Chapter 3 Uniform Plane Waves - University of Houstoncourses.egr.uh.edu/ECE/ECE3317/SectionLong/docs/EMII_CH3... · 2006. 8. 21. · 3-1 Chapter 3 Uniform Plane Waves Dr. Stuart Long.
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3-1
Chapter 3 Chapter 3 Uniform Plane WavesUniform Plane Waves
Dr. Stuart Long
3-2
What is a “wave” ?
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 is NOT propagated
3-3
2 2
2 2 2
1 ( , ) 0
(
Given
A solutio
n
Unique solution depends on
,0)
( )
1( , ) [ ( )
physical pr
( ]
o
)2
p x tx v t
p x f x
p x t f x vt f x vt
⎡ ⎤∂ ∂− =⎢ ⎥
∂ ∂⎢ ⎥⎣ ⎦
=
= − + +
2 22
2 2
2 2
2 2
time harm
( , ) ( , )
onic case
blem
p x t f p x t v fx t
jt
x v
ω
ω
∂ ∂′′ ′′= =∂ ∂
∂∂
∂+
∂
⇒
( ) 0p x⎡ ⎤
=⎢ ⎥⎢ ⎥⎣ ⎦
One Dimensional Wave Equation
3-4
0 0
Source Free
Time Harmonic case Time dependent
Linear medium
;
0
;
0
e
v
j t
ρ
ω
μ ε
= =
=
⇒
=
⇒
⇒
J
B H D E
0
0
00
jj
ω μω ε
× = −
× ===
E HH E
H E
∇
∇∇∇ii
Maxwell’s Equations
3-5
2
20 0
20(
Vector Identity ( ) ( )
)
)
(
0 (
)j
j j
ωμ
ωμ ωε
− ×
× × = −
= −
=− −
E E E
H E E
E E
∇ ∇ ∇ ∇ ∇
∇ ∇∇∇
∇
i
i
2 20 0
20
2
2 0
Wave equation for
ˆ for E and E ( )
0
E
E
x x
xx
E
z
z
ω μ ε
ω μ ε
+
=
+ =∂
=
∂
E EE∇
x
200 0
02
(1-dim. case)
ˆ try soln of form [ ]
0
0
jkzE eEk ω μ ε
−=
+ =−
E x
{ } 0
2 20 0 Dispersion Relation
ˆ ( , ) Re cos( )
j tE z t e E t kz
k
ω ω
ω μ ε
= = −
=
E x
3-6
0 pi/2 pi-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x component of the electric field at z=0 as
a function of time
tω
Ex Tω
pi 2
0E cos( )x E tω=
periodic in time period T
spatially repeating every wavelength λ
angular frequency ω=2πf
2π π 2π3
2π
3-7
zvt k
λ ωπ
ω
Δ=
Δ2= =
Electric field as a function ofz 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
tω π=Tω
z
E x
π 2π32π
2π
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
2t πω =E x
z
π 2π32π
2π0 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 0tω = E x
z
π 2π32π
2π
kπλ =
2
3-8
zvt k
λ ωπ
ω
Δ=
Δ2= =
0E cos( )x E t kzω= −
2π32π
2π
4π 3
4π π 7
4π5
4π0
1−
1
z
Electric field as a function ofz at different times
0
3-9
- The wave spacially repeats at point where .
- The quantity , where is called the wavelenght.
- The number of wavelengths contained in a spatial distribution of
22
is given 2 y b
z k
k
λ λ ππλ λ
π
= =
=
and it is called the wavenumber.
