Slide 1Dr. Stuart Long
3-2What is a “wave” ?
Wave – a mechanism by which a disturbance is propagated from one place
to another .....
water, heat, sound, gravity, and EM (radio, light, microwaves, UV, IR)
Notice how the media itself (UH fans in this case) is NOT propagated
Animation courtesy of Dr. Dan Russell, Kettering University
GO COUGARS!!!
=
= −
x v t
Unique
∂ ∂′′ ′′= = ∂ ∂
ω ∂ + =
ω∂ ∂
Linear medium
(1-dim. case)
ˆ try soln of form [ ]
Dispersion Relation
−=
+ =
⇒
−
= =
E
x
{ } 0ˆ ( , ) Re cos( ) j tE z t e E t kzω ω= = −E
x
∇ ∇ ∇ ∇ ∇
as a function of time
tω
Ex
angular frequency ω=2πf
2 π π 2π3
3-7
∴ =
= =
Electric field as a function of z at different times0E cos( )x E t kzω= −
0 pi/2 pi 3pi/2 2pi -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
kz
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
( )
( )
ππ
3-9Quick Review
• The wave spatially repeats at a point z = λ where kλ = 2π.
• The quantity λ , where , is called the wavelength.
• The number of wavelengths contained in a spatial distribution of 2π is given by and it is called the wavenumber.
• The velocity of the peak of the wave (position of constant phase) requires that wt –kz = [a constant], (let’s call that constant zero). So the velocity of propagation is then given by
• The velocity in free space is given by
3-10
8
v f k
[m] Not
= =
≈=
Also, remember that the orientation of the E field of a uniform plane electromagnetic wave is perpendicular to the H field for that wave, and that both are perpendicular to the direction in which the wave propagates
Thus far we have a number of useful definitions:
3-11
ˆ
r
x
x
0
0Recall
Where the field of a uniform plane wave is given by
The magnetic
Uniform Plane Waves
Waves whose constant phase fronts are planar (plane waves) and whose amplitude (E0 ) is
uniform
z
y
http://phys23p.sl.psu.edu/phys_anim/EM/indexer_EMC.html
3-12
Similarly
ˆ ˆ( , ) Re cos( )
t kz
x
y
z
http://phys23p.sl.psu.edu/phys_anim/EM/indexer_EMC.html
3-13
Source Freq.[Hz] Freq. (common units) Wavelength [m] Wavelength (common units)
U.S AC Power 60 60 Hz 5x106 5000 Km
ELF Submarine Communications 500 500 Hz 6x105 600 Km
AM radio 106 1000 Hz 300 300 m
CB radio 2.7x107 27 MHz 11 11 m
Early Cordless phone 4.9x107 49 MHz 6.1 6.1 m
TV ch. 2 (digital) 5.4x107 54 MHz 5.5 5 m
FM radio 108 100 MHz 3 3 m
TV ch. 8 (digital) 1.8x108 180 MHz 1.7 1.7 m
UHF Aircraft Comm. 5x108 500 MHz .6 60 cm
TV ch. 39 6.2x108 620 MHz .48 48 cm
Early Cell phone 8.7x108 870 MHz .34 34 cm
μ-wave oven 2.45x109 2.45 GHz .12 12 cm
"C" band 6x109 6 GHz .05 5 cm
Police radar (X-band) 1.05x1010 10.5 GHz .0285 2.85 cm
mm wave 1011 100 GHz .003 3 mm
He-Ne Laser 4.7x1014 470 THz 6.3x 10-7 6300 Å
Light 1015 1 PHz 3x10-7 3000 Å
X-ray 1018 1 EHz 3x10-7 3 Å
The Electromagnetic Spectrum
3-14The Electromagnetic Spectrum
3-16
3-17
The polarization of a wave is described by the locus of the tip of the E vector as time progresses at a fixed
point in space.
If locus is a circle the wave is said to be
Circularly Polarized
If locus is a straight line the wave is said to be Linearly Polarized
If locus is an ellipse the wave is said to be
Elliptically Polarized
Polarization
3-18
If locus is a straight line the wave is said to be Linearly Polarized
x
y
z
http://phys23p.sl.psu.edu/phys_anim/EM/indexer_EMC.html
Polarization
3-19
If locus is a circle the wave is said to be
Circularly Polarized x
y
z
http://phys23p.sl.psu.edu/phys_anim/EM/indexer_EMC.html
Polarization
3-20
If locus is an ellipse the wave is said to be
Elliptically Polarized x
Consider a plane wave propagating in the positive z direction.
The associated electric field can be expressed in the form of
cos( ) cos( )
x a
y b
ω φ ω φ
where the two components are, in general terms,
The complex representation is given can be expressed by
Polarization
( - )( - )- -ˆ ˆa bj kzj kzae be φφ= +E x y
3-22
os si
φ φ φ
A B C D E F aba b
φ φ
where
n
lips
os
e
1
A B C D E F aba b
B AC
φ φ
φ φ
E E EE a ab b
E E E EE E a ab a
b
yx y x
= ⇒ =
rization adius " ")
a
2 cos sin
x y yx E E EE a ab b
a a
φ φ − + =
Consider a plane wave propagating in the positive z direction.
The associated electric field can be expressed in the form of
cos( ) cos( )
x a
y b
ω φ ω φ
= − +
= − +
ˆ ˆx yE E= +E x y where the two components are, in general terms,
Polarization
0 cos( )E t kzω= −E
The polarization of this plane wave is determined by the quantity
y
x
< <
− < <
If field is traveling in the , or direction can be found respectively by
ˆ ˆ ˆ
z z y y x x
A
φ
∠
∠∠ ∠ ∠ ∠ ∠
t t kz t kz
jx y e
3-32Polarization Example
3-33
at y
x z
0tω =
LHEP
( ) ( ) ( ) ( )( ) ( ) ( )( )
( ) ( ) ( ) ( )
-
2 3
j x j
z e
ω ω
3-35
z
z
(c)
c
c
0
m
where conduction current ; conductivity
j
In a source free dissipative medium Ampere’s Law states
As derived earlier, the wave equation is given by
As we have seen, ε is complex for a dissipative medium.
~

