maxwell’s equations plane waves in a vacuum

0 Bv Dt

x

D

jH

tx

B

E

Maxwell’s equations

0 Bv D

t

EεjxH o

t

HμxE o

Plane waves in a vacuum

Plane waves in a vacuum

t

HμxE o

t

xHμxEx o

t

EεxH o

Plane waves in a vacuum

t

xHμxEx o

ttE

εμ

o

o

EE 2

Plane waves in a vacuum

0t

EεμE

2

2

oo2

BvEQF HvEQ

2

2

2 L

QQE

T

LM

3

22

22

2

ML

TQ

LTL

M

Q

TL

Q

T

LQ

T

LM

2 2

2

2

2

Q

ML

TQTL

M

ML

Q

TQ

ML 2

22

31

2

T

L

02

22

too

EE

22

2

secondmeter

volts

t

E

22

metermeter

volts E

2

meter

second

oo

oo

c

1

97 10

361

104

1

c

second103 8 metersc

01

2

2

22

tc

EE

second103 8 metersc

1638 - Galileo attempts to measure the speed of light by a lantern relay between distant hilltops. He gets a very large answer.

Albert Einstein, when asked to describe radio, replied: "You see, wire telegraph is a kind of a very, very long cat. You pull his tail in New York and his head is meowing in Los Angeles. Do you understand this? And radio operates exactly the same way