2008
8
One-dimensional Wave equation
1. Wave experiments in other disciplines (mechanics)
A plunger (wave-maker) in a water tank can move up and down. The repetition
frequency of the plunger’s motion is slow enough to excite waves not interfering
with each other. The water (mechanical) waves propagate slowly compared to EM.
ELEN 3371 Electromagnetics
Fall 2008
9
One-dimensional Wave equation
These waves are transversal
We can evaluate the velocity
of wave propagation (the
speed at which wave crest is
traveling) and the trajectory of
wave propagation.
 is perpendicular to the direction of
propagation.
The velocity of propagation depends
on the surface tension and mass
density of water.
ELEN 3371 Electromagnetics
Fall 2008
10
One-dimensional Wave equation
A string is stretched
between two points. A
small perturbation is
launched at one end of
it and it propagates to
the other end. We
neglect any reflections.
The velocity of
propagation depends
on the tension on the
string and its mass
density.
These waves are transversal:  is perpendicular to the direction of
propagation
ELEN 3371 Electromagnetics
Fall 2008
11
One-dimensional Wave equation
A spring is stretched
between two walls. If
one of the walls is
suddenly moved, a
perturbation in the
spring compression
propagates to the other
end of the spring.
We can find the
trajectory of the
propagation and the
velocity of propagation.
The velocity of propagation depends on the elasticity and the mass density of the
spring. The wave in this experiment is longitudinal :  is in the direction of
propagation (parallel to it). Longitudinal EM waves do not exist!
ELEN 3371 Electromagnetics
Fall 2008
12
One-dimensional Wave equation
2. Analytical solution of a 1D equation – traveling waves.
Recall that in 1D the wave equation is:
 2 Ey
z
or
2
 0 0
 2 Ey
z
2
 2 Ey
t
2
0
2
1  Ey
 2
0
2
c t
(7.12.1)
(7.12.2)
The most general solution for this equation is:
Ey ( z, t )  F ( z  ct )  G( z  ct )
(7.12.3)
Here F and G are arbitrary functions determined by the generator exciting the wave.
ELEN 3371 Electromagnetics
Fall 2008
13
One-dimensional Wave equation
Ey ( z, t )  F ( z  ct )  G( z  ct )
The solution F(z – ct) is a wave traveling in the +z direction.
The solution G(z + ct) is a wave traveling in the -z direction.
Two possible ways to create both waves simultaneously:
1) Two generators (sources) at z = - and at z = +;
2) A source at z = - and a reflecting boundary, say, at z = 0:
an incident and a reflected waves!
Let us verify the validity of the general solution…
ELEN 3371 Electromagnetics
Fall 2008
14
One-dimensional Wave equation
First, we introduce two new variables:
  z  ct
  z  ct
(7.14.1)
(7.14.2)
And use the chain rule of differentiation:
E y
 dG  


t
 d  t
E y dF    dG  

 

z
d  z  d  z
ELEN 3371 Electromagnetics

dF  

d  t
Fall 2008
dG
 dF


c



c

d

d


dG
 dF

1



1

d
 d
(7.14.3)
(7.14.4)
15
One-dimensional Wave equation
The second derivatives are:
2
1  Ey
1  d 2F
d 2G 2 
d 2 F d 2G
2
 2
  2  2  c  
c
 2 
2
2   
c t
c  d
d
d
d 2

 2 E y d 2 F 2 d 2G 2 d 2 F d 2G

1 
1 

2
2  
2  
2
z
d
d
d
d 2
Therefore:
 2 Ey
z
2
2

1 Ey
 2
c t 2
Which proves that (7.12.3) is a general form of a solution for (7.12.2).
ELEN 3371 Electromagnetics
Fall 2008
(7.15.1)
(7.15.2)
(7.15.3)
16
One-dimensional Wave equation
Let us find a particular solution for the following assumptions:
1. The solution is a known function P(z) at the time t = 0;
2. The derivative of the solution is a known function Q(z) at the time t = 0.
E y ( z,0)  P( z ) 