- The velocity of the peak of the wave (position of constant phase) requires that so the velocity of propagation is
given by
2
- constant
k
t kzz vt k
πλ
ωω
=
=∂
= =∂
8
0 0 0 0
[m/sec]
[m/sec]
- The velocity in free space is given by
1 3 10 vkω ω
ω μ ε μ ε= = = ≈ ×
Quick Review
3-10
8
[sec] m/sec]
[rad]
[
[m/s ec]
Period Phase Velocity
Angular Velocity in
1T
2 c 3 10 Frequency
vf k
f
ω
ω π
= =
= ≈ ×
0 0[Hz] [1 m]
[m] Not
e:
free space
Frequency Waven1 T
2
umber
Wa 3velength [ ] [ ] 0GHz c
f
fk
m
k ω μ ε
πλ λ
= =
≈= i
Also, remember that the orientation of the E field of a uniform plane electromagnetic wave is perpendicular to the H field of that wave and that both are perpendicular to the direction from which the wave propagates
So far we have come across some useful expressions such as:
3-11
ˆ field is in direction ˆ field is in direc tion
E H
xy
ˆ Wave propagating in dire on cti+ z
0
0
Recall
Where the field of a uniform plane wave is given by
The magnetic fi
eld is the
-
ˆ
n
jk z
j
E e
ωμ
−
∇ × =
=
E H
E
E x
0
0ˆ
jk zE e
η
−= H y
Uniform Plane Waves
Waves with constant phase fronts (plane waves) and whose
amplitude (E0 ) is uniform
http://www.elec.york.ac.uk/cpd/img/em-wave.png
3-12
FPermittivity m
HPermeability m
90
70
1 10 36
4 0
1
−
−
⎡ ⎤⎣ ⎦
⎦× ⎡ ⎤= ⎣
≈ ×επ
μ π
{ } 0
0
0 0
0
00
0[Ohms]
Or in the time domain
Similarly
Where the is the intrinsic impedance of
ˆ ˆ( , ) Re cos( )
ˆ ˆ( , ) Re co
free sp
s(
ac
)
e
= 120 377
j t
j t
z t e E t kz
Eez t t kz
ω
ω
ω
ωη η
η
μη πε
= = −
⎧ ⎫⎪ ⎪= = −⎨ ⎬⎪ ⎪⎩ ⎭
= ≈
E
E
E x x
H y y
http://www.elec.york.ac.uk/cpd/img/em-wave.png
3-13The Electromagnetic Spectrum
3 Å3x10-71018X-ray
3000 Å3x10-71015Light
6300 Å6.3x 10-74.7x1014He-Ne Laser
3 mm.003100 GHz1011mm wave
2.85 cm.028510.5 GHz1.05x1010Police radar
5 cm.056 GHz6x109"C" band
12 cm.122.45 GHz2.45x109μ-wave oven
34 cm.34870 MHz8.7x108Cellular phone
48 cm.48620 MHz6.2x108TV ch. 39
60 cm.6500 MHz5x108UHF Aircraft Comm.
1.7 m1.7180 MHz1.8x108TV ch. 8
3 m3100 MHz108FM radio
5 m560 MHz6x107TV ch. 2
6.1 m6.149 MHz4.9x107Cordless phone
11 m1127 MHz2.7x107CB radio
300 m3001000 Hz106AM radio
600 Km6x105500 Hz500ELF Subm. Comm.
5000 Km5x10660 Hz60U.S A-C Power
Wavelength (common units)Wavelength [m]Freq. (common units)Freq.[Hz]Source
3-14
http://www.impression5.org/solarenergy/misc/emspectrum.html
The Electromagnetic Spectrum
3-15The Electromagnetic Spectrum
3-16
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 beLinearly Polarized
If locus is an ellipse the wave is said to be
Elliptically Polarized
PolarizationPolarization
3-17
If locus is a straight line the wave is said to beLinearly Polarized
PolarizationPolarization
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/imgpho/pollin.gif
3-18
If locus is a circle the wave is said to be
Circularly Polarized
PolarizationPolarization
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/imgpho/pollin.gif
3-19
If locus is an ellipse the wave is said to be
