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.

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


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



3-19

If locus is an ellipse the wave is said to be

Elliptically Polarized



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


0 cos( )E t kzω= −E

3-21

Polarization Polarization ClassificationClassification

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



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


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

+ + + + + =

= = − = = = = −


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

⇒ =

⎡ ⎤⎡ ⎤= − ⎢ ⎥⎢ ⎥ −⎣ ⎦ ⎣ ⎦


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


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∠ = ⇒∠


3-31

y

-

y

z

z


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


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


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


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


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


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


( )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⇒


3-53


http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/littlewavepackets.gif

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.

Documents