E y
( z,0)  Q( z ) 
t

(7.16.1)
A particular solution of (7.12.2) that satisfies (7.16.1) is
z  ct
1
1
E y ( z, t )   P( z  ct )  P( z  ct )  
Q( z ')dz '

2
2c z ct

1
1
 P( z  ct )  P( z  ct )    R( z  ct )  R( z  ct ) 
2
2
z
Were
1
R( z )   Q( z ')dz ' is an auxiliary function.
c0
ELEN 3371 Electromagnetics
Fall 2008
(7.16.2)
17
One-dimensional Wave equation
 ( z ct )2
Example 7.3: Show that the function F ( z  ct )  F0e
is a
solution of the wave equation. That’s a Gaussian-pulse traveling wave.
Let z’ = z – ct; therefore
F ( z  ct )  F0e


 z '2
and G(z + ct) = 0. The chain rule:


F dF z '
 z '2

 F0  2 z ' e
 (1)
z dz ' z
F dF z '
 z '2

 F0  2 z ' e
 ( c )
t dz ' t
and
ELEN 3371 Electromagnetics
2 F
2
 z '2 
2


F


2

4
z
'

e

(1)


0 

z 2
1 2 F
1
2
 z '2 

 2 2   2 F0   2  4 z '  e
 (c) 2


c t
c
Fall 2008
(7.17.1)
(7.17.2)
(7.17.3)
(7.17.4)
18
One-dimensional Wave equation
Combining the second derivatives, we arrive at:
1
2
 z '2 
2
2
 z '2 


F0  2  4 z '   e
 (1)  2 F0   2  4 z '  e
 ( c ) 2  0




c
A sequence of pulses taken at
successive times illustrates the
propagation of the pulses. The
velocity of propagation is c.
ELEN 3371 Electromagnetics
Fall 2008
(7.18.1)
19
One-dimensional Wave equation
We can also define the propagation of Gaussian pulse as an initial value problem:
P ( z )  F ( z ') t 0  F0e
 z2
(7.19.1)
F dF z '
 z '2


 2 F0 cz ' e
t dz ' t
(7.19.2)
F
 z2
Q( z )  ( z ') t 0  2 F0cze
t
(7.19.3)
The auxiliary function will be:
z
z
1
1
 z '2
 z2 

R( z)   Q( z ')dz '  2cF0  z ' e dz '  F0 1  e
 F0  F ( z)


c0
c
0
ELEN 3371 Electromagnetics
Fall 2008
(7.19.4)
20
One-dimensional Wave equation
Therefore, the particular solution will be:
1
1
1
1
E y ( z, t )  F ( z  ct )  F ( z  ct )   F0  F ( z  ct )    F0  F ( z  ct ) 
2
2
2
2
 F ( z  ct )
Which is a Gaussian wave traveling in the +z direction.
ELEN 3371 Electromagnetics
Fall 2008
(7.20.1)
21
Matlab solution of a 1D wave eqn.
The main weakness of numerical solutions is that they do not really give any
inside to the underlying physics of the problem as theoretical solutions do.
However, Matlab allows to plot the numerical solutions, which, to some extend,
overcomes this limitation…
Recall that a 1D WE:
 2 Ey
z
2
2

1 Ey
 2
0
2
c t
(7.21.1)
To develop a numerical solution to (7.21.1), we first need to solve a first-order
PDE sometimes called the advection equation:
 1 