Elliptically Polarized
PolarizationPolarization
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/imgpho/pollin.gif
3-20
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
E a t kzE b t kz
ω φω φ
= − +
= − +
y
x
EA
Eφ= ∠
ˆ ˆx yE E= +E x y
where the two components are, in general terms,
The polarization of this plane wave is determined by the quantity
Where
a | |
=| |
yb
x
E bA
E aφ φ φ= = −
and
PolarizationPolarization
0 cos( )E t kzω= −E
3-21
Polarization Polarization ClassificationClassification
Linear Polarization (LP)
Linear Polarization (LP)
Left-Hand Circular Polarization (
LHCP)
A 0 0 or
A
A 1
;
; 2
A 1
;
φ π
πφ
φ
= = ±
→ ∞
= =
= = − Right-Hand Circular Polarization (RHCP)
Left-Hand Elliptical Polarization (LHEP)
Right-Hand Elliptical Polarization (RHEP)
2
0
0
π
φ π
π φ
< <
− < <
If field is traveling in the , ordirection can be found respectively by
ˆ ˆ ˆ
or or
y yx x z z
z z y y x x
A
EE EE E E
φ
φφ φφ φ φ
∠
∠∠ ∠∠ ∠ ∠
positiveE y x z
3-22
Linear Polarization (LP)
Linear Polarization (LP)
Left-Hand Circular Polarization (
LHCP)
A 0 0 or
A
A 1
;
; 2
A 1
;
φ π
πφ
φ
= = ±
→ ∞
= =
= = − Right-Hand Circular Polarization (RHCP)
Left-Hand Elliptical Polarization (LHEP)
Right-Hand Elliptical Polarization (RHEP)
2
0
0
π
φ π
π φ
< <
− < <
If field is traveling in the , ordirection can be found respectively by
ˆ ˆ ˆ
or or y z x yz x
x z z y y x
A
E EEE E E
φ
φ φφφ φ φ
∠
∠ ∠∠∠ ∠ ∠
negativE e y x zPolarization Polarization ClassificationClassification
3-23
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
E a t kzE b t kz
ω φω φ
= − +
= − +
ˆ ˆx yE E= +E x y
where the two components are, in general terms,
The complex representation is given can be expressed by
PolarizationPolarization
0 cos( )E t kzω= −E
( - )( - )- -ˆ ˆa bj kzj kzae be φφ= +E x y
3-24
222
0 0
cos
cos( )
Look at and ;
Recall that the general quadratic equation is given by
2 cos
sin
b a
x
y b
x y yx
z
E a t
E b t
E E EEa ab b
φ φ φ
ω
ω φ
φ φ
=
=
= +
⎛ ⎞ ⎛ ⎞⎛ ⎞ − + =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
= =
2 2
22 2
0
1 2cos 1
0 0 si
where
; ; ; ; ; n
Ax Bxy cy Dx Ey F
A B C D E Faba b
φ φ
+ + + + + =
= = − = = = = −
PolarizationPolarization
3-25
( )
2 2
22 2
2
22 2
2 2 2 2
0
1 2cos 1 ; ; ; 0 ; 0 ; si
where
If this becomes equ of an ellipse
n
4 0
2 1 1 4cos 4 cos 0
r t
1
o
Ax Bxy cy Dx Ey F
A B C D E Faba b
B AC
ab a b a b
φ φ
φ φ
+ + + + + =
= = − = = = = −
− <
⎛ ⎞⎛ ⎞⎛ ⎞− − = − ≤⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
2 2
-cot 2
1 1
ated b
c
y an a
ot 2
ngle
os
2c
A CB
aba b
θ θ
θφ
⇒ =
⎡ ⎤⎡ ⎤= − ⎢ ⎥⎢ ⎥ −⎣ ⎦ ⎣ ⎦
PolarizationPolarization
3-26
222
22
0
2 cos sin
Let ( )
Lin
2 0 0
ear polar
izatio
x y yx
x y y yx x
yxy x
E E EEa ab b
E E E EE Ea ab
or
b a b
EE bE Ea b a
φ φ π
φ φ
= =
⎛ ⎞ ⎛ ⎞⎛ ⎞ − + =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ − + = ⇒ − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
= ⇒ =
n (line of slope )b a
ExampleExample
3-27
222
22
2
2 cos sin
Let ;