µη ε=
3-38
The wave number and the intrinsic impedance can also be written as
The electromagnetic fields of a uniform plane wave in a dissipative
k -
e
3-39
0
0
written as
k zI
)
φ
ω
From the electromagnetic fields we can observe that
1) The in the direction with a velocity
where is now the wavenumber.
wave tr
2) The
k
ω =
z
exponentially at the rate nepers per meter, where is the attenuation constant.
3) The magnetic field is
attenuated
-The attenuation in nepers after length d is given by
-The relationship between dB and nepers is given by
1[neper] = 8.686[dB]
3-42Example
[ ]F
I
1) The electric field is decreased by a factor of 0.707. Find the attenuation in nepers and dB
Eln ln 0.707 0.3467 [nepers] E
dB 0.3467[nepers] 8.6
= = −

If dealing with power u
E 20log E
P 10log P
E = 0.707E
then
]
= =
⇒
−
Where for a conducting
σω με j ωωε
<< If then
3-46
Good Conductor 1( ) σ ωε
>>
3-47
Behavior of k I and k R as a Function of Loss Tangent
0
100
200
300
k I o
r k R
)
Exact kr Exact ki Good Conductor appx. kr=ki Good Dielectric appx. kr Good Dielectric appx. ki
0 0
81 4
mho ; ; m
= = =
σ ωε
Behavior of kI and kR as a Function of σ ωε
3-48
electric field can exits
values of σ approximate
Silver 6.2 10 mho/m Copper 5.8 10 mho/m Gold
σ
7
7
7
4.1 10 mho/m Aluminum 3.8 10 mho/m Brass 1.5 10 mho/m Solder
σ σ σ
Graphite 7 10 mho/m Silicon
σ σ
σ σ
σ σ σ
σ σ σ

Can dissipate energy in oscillations of bound charge in a dielectric.
Lossy Dielectrics
0
Can define an effective conductivity Same effect as but from a different source
Table gives
′′=
′′ = =
′ ′
′ =
→ →
03 Steak 40 0.3
Phase Lag caused by bound charge not “keeping up” with E Field
3-50
The skin effect is the tendency of an alternating electric current to distribute itself within a conductor so that the current density near the surface of the conductor is greater than that at its core. That is, the electric current tends to flow at the "skin" of the conductor.
0
0
Since
For EM a
; Plasma freq.
For
1
ω
p
= −
=
Plasma is a collection of (+) and (-) charged particles for which <ρv>=0
Plane Waves in a Plasma
2
; Plasma freq.
For
1
ω
p
p 1
2 2
Then
and
α
α
ω
dissipated
http://www.optoiq.com/
3-53
The phase velocity is the speed of the individual wave crests, whereas the group velocity is the speed of the wave packet as a whole (the envelope).
In this case, the phase velocity is greater than the group velocity.
http://www.geneseo.edu/~freeman
Phase Velocity =
Group Velocity =
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