0
z c t
(7.21.2)
(7.21.2) describes the transport of a conserved scalar quantity in a vector field:
for instance, a pollutant spreading through a flowing stream. It’s hard to solve
numerically in the general case.
ELEN 3371 Electromagnetics
Fall 2008
22
Matlab solution of a 1D wave eqn.
For the initial condition
 ( z, t  0)  F ( z )
(7.22.1)
The analytical solution of the advection equation is given by
 ( z, t )  F ( z  ct )
(7.22.2)
Which is also a solution of the wave equation…
Remark: the wave equation and the advection equation are hyperbolic
equations, the diffusion equations are parabolic equations, and
Laplace’s and Poisson’s equations are elliptic equations.
ELEN 3371 Electromagnetics
Fall 2008
23
Matlab solution of a 1D wave eqn.
We assume that the space z and
time t can be represented in a 3D
figure. The amplitude  of the wave
is specified by the third coordinate.
Next, we set up a numerical grid:
we divide the region L, in which the
wave propagates, into N sections
(N = 4 in the figure).
Therefore, the step in space:
L
h
N
(7.23.1)
Assuming that the velocity of the
propagation is c and it takes time 
for the wave to cover h:
ELEN 3371 Electromagnetics
h  c
Fall 2008
(7.23.2)
24
Matlab solution of a 1D wave eqn.
We assume the solution to be stable and use the periodic boundary conditions:
once a numerically calculated wave reaches the boundary at z = +L/2, it
reappears at the same time at the boundary z = -L/2 and continues to propagate
in the region –L/2  z  +L/2. However, instead of evaluating the wave at the
edges, it is estimated at ½ of a spatial increment from them.
Using the forward difference method:
  ( zi , tn   )   ( zi , tn )

t

where
ELEN 3371 Electromagnetics
L
 1
zi   i   h 
2
 2
tn  (n  1)
Fall 2008
(7.24.1)
(7.24.2)
(7.24.3)
25
Matlab solution of a 1D wave eqn.
Using the central difference method:
  ( zi  h, tn )   ( zi  h, tn )

z
2h
(7.25.1)
Therefore, the advection equation will be:
 ( zi  h, tn )   ( zi  h, tn ) 1  ( zi , tn   )   ( zi , tn )

0
2h
c

(7.25.2)
In (7.25.2), all terms except for one are given for the present time, and one
term specifies the future value of the wave:
c
 ( zi ,tn   )   ( zi ,tn )   ( zi  h,tn )   ( zi  h,tn ) 
2h
That’s a Finite Difference Time Domain (FDTD) method.
ELEN 3371 Electromagnetics
Fall 2008
(7.25.3)
26
Matlab solution of a 1D wave eqn.
The expression in (7.25.3) is valid in the interior range: 2  n  N-1. At the edges
employing the periodic boundary conditions:
c
 ( z2 tn )   ( z N ,tn )
2h
c
 ( z N ,tn   )   ( z N ,tn )   ( z1tn )   ( z N 1 ,tn ) 
2h
 ( z1 ,tn   )   ( z1 ,tn ) 
(7.26.1)
(7.26.2)
For unstable problems, the Lax method is used:
 ( zi ,tn   ) 
1
c
 ( zi  h,tn )   ( zi  h,tn )   ( zi  h,tn )   ( zi  h,tn ) 
2
2h
1
c
 ( z1 ,tn   )   ( z2 ,tn )   ( z N ,tn )    ( z2 tn )   ( z N ,tn ) 
2
2h
1
c
 ( z N ,tn   )   ( z1 ,tn )   ( z N 1 ,tn )    ( z1 tn )   ( z N 1 ,tn ) 
2
2h
ELEN 3371 Electromagnetics
Fall 2008
(7.26.3)
(7.26.4)
(7.26.5)
27
Time-harmonic plane waves
1. Plane waves in vacuum.
Assuming that a time-harmonic propagating
wave is polarized in the y-direction.
E( z, t )  Ey ( z, t )  Ey ( z)e jt
(7.27.1)
a phasor
In a vacuum, the phase velocity of the wave
equals to the velocity of light c.
Therefore, the 1D wave equation is:
 2 Ey ( z)
z
2
2
2
2

 jt

E
(
z
)
d
E
(
z
)
j 

1
y
y
 2
 0
 2 Ey ( z) e  0
2
2
c
t
c
 dz

ELEN 3371 Electromagnetics
Fall 2008
(7.27.2)
28
Time-harmonic plane waves
or
d 2 E y ( z )   2