Circular polarization (circle of
radius
1
" ")
x y yx
yx
a b
E E EEa ab b
EEa a
a
πφ
φ φ
= =
⎛ ⎞ ⎛ ⎞⎛ ⎞ − + =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞⎛ ⎞ + =⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
ExampleExample
3-28
222
22
Let ;
Elliptical polarization (equ of an ellipse with
2 2
2 cos sin
12
major radius =
2
x y yx
yx
b a
E E EEa ab b
EEa a
a
πφ
φ φ
= =
⎛ ⎞ ⎛ ⎞⎛ ⎞ − + =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞⎛ ⎞ + =⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
and minor radius )a=
ExampleExample
3-29
PolarizationPolarizationExampleExample
x
-
x
y
y
Find the polarization of the following field:
E (1 90
1
ˆ ˆ(a) ( )
)
( 90 )E 1(1
| |9
E (1 )
E (1 ) 10 |) |
90
jkz
A
A
kz
kzkz
kz
y
kz z
jx e
k
RHCP
φ
φ
= +
= =
= ∠
∠
∠
= ∠ −
=
= ∠ − +
− +∠ −
∠
∠
+
−
⇒−
−−
E
3-30
z
x
z
-
x
Find the polarization of the following field:
ˆ ˆ(b) ((2 ) (3 ) )
E ( 10 18.4349 )
18| | ( )|
.4349E 10 |
E ( 5 26.5651 )
E ( 5 26.5651 ) 5 26( 10 18.434
.56 19
5)
jky
A
j x
ky
ky
j
k
z
ky
kyky
e
y
A
φ
= ∠− +
∠− +
= + + −
= = = ∠ −
= ∠− −
−−
−∠ −−
+∠
E
1 452
LHEPφ∠ = ⇒∠
PolarizationPolarizationExampleExample
3-31
y
-
y
z
z
Find the polarization of the following field:
E ( 2 45
E ( 2 45)
E ( 2
)
ˆ ˆ(c) ((1 ) (1 ) )
| | ( )| |
1 90
45E 2( 2 45
45 ) 2 5)
4
jkx
kx
kx kxA
k
j y j z e
x
kxk
A
x
RHCP
φ
φ
= ∠ − −
∠
= + +
= ∠ − +
− −
−
= = = ∠ −
⇒
∠ −∠
∠ =
+
− −−
∠ −
−
E
PolarizationPolarizationExampleExample
3-32
Plane Waves in Dissipative Media
c
c
0
m
For isotropic conductors Ohm's Law states that
where conduction current ; conductivity
source current
Consequentl
⎡ ⎤⎣
= σ
⎦σ
J E
J
J
c 0
0
y Ampere's Law becomes
Where Compl
j
j j
j
ω
ω εω
εω
∇× = + +
σ⎡ ⎤∇× = − +⎢ ⎥⎣ ⎦
σ−=
H D J J
H E J
ε ex Permittivity
3-33
( )
( )2 2
In a source free conducting medium Ampere's Law states
As derived earlier, the wave equation is given by
=0
0
jω
ω μ
∇ × =
∇ + =
J
H E
E
ε
ε
2 2
As we have seen, is complex for a conducting medium.
The wave number and the intrinsic impedance are
n
Not
ow
e:
μω μ= =
complex numbers.
ε
k ε ; ε η
Plane Waves in Dissipative Media
3-34
The wave number and the intrinsic impedance can also be written as
The electromagnetic fields of a uniform plane wave in a dissipative
-
e
R I
j
k jk
φ
=
=
k
η η
0
0
medium are
ˆ
ˆ
given by
j z
j z
xE e
E ey
−
−
=
=
k
k
E
Hη
Plane Waves in Dissipative Media
3-35
0
0
The electro
magnetic fields can also be
written as
ˆ
ˆ
k z jk zI R
k z jI
xE e e
E e ey
− −
− −
=
=
E
H
( )
( )
0
0
e
( , ) cos
cos (
Or in the time domain
, )
k z jR
k zIx R
k zIR
y
E z t E e t k z
E e t k zH z t
φ
ω
ω φ
−
−
= −
− −=
η
η
Plane Waves in Dissipative Media
3-36
From the electromagnetic fields we can observe that
1) The wave travels in the direction with a velocity
where is called the wavenumber.
2) The a
ˆ+
mplitude
vR
R
kk
ω=
z
is attenuated exponentially at the rate nepers per meter, where is the attenuation constant.
3) The magnetic field is out of pha
se by
.
