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Chapter 10
Wave Propagation
1
Waves
• What is a wave?
Wave incident at a boundary between two media
Boundary between
two media
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Waves
• A wave is function of space and time
• A wave function is the solution to a differential
equation known as the wave equation
• The scalar wave equation may be written as
2
2E
2  E
u

0
t 2
z 2
2E
 u 2 2 E  0
2
t
: 1D
: 3D
3
Waves
• Solutions to the 1D wave equation look like:
E   f ( z  ut )
E   g ( z  ut )
Traveling in the +z direction
Traveling in the -z direction
• Note these functions satisfy the wave equation
• Examples of such functions are:
cos[ k ( z  ut )]
sin[ k ( z  ut )]
exp[ jk ( z  ut )]
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Waves
• Substituting ( E ( z , t )  E ( z )e jt ) for time
harmonic fields yields Helmholtz equation
where time and are space decoupled
2
2E   
  E  0
z 2  u 
• Solution looks like
E  Ae j (t  z )  Be j (t  z )
where
 /u
is known as the propagation constant
5
Waves
• Wavelength

2

• Phase Speed
u

 f

• Note phase speed represents the speed at
which a constant phase front moves in space
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Electromagnetic Wave Frequency Bands
Band
Frequency
AC
10‐300 Hz
Audio
0.3‐30 kHz
RF
0.03‐300 MHz
Microwave
0.3‐30 GHz
Millimeter wave
Submillimeter wave
Far Infrared
Visible
Ultraviolet
X‐Rays
30‐300 GHz
0.3‐3 THz
1‐10 THz
Thermal Infrared
Near Infrared
Wavelength
10‐100 THz
100‐430 THz
430‐750 THz
750‐30,000 THz
1000‐30,000 km
10‐1000 km
1 m‐10 km
1‐100 cm
1‐10 mm
0.1‐1 mm
30‐300 µm
3‐30 µm
0.7‐3 µm
0.4‐0.7 µm
10‐400 nm
>30,000 THz
<10 nm
7
Electromagnetic Waves
• Assume (a) charge free region and (b) time‐
harmonic field
D  0
B  0
  E   jH
  H  E  jE
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Electromagnetic Waves
• Taking the curl of Faraday’s law (third equation)
    E   j (  H )   j   j E
• Using the identity
  (   A)    (   A)   2 A
  (  E )   2 E   j   j E
=0
Gauss’s law ‐ charge free region
 2 E  j   j E   2 E   2 E  0
9
Electromagnetic Waves
• For time‐harmonic signals, Maxwell’s equations reduce
to the Helmholtz equation in a charge free region
2E   2E  0
: 3D
2E
  2E  0
2
z
: 1D
• The complex propagation constant is given by
 2  j (  j )
• Solutions in 1D are given by
E ( z )  Ae  z  Be z
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Electromagnetic Waves
• Solution to Maxwells equations in 1D – electric field
E ( z )  Ae  z  Be z
• A similar solution may be obtained for magnetic field
H ( z )  Ce  z  De z
• Relating E and H fields using Maxwell’s equations
A
B
H ( z )  e  z  e z


where the wave impedance is given by

j
  j
11
Electromagnetic Waves
• The complex propagation constant can be written as
    j
where the attenuation constant, , is given by
 
 
2

  
1 
  1
2 

  


and the propagation constant,  , is given by
 
 
2

  