   Ey ( z )  0
2
dz
c
(7.28.1)
We introduce a new quantity called a wave number:
k

c

2 f 2

c

(7.28.2)
Therefore, the 1D wave equation (the Helmholtz equation) is:

ELEN 3371 Electromagnetics
d 2 Ey ( z)
dz
2
 k 2 Ey ( z)  0
Fall 2008
(7.28.3)
29
Time-harmonic plane waves
A solution of the second-order ODE (7.28.3) is in a form:
Ey ( z)  ae jkz  be jkz
(7.29.1)
where a and b are the integration constants. Incorporating (7.27.1), we obtain:
Ey  ae j (t kz )  be j (t kz )
(7.29.2)
The real part of the solution will be:
Ey  acos(t  kz)  bcos(t  kz)
Note: instead of the real, we could use the imaginary part – sin function.
The first term in (7.29.3) is a wave moving in a +z direction; the second
term is a wave moving in the –z direction (incident and reflected waves).
ELEN 3371 Electromagnetics
Fall 2008
(7.29.3)
30
Time-harmonic plane waves
Since the waves are propagating in vacuum, the phase velocities for
these traveling waves are:
v  

k
 c
(7.30.1)
In general, the phase velocity is a vector since it has both a magnitude and a
direction. It can have a value greater than the light speed! However, there is no
energy (or particles) transferred at that speed.
The wave number may also be a vector and, therefore, indicate the direction,
in which the wave is traveling. In this case, it is frequently called a wave vector
and the quantity kz can be replaced by
 
k r  kr kˆ rˆ
ELEN 3371 Electromagnetics
Fall 2008
(7.30.2)
31
Time-harmonic plane waves
Example 7.4: A polarized in the y direction electric field that propagates in vacuum
was simultaneously measured at z = 0 and one wavelength away at z = 2 cm.
The amplitude is 2 V/m. Find the frequency of excitation, and write an
expression that describes the wave if it’s moving in the +z direction.
The wavelength is  = 0.02 m, therefore:
2 f 2
c 3 108
k
  f  
 1.5 1010 Hz15GHz
c

 0.02
The wave number:
The wave is:
ELEN 3371 Electromagnetics
2
2
k

 100
 0.02
E ( z, t )  2 106 cos 2 15 109 t  50 z  u y
Fall 2008
(7.31.1)
(7.31.2)
(7.31.3)
32
Time-harmonic plane waves
Example 7.5: Show that a linearly polarized plane wave can be resolved into two
equal amplitude circularly polarized waves: i.e. waves that rotate about the z axis.
The linearly polarized wave
Ey ( z, t )  E0u y e j t kz 
can be written as a sum of two components:
where
E y ( z , t ) 
Since
We obtain:
Ey ( z, t )  Ey ( z, t )  Ey ( z, t )
E0
E0
j t  kz 
j t  kz 

u

ju
e
;

E
(
z
,
t
)

u

ju
e




y
x
y
y
x
2
2
ux  u sin ;u y  u cos 
uy  jux  u e j ;uy  jux  u e j
(7.32.1)
(7.32.2)
(7.32.3)
(7.32.4)
(7.32.5)
Which demonstrates that the first and second waves rotate in opposite directions.
ELEN 3371 Electromagnetics
Fall 2008
33
Time-harmonic plane waves
2. Magnetic field intensity and characteristic impedance.
The magnetic field intensity can be found via the Faraday’s law:
 E( z, t )   jB( z, t )   j0 H ( z, t )
u
 x
1 
H ( z, t )  
j0  x