I I
y
k k
H φ
Plane Waves in Dissipative Media
3-37
( )
start
end
-0-
0
One neper attenuation if
The attenuation in nepers after l
Amplitudeln 1Amplitude
d
Attenuation[nepers] ln
enght is given by
The relationship between
I
I
k z
Ik z d
E e k dE e +
⎡ ⎤=⎢ ⎥
⎣ ⎦
⎡ ⎤= =⎢ ⎥
⎣ ⎦
nepers and dB is given by
[nep 1 8.6er] [dB
86 ]
=
Attenuation
3-38
Example
[ ]F
I
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] 8.686 3.01 [dB]neper
or
s
⎡ ⎤= = −⎢ ⎥
⎣ ⎦
⎡ ⎤− = −⎢ ⎥
⎣ ⎦i
( )F
I
E 20log 20log 0.707 3.01 [dB]
E
⎛ ⎞= = −⎜ ⎟
⎝ ⎠
3-39
Note on dB Scale
F
I
F
I
2
If dealing with electric field use
If dealing with power u
E 20logE
P 10logP
se
This is because
when
P
~
E
⎡ ⎤⎢ ⎥⎣ ⎦
⎡ ⎤⎢ ⎥⎣ ⎦
[ ] [ ]
F I
2F I F I
E = 0.707E
P = 0.707 P P = 0.5P
then
20log 0.707 10log 0.5 3.01 [dB
]
= =
⇒
−
3-40
( ) ( )z 0
I
The penetration depth ( ) such that is given by
Where for a conducting
1
=
media
-
1
11
pp z d
p
R
de
j
k d
σμε j k kωε
σω με j ωωε
==⎛ ⎞= ⎜ ⎟⎝ ⎠
⎡ ⎤⎢ ⎥⎣ ⎦
=
= −− =k
E E
Keep in mind that
I
1
If thena >>1
12
( )aja j+ ≈ +
General Medium
orIf th en
1 1a< <
2
1aja j+ ≈ +
3-41
Good Dielectric
-
( ) 1
;
1 2
2;
2
jR I I
p R
σωε
σω με j ωε
σk k k
d k ω μεσ
με
εμ
<<
⎡ ⎤⎢ ⎥⎣ ⎦
≈ −
= =
= =
k
k
Slightly Conducting Media
3-42
( )
Good Conductor( )
12
-
;
1
2
2 2
;
μ j
jR
σω
I I
p R
ε
σ
σσ
ω
k k k
d k
σ
ω μ
ω μδω μ
>>
−
≡
≈
= =
= =
k
k
Highly Conducting Media
Also called the skin depth δ
3-43
Behavior of k I and k R as a Function of Loss Tangent
0
100
200
300
4000.1 1.0 10.0 100.0
kI o
r kR
(1/m
)
Exact krExact kiGood Conductor appx. kr=ki
Good Dielectric appx. krGood Dielectric appx. ki
0 0
8 1 4
m h o ; ;
m
S e a w a te rμ μ ε ε σ ⎡ ⎤
⎢ ⎥⎣ ⎦= = =
σωε
3-44
Conductors
3-45 OHM'S LAW
σ σ ∞⇒ →=J E
A perfect conductor is an idealized material in which no
electric field can exits
Ordinary metal with very high
values of σapproximate
“perfect”conductors
[ ][ ][ ]
20
7
7
Superconductive lead 2.7 10 mho/m
Silver 6.2 10 mho/mCopper 5.8 10 mho/mGold
σ
σσ
= ×
= ×= ×
[ ][ ]
[ ]
7
7
7
4.1 10 mho/mAluminum 3.8 10 mho/mBrass 1.5 10 mho/mSolder
σσ
σ
= ×= ×
= ×
[ ][ ]
[ ]
7
7
4
0.7 10 mho/mStainless steel 0.1 10 mho/m
Graphite 7 10 mho/mSilicon
σσ
σ
= ×= ×
= ×
[ ][ ][ ]
3
4
1.2 10 mho/mSea water 4 mho/m
Distilled water 2 10 mho/mSandy soil
σσ
σ −
= ×=
= ×
[ ][ ][ ]
5
6
9
10 mho/mGranite 10 mho/mBakelite 10 mho/mDiamond
σσσ
−
−
−
===
[ ][ ][ ]
13
16
17
2 10 mho/mPolystyrene 10 mho/mQuartz 10 mho/m
σσσ
−
−
−
= ×==
“Good” Conductor
3-46 j ε ε
=′ ′′= −
D Eεε
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
[ tan ]
ta