1 

1

2 
 




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Wave Polarization
• Polarization is the trace of the directional vector that
describes the electric field.
Linear
Circular
Elliptical
13
Wave Polarization
• Linear Polarization
𝑬 𝒛, 𝒕
𝑯 𝒛, 𝒕
𝑬𝒙 𝐜𝐨𝐬 𝝎𝒕
𝑬𝒙 /𝜼 𝐜𝐨𝐬 𝝎𝒕
𝜷𝒛 𝐚𝒚
𝜷𝒛 𝐚𝐱
𝑯𝟎 𝐜𝐨𝐬 𝝎𝒕
𝜷𝒛 𝐚𝒚
The electric and magnetic field components are normal to each
other and to the direction of propagation – z‐direction
• Elliptical Polarization
𝑬 𝒛, 𝒕
𝑯 𝒛, 𝒕
𝑬𝒙 𝐜𝐨𝐬 𝝎𝒕
𝜷𝒛 𝐚𝐱
𝑬𝒚 𝐜𝐨𝐬 𝝎𝒕
𝜷𝒛
𝑬𝒙 /𝜼 𝐜𝐨𝐬 𝝎𝒕
𝜷𝒛 𝐚𝒚
𝑬𝒚 /𝜼 𝐜𝐨𝐬 𝝎𝒕
𝝅/𝟐 𝐚𝒚
𝜷𝒛
𝝅/𝟐 𝐚𝒙
The two field components are 90 out of phase
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EM Waves in Lossless Dielectrics
• In lossless dielectrics, the conductivity σ = 0, and the fields are
given by
𝑨𝒆 𝜶𝒛 𝒆 𝒋𝜷𝒛 𝑩𝒆𝜶𝒛 𝒆𝒋𝜷𝒛
𝑬 𝒛
𝑨 𝜶𝒛 𝒋𝜷𝒛 𝑩 𝜶𝒛 𝒋𝜷𝒛
𝑯 𝒛
𝒆 𝒆
𝒆 𝒆
𝜼
𝜼
 0
   
&
• The wave impedance is purely real and given by



E

x
z
H = E/
y
Plane Wave
• Air is generally treated as a lossless dielectric with  and 
given by the free space values 0 and 0 .
15
EM Waves in Good Conductors
• In good conductors, σ   ,
𝑨𝒆 𝜶𝒛 𝒆 𝒋𝜷𝒛 𝑩𝒆𝜶𝒛 𝒆𝒋𝜷𝒛
𝑨 𝜶𝒛 𝒋𝜷𝒛 𝑩 𝜶𝒛 𝒋𝜷𝒛
𝑯 𝒛
𝒆 𝒆
𝒆 𝒆
𝜼
𝜼
•  and  are given by
𝑨𝒆 𝒛/𝜹 𝒆 𝒋𝜷𝒛 𝑩𝒆𝒛/𝜹 𝒆𝒋𝜷𝒛
𝑨 𝒛/𝜹 𝒋𝜷𝒛 𝑩 𝒛/𝜹 𝒋𝜷𝒛
𝒆
𝒆
𝒆 𝒆
𝜼
𝜼
𝑬 𝒛
  

2
• The intrinsic impedance is given by
 
j 

• The skin depth is defined as
 
1

𝟐
𝝎𝝁𝝈
which represents the depth of penetration into the medium
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Wave Propagation Properties
Lossless Medium
Wave Property
𝜶
Complex Propagation
Constant
𝜸 𝜶 𝒋𝜷
𝜷
Wavelength
Skin Depth
Phase Speed
Intrinsic Impedance
Lossy Medium
𝟎
 
𝝎 𝝁𝜺
 
𝟐𝝅
𝜷
𝝀
 
2 

2

  
1 
  1

  

2

 
 1  
  1
2 

  


 
Good Conductor
𝟏
→∞
𝜶
𝒖
𝝎
𝜷
𝟏
𝝁𝜺
𝜼
𝝁
𝜺
𝜹
𝟏
𝜶
𝟐
𝝎𝝁𝝈
𝝎
𝜷
𝒖
𝝎
𝜷
𝟐𝝎
𝝁𝝈
𝒖

2
𝟐𝝅
𝜷
𝟏
𝜹
𝜶
𝝀
𝜹

  
j
  j
𝝀

𝟐𝝅
𝜷
j

17
Wave Propagation
Interference due to the wave nature of signals
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Wave Propagation & Refraction
Wave Refracting and Propagating
in Lossless Media (e.g. dielectric)
Lossless
Media
Wave Refracting and Propagating
in Lossy Media (e.g. conductor)
Lossy
Media
Boundary between media
19
Power and Poynting Vector
• Over any closed volume,