 0
Since:
Therefore:
ELEN 3371 Electromagnetics
H ( z, t )  
1
  E0 e j (t kz ) 
j0
z
Fall 2008
uy

y
E0 e j t kz 
uz 


z 

0 
 k 
j (t  kz )
(u x )   
ux
 E0e
 0 
(7.33.1)
(7.33.2)
(7.33.3)
34
Time-harmonic plane waves
We introduce a new quantity called a characteristic impedance of the medium:
Zc 
For a free space:
Therefore:

k
Z0 
H x ( z, t )  
The Poynting vector:

kc


 c 


k


0
E ( z, t )

 120  376.73
H ( z, t )
0
1
1
E0e j (t kz )ux   uz  E y ( z, t )  A m]
Zc
Zc
S  E( z, t )  H ( z, t )  Suz
If we know the value of one of the field components and the characteristic
impedance, we can find the value of the other field component.
ELEN 3371 Electromagnetics
Fall 2008
(7.34.1)
(7.34.2)
(7.34.3)
(7.34.4)
35
Time-harmonic plane waves
Example 7.6: Find the magnetic field intensity for the following electric field in
vacuum:
E ( z, t )  2 106 cos 2 15 109 t  50 z  u y
E ( z, t ) 2 106
( z, t ) 

cos 2 15 109 t  50 z   uz  u y 
Z0
120
( z, t )  5.3 109 cos  2 15 109 t  50 z   u x  A m
This direction of the magnetic field intensity is required so the power will flow
in the +z direction:
S  EH
uz
ELEN 3371 Electromagnetics
uy
?
Fall 2008
36
Time-harmonic plane waves
Example 7.7: Find the average power in a circular area in a plane defined by z =
constant, whose radius is 3 m if the electric field in a vacuum is:
E( z, t )  10cos t  kz  ux
In a complex form:
E( z, t )  10e
j t kz 
ux
Since the field is in a vacuum, Z0 = 120 .
10 j t  kz 
H ( z, t ) 
e
uy
120
The average power:
 1
1 
10
*
Pav  Re   E ( z, t )  H ( z, t ) ds   10 
   32  3.75W
2 s
120
 2
ELEN 3371 Electromagnetics
Fall 2008
37
Plane wave propagation in a
dielectric medium
1. Plane wave in a lossless homogeneous dielectric.
The wave number for the wave propagating in a vacuum is a function of
permittivity and permeability of free space:
k0  2 f  0 0
(7.37.1)
Naturally, for a dielectric medium that may have different constants, the wave
number will be
k  2 f  0 r 0 r
ELEN 3371 Electromagnetics
Fall 2008
(7.37.2)
38
Plane wave propagation in a
dielectric medium
If a plane wave
generated by a
signal generator
propagates through
two different
dielectrics…
Say, with the same magnetic constants but different permittivities, the wave
numbers will be for these two media:
k1  2 f 10
k2  2 f  2 0
(7.38.1)
Both signals travel the same distance z but will have different phase velocities:
v1   k1 ;v2   k2
ELEN 3371 Electromagnetics
Fall 2008
(7.38.2)
39
Plane wave propagation in a
dielectric medium
This difference in velocities delays the arrival of one signal with respect to the
other and causes a phase difference that can be detected:
  k1z  k2 z  2 f


10   2 0 z
(7.39.1)
If the total phase change in the signal passing through one of the paths is known:
1  2 f 10 z
The relative phase difference is