Ic
n
e
e
e
r
σ ωεσ
σεδε ωε
ε ε δε
′′=
′′= =
′ ′
′=
4.2 0.1 Dry soil 2.8 0.07Distilled water 80 0.04Nylon 4 0.01Teflon 2 0.0003Glass 4 7 0.0002Dry wood 1.5 4 0.01Styrofoam 1.03 0.000
→→
03Steak 40 0.3
Phase Lag caused by
bound charge not “keeping
up” with E Field
3-47
The skin effect is the tendency of an alternating electric current to distribute itself within a conductorso 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
Current is exponentially damped into material
For EM a
w ves
ˆ
ˆ
I R
I R
k z jk z
k z jk z
E e e σ
E e eσ
− −
− −
= =
=
E J E
J
x
x
http://www.ee.surrey.ac.uk/Workshop/advice/coils/power_loss.html
Skin Effect
3-48
2
0 0 p2
p
For low density plasma (few collisions)
; Plasma freq.
"Cold Plasma"Note: is a function of Dispersive medium
For
1
pωμ μ ε ε ω
ω
ε ω
ω ω
→⎡ ⎤⎢ ⎥= = −⎢ ⎥⎣ ⎦
>
⇒
12 2
0 0 2 k 1
p
vk
ωω μ ε
ω
ω
⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦
=
Plasma is a collection of (+) and (-) charged particles for which <ρv>=0
Plane Waves in a Plasma
3-49
Plane Waves in a Plasma
p1
2 2
0 0 2
0 0
00
1
For the wavenumber becomes imaginary
Then
and
Since and are both imagin
ˆ ˆ
ˆ
1 R2
ary
p
j z z
z
j j
(z) xE e xE e
(z) y E ej
α
α
ω ω
ωα ω μ ε
ω
αωμ
− −
−
<
⎡ ⎤⎢ ⎥= − = − −⎢ ⎥⎣ ⎦
= =
=
=
k
k
E
H
E H
s
e 0⎡ ⎤× =⎣ ⎦E H*
Evanescent Waves
Attenuation occurs but no real power is
dissipated
3-50
The phase velocityis 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/animations/phaseani_comp.avi
Phase vs. Group Velocity
3-51
Phase vs. Group Velocity
( )0
1 0 2 0
1 0 2 0
Consider a plane wave propagating in the + direction
with two frequencies and
and with wavenumbers and
F
ˆ
, c
or
o
s(x t) E t kx
k k k k k k
ω
ω ω ω ω ω ω
= −
= − Δ = + Δ
= − Δ = + Δ
x
E
( )
( )
( ) ( ){ }
1 0 0 0
2 0 0 0
0 0 0 0 0
For
Sum to get t
cos ( ) ( )
cos ( ) ( )
,
otal field
cos ( ) ( ) cos ( )
( )total
E t k k x
E t k k x
(x t) E t k k x t k k x
ω ω ω
ω ω ω
ω ω ω ω
− Δ − − Δ
+ Δ − + Δ
= − Δ
⇒
− − Δ + + Δ − + Δ
⇒
E
3-52
( ) ( )
st
0 0 0
00 0
0
Using trig identities
The 2 cosine factors give a slow variation superimposed over a more rapid one
Constant phase on rapid (1 cos) term
consta
, 2 cos cos
nt
total(x t) E t k x t kx
xt k x vt k
ω ω
ωδωδ
= − Δ
− =⇒
− Δ
= =
E
nd
Phase Velocity
Constant argument on 2 slower variation
consta
nt Group Ve
locity
p
gxt kx vt k k
δ ω δωωδ δ
ΔΔ − Δ = = = =
Δ⇒
Phase vs. Group Velocity
3-53
Phase vs. Group Velocity
http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/littlewavepackets.gif
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