1
1

 ( E  H )  dS   t   2 E  2 H  dv    E dv
S
Total power leaving the volume
2
2
V
Rate of change of stored energy
in electric and magnetic fields
2
v
Dissipated power
• The vector quantity
P  EH
is known as the Poynting vector and
represents the power density leaving the
volume. The direction is given by the direction
of the vector. Unit of P is W/m2.
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Power and Poynting Vector
• Average Power Density of Plane Waves
2
E
1
1
2
P  E H 
 H
2
2 2
• Total Power
𝑷𝑻
∮ 𝑷 · 𝒅𝑺
P

PT  P  A sin 
A
21
Boundary Conditions
• Tangential Fields
(1) Tangential electric field is continuous
nˆ  ( E1  E 2 )  0 or
Et 1  Et 2
(2) Tangential magnetic field is discontinuous
nˆ  ( H 1  H 2 )  J s
or
H 1t  H 2 t  J s
At a dielectric/dielectric interface, the tangential magnetic
field will be continuous, because Js = 0
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Boundary Conditions
• Normal Fields
(1) Normal electric flux is discontinuous
nˆ  ( D1  D2 )   S
D1n  D2 n   S
or
(2) Normal magnetic flux is continuous
nˆ  ( B1  B2 )  0 or
B1 n  B2 n = 0
At a dielectric/dielectric interface, the normal electric flux
will be continuous, because s = 0
23
Reflection and Transmission
• Normal Incidence – Lossless Media
E i ( z )  aˆ x E i 0 e  j 1 z
x
E r ( z )  aˆ x E r 0 e  j 1 z  aˆ x  E i 0 e  j 1 z
E t ( z )  aˆ x E t 0 e  j 2 z  aˆ xE i 0 e  j 2 z
H i  aˆ y
Ei0
1
H r   aˆ y
H t  aˆ y
e
 j 1 z
E i 0
1
e  j 1 z
E i 0  j z
e
2
Ei
z
Hi o
Et
Er
o
Ht

Hr
μ1 1
μ2 2
2
Plane Wave Incidence
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Reflection and Transmission
• Normal Incidence
Apply boundary conditions @ z = 0
nˆ  ( E i  E r )  nˆ  E t
nˆ  ( H i  H r )  nˆ  H t
1  

1
(1   ) 
Reflection Coefficient

1
2
Er
H
  1
 r  2
Ei
H i  2  1

Transmission Coefficient
Et 2 H t
2 2

 1  
E i 1 H i
 2  1
25
Reflection and Transmission
• Normal Incidence ‐ Power


E
1
S i  Re E i  H i*  aˆ z i 0
2
21
2
E i 0
1
S r  Re E r  H r*   aˆ z
2
21


2
2
  aˆ z  S i
2
E i 0
1
2
S t  Re E t  H t*  aˆ z
 aˆ z (1   ) S i
2
2 2


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Reflection and Transmission
• Normal Incidence – Standing Waves
In medium 1, the total electric field is
E ( z )  E i ( z )  E r ( z )  aˆ x E i 0 ( e  j 1 z   e j 1 z )
The SWR is given by
SWR 
E max
E min
When μ1 = μ2

i
j
SWR 
1 
1 
i   j
27
Reflection and Transmission
• Oblique Incidence – Perpendicular Pol. (Horizontal)
x
E i  aˆ y E i 0 e  j 1 ( x sin  i  z cos  i )
z
E r  aˆ y  E i 0 e  j 1 ( x sin  r  z cos  r )
Hr
E t  aˆ yE i 0 e  j 2 ( x sin  t  z cos  t )
H i ( z )  (  aˆ x cos  i  aˆ z sin  i )
H r ( z )  ( aˆ x cos  r  aˆ z sin  r )
Ei0
1
E i 0
e  j 1 ( x sin  i  z cos  i )
e
 j 1 ( x sin  r  z cos  r )
1
E
H t ( z )  (  aˆ x cos  t  aˆ z sin  t ) i 0 e  j ( x sin   z cos  )
2
2
t
r o Er
i
Et
Ei
Ht
Hi
μ1 1
μ2 2
o
o
t
t
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Reflection and Transmission
• Oblique Incidence
Apply boundary conditions @ z = 0
nˆ  ( E i  E r )  nˆ  E t
nˆ  ( H i  H r )  nˆ  H t
e  j 1 x sin  i   e  j 1 x sin  r  e  j 2 x sin  t
 cos  e