 2 f 10   2 0 z

 1 2
1
1
2 f 10 z
Therefore, if the properties of one of the regions are known and the phase
difference is measured, we can identify the other material.
ELEN 3371 Electromagnetics
Fall 2008
(7.39.2)
(7.39.3)
40
Plane wave propagation in a
dielectric medium
The ratio of the phase velocity in a vacuum to the phase velocity in a
dielectric is called the index of refraction for the material:
n
Optical materials are usually
characterized by their index
of refraction.
ELEN 3371 Electromagnetics
c
v p ,diel
 r
Material
Vacuum
Air at STP
Ice
Water at 20 C
Acetone
Ethyl alcohol
Sugar solution(30%)
Fluorite
Fused quartz
Glycerine
Sugar solution (80%)
Typical crown glass
Crown glasses
Fall 2008
(7.40.1)
Index
1.00000
1.00029
1.31
1.33
1.36
1.36
1.38
1.433
1.46
1.473
1.49
1.52
1.52-1.62
Material
Spectacle crown, C-1
Sodium chloride
Polystyrene
Carbon disulfide
Flint glasses
Heavy flint glass
Extra dense flint, EDF-3
Methylene iodide
Sapphire
Rare earth flint
Lanthanum flint
Arsenic trisulfide glass
Diamond
Index
1.523
1.54
1.55-1.59
1.63
1.57-1.75
1.65
1.7200
1.74
1.77
1.7-1.84
1.82-1.98
2.04
2.417
41
Plane wave propagation in a
dielectric medium
Example 7.7: Find the phase difference if one region is filled with a gas with r
= 1.0005 and the other region is a vacuum. The frequency of oscillation is 10
GHz and the length is z = 1m.
The phase difference is
  2 f




 0 0   0 r 0 z  2 f  0 0 1   r z
2 1010
0
 
1

1.0005

1


0.052

radians


3
3 108


This difference is small but can be detected. Also, if the travel distance is
increased, the resolution (i.e. detectable phase difference) will be higher.
ELEN 3371 Electromagnetics
Fall 2008
42
Plane wave propagation in a
dielectric medium
2. Plane wave in a lossy homogeneous dielectric.
A dielectric material can be lossy, i.e. exhibit a nonzero conductivity . In this
situation, a conduction current must be added to the displacement current
when considering the Ampere’s law.
 E (r )   j H (r )
(7.42.1)
 H (r)  J (r)  j E(r)    j  E(r)
(7.42.2)
Assuming, as before, no free charges (v = 0) and following the same procedure:
2 E(r)  j   j  E(r )  0
ELEN 3371 Electromagnetics
Fall 2008
(7.42.3)
43
Plane wave propagation in a
dielectric medium
Assuming, as previously, that the electric field is linearly polarized in the y
direction and the wave propagates in the z direction, we arrive to:
d 2 Ey ( z)
dz
2
 j   j  E y ( z )  0
d 2 Ey ( z)
or
dz
2
  2 Ey ( z)  0
(7.43.1)
(7.43.2)
Where the propagation constant:

 
  j   j    j   1 

j



2
2
ELEN 3371 Electromagnetics
Fall 2008
(7.43.3)
44
Plane wave propagation in a
dielectric medium
The propagation constant is complex:
    j
(7.44.1)
In a vacuum,  = 0 and  = k. In a general case, the real and imaginary parts
are nonlinear functions of the frequency:
 ( )  
 ( )  
ELEN 3371 Electromagnetics

2

  

2 1  1  
 
   

 
2
  

1  1  
 
2 
   

Fall 2008
(7.44.2)
(7.44.3)
45
Plane wave propagation in a
dielectric medium
As a result, the phase velocity may depend on the wave’s frequency.
This phenomena is called dispersion and the medium in which wave
is propagating, is called a dispersive medium (every lossy medium).
v p ( ) 



1
(7.45.1)
 
2
  
1  1  
 
2 


 
Another quantity we recall here is the group velocity:
vg ( ) 
ELEN 3371 Electromagnetics
 
1 1 

  
2
 
1 

  
2

2

2
2
2

 


1

 




1  1  
1 

  



2


 





Fall 2008
(7.45.2)
46
Plane wave propagation in a
dielectric medium
The component of the electric field propagating in the +z direction is:
Ey ( z, t )  Ey 0e jt  z   Ey 0e z e j t  z 
(7.46.1)
Wave propagates with a phase constant  but the amplitude
decreases with an attenuation constant .
Units of  are radians/m.
Units of  are nepers/m [Np/m]. If  = 1 Np/m, the amplitude of the wave
will decrease e times at a distance 1 m. 1 Np/m  8.686 dB/m.
The characteristic impedance is:
Zc ( ) 
ELEN 3371 Electromagnetics
Ey ( z)
H x ( z)

j
 ( )  j  ( )
Fall 2008
(7.46.2)
47
Plane wave propagation in a
dielectric medium
Example 7.8: A 10 V/m wave at the frequency 300 MHz propagates in the +z
direction in an infinite medium. The electric field is polarized in the x direction.
The parameters of medium are r = 9, r = 1, and  = 10 S/m. Write the
complete time domain expression for the electric field.
We can find the attenuation constant as
 


  

2 1  1    
   

2
 10
4 107 1 36

10  36


2  9 10 1  1  
6
9 
2


300

10

9

10



9
2



 108.01Np / m
The phase constant is:
2
2
7
9 
 
4


10

1

9

10
10

36

rad
 


8


 
1

1


2


3

10
1

1


109.65


 


2 
2  36
2  3 108  9 109  
m

   



ELEN 3371 Electromagnetics
Fall 2008
48
Plane wave propagation in a
dielectric medium
The complex propagation constant is   108 + j110 and the electric field is
E ( z, t )10e108 z cos  2  3 108  t  110 z u x V m
Example 7.9: Plot the phase
velocity and the group velocity for
the medium with r = 9, r = 1, and
 = 10 S/m
The velocities can be described
by (7.45.1) and (7.45.2).
Both velocities increase with the
frequency. The limit is the same:
v 
c
r
ELEN 3371 Electromagnetics
 108 m s
Fall 2008
49
Plane wave propagation in a
dielectric medium
Two approximations are frequently used:
A) A dielectric with small losses ( << ) with a high-frequency
approximation:
 
1 1 
 2
  
2
(7.49.1)
The approximate values for attenuation and propagation constants are:


2


   
ELEN 3371 Electromagnetics
Fall 2008
(7.49.2)
(7.49.3)
50
Plane wave propagation in a
dielectric medium
v  vg 
1

(7.50.1)
In this situation, the phase and the group velocities are the same. Also, some
attenuation is introduced.
“A pizza in a microwave oven”: water in the pizza acts as a conductor turning
pizza into a complex impedance. The wave passing through it decays,
therefore, the energy is absorbed and must be converted into heat.
B) A dielectric with large losses ( >> ) with a low-frequency
approximation:
The conduction current is much greater than the displacement current, therefore:
 H (r )  J (r )  j E(r )   E(r )
ELEN 3371 Electromagnetics
Fall 2008
(7.50.2)
51
Plane wave propagation in a
dielectric medium
In this situation:

 
1 1 
 

  
2
Therefore:
  

(7.51.1)
(7.51.2)
2
We introduce a skin depth of the material:

1


2


1
[m]
 f 
(7.51.3)
The skin depth decreases with the increasing frequency – skin effect.
ELEN 3371 Electromagnetics
Fall 2008
52
Plane wave propagation in a
dielectric medium
Example 7.10: Estimate the skin depth of copper at a frequency of 3 GHz. The
conductivity of copper is  = 5.8  107 S/m; r = 1, and r = 1.

1
1

 1.21106 m
 f 
  3 109  4 107  5.8 107
Plot the magnitude of the electric field of a plane wave for t = 0 and Ey0 = 10 V/m.
10
Check the validity of the lowfrequency approximation first:
y
E (z), V/m
  0.1667 
8
We can use the approximation.
6
4
2
0
ELEN 3371 Electromagnetics
0
Fall 2008
1
2
3
4
5
z, m
6
7
8
9
10
53
Plane wave propagation in a
dielectric medium
Imagine: if the “pizza in a microwave oven” was covered by a good conductor…
The skin effect would lead to all energy being absorbed by a tiny layer of the
conductor and the pizza would be cold.
Finally, the characteristic impedance for an imperfect conductor is:
Z cond

 (1  j )
2
If the conductivity is large, Z approaches zero.
??QUESTIONS??
ELEN 3371 Electromagnetics
Fall 2008
(7.53.1)
54
Polarization
Polarization is the property of electromagnetic waves,
such as light, that describes the direction of the transverse
electric field. More generally, the polarization of a
transverse wave describes the direction of oscillation in
the plane perpendicular to the direction of travel.
Longitudinal waves such as sound waves do not exhibit
polarization, because for these waves the direction of
oscillation is along the direction of travel.
ELEN 3371 Electromagnetics
Fall 2008
55
Polarization types
Linear
ELEN 3371 Electromagnetics
Circular
Fall 2008
Elliptical
56
Linear polarization
In such arrangement
with two parallel
infinite plates, when
applying a sinusoidal
input to the system,
the phase difference
between voltages on
the top and bottom
plates will be 1800.
The resulting electric field between the plates appears to have only one
non-zero component and, therefore, said to be linearly polarized in the
uy direction.
Back
ELEN 3371 Electromagnetics
Fall 2008
57
Types of waves
A transverse EM wave is a wave
whose E and H vectors are
perpendicular to the direction of
wave’s propagation: light, radiowaves.
A longitudinal EM wave would
be a wave whose E and H
vectors are parallel to the
direction of wave’s propagation:
sound.
ELEN 3371 Electromagnetics
Fall 2008
58
Types of transversal waves
A plane wave is a constant-frequency wave
whose wavefronts (surfaces of constant phase)
are infinite parallel planes of constant amplitude
normal to the phase velocity vector.
By extension, the term is also used to describe
waves that are approximately plane waves in a
localized region of space. For example, a
localized source such as an antenna produces a
field that is approximately a plane wave in its
far-field region.
A wavefront
ELEN 3371 Electromagnetics
Fall 2008
59
Types of transversal waves
A spherical wave is a
constant-frequency wave
whose wavefronts (surfaces
of constant phase) are
parallel concentric spheres of
constant amplitude normal to
the phase velocity vector.
When the distance from the
source is very large, a spherical
wave can be locally
approximated as a plane wave.
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ELEN 3371 Electromagnetics
Fall 2008
60
Phase and group velocities
The phase velocity of a wave is the rate at which the phase of the
wave propagates in space. This is the velocity at which the phase of
any single frequency component of the wave will propagate. We can
pick one particular phase of the wave (for example the crest) and it
would appear to travel at the phase velocity.
Vp 

k
Here,  is a radial frequency and k is the wave number.
The group velocity of a wave is the velocity with which the variations
in the shape of the wave's amplitude (modulation or envelope of the
wave) propagate through space.

Vg 
k
ELEN 3371 Electromagnetics
Fall 2008
61
Phase and group velocities
In the simple case of a pure traveling sinusoidal wave we can imagine a "rigid"
profile being physically moved in the positive x direction with speed v.
Clearly, the wave function depends on both time and position. At any fixed instant
of time, the function varies sinusoidally along x, whereas at any fixed location on
the x axis the function varies sinusoidally with time. If the wave profile is a solid
entity sliding to the right, then obviously the phase velocity is the ordinary speed
with which the actual physical parts are moving. However, we could also imagine
the “magnitude" as the position along a transverse space axis, and a sequence of
tiny massive particles along the x axis, each oscillating vertically. In this case the
wave pattern propagates to the right with phase velocity vp, just as before, and yet
no material particle has any motion at all. This illustrates that the phase of a
traveling wave may or may not correspond to a particular physical entity.
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ELEN 3371 Electromagnetics
Fall 2008