1
i
 j 1 x sin  i

 cos  r  e  j 1 x sin  r   cos  t
1
  j x sin 
e
2
2
t
2 Complex Equations – 4 Real Unknowns
29
Reflection and Transmission
• Oblique Incidence
r  i
 1 sin  i   2 sin  t
 2 cos  i  1 cos  t
 2 cos  i  1 cos  t
2 2 cos  i

 2 cos  i  1 cos  t

Z 
Ey
Hx


cos 
Z 1 
1
cos  i
Z 2 
2
cos  t
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Reflection and Transmission
• Oblique Incidence – Parallel Polarization (Vertical Pol)
E i  ( aˆ x cos  i  aˆ z sin  i ) E i 0 e  j 1 ( x sin  i  z cos  i )
E r  ( aˆ x cos  r  aˆ z sin  r ) E i 0 e  j 1 ( x sin  r  z cos  r )
x
E t  ( aˆ x cos  t  aˆ z sin  t )E i 0 e  j 2 ( x sin  t  z cos t )
H i  aˆ y
Ei0
1
H r   aˆ y
H t  aˆ y
e
 j 1 ( x sin  i  z cos  i )
E i 0
1
z
Er
Et
 r  Hr
o
Ei i
o
Ht
μ1 1
μ2 2
e  j 1 ( x sin  r  z cos  r )
Hi
E i 0  j ( x sin   z cos  )
e
2
2
t
t
t
31
Reflection and Transmission
• Oblique Incidence
Apply boundary conditions @ z = 0
nˆ  ( E i  E r )  nˆ  E t
nˆ  ( H i  H r )  nˆ  H t
cos  i e  j 1 x sin  i  cos  r  e  j 1 x sin  r  cos  te  j 2 x sin  t
e

1
1
 j 1 x sin  i
  e
  e  j 1 x sin  r 
 j 2 x sin  t
2
2 Complex Equations – 4 Real Unknowns
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Reflection and Transmission
• Oblique Incidence
r  i
 1 sin  i   2 sin  t
 2 cos  t  1 cos  i
 2 cos  t  1 cos  i
2 2 cos  i

 2 cos  t  1 cos  i

Z ||  
Ex
  cos 
Hy
Z 1||  1 cos  i
Z 2||   2 cos  t
33
Reflection and Transmission
• Total Transmission – Brewster Angle
‐ Perpendicular Polarization
 cos  i  1 cos  t
 2
0
 2 cos  i  1 cos  t
sin  i 
 2 2

 1 1
1  2

 2 1
 2 1

 1 2
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Reflection and Transmission
• Total Transmission – Brewster Angle
‐ Parallel Polarization

 2 cos  t  1 cos  i
0
 2 cos  t  1 cos  i
sin  i 
 2 2

 1 1
 2 1

1  2
If 1   2
 1 2

 2 1
2
1
tan  B 
35
Reflection and Transmission
• Critical Angle
 1 sin  i   2 sin  t
1 1 sin  i   2 2 sin  t
  2 2 




1 1 

Total internal
reflection when
 i   c  sin 1 
1   2
 i   c  sin 1 
 2 



 1
36
18
2/3/2025
Reflection and Transmission
• Reflection Coefficient:
Lossless Medium
Lossy Medium
37
Reflection and Transmission
Wave Refraction
Boundary between
two media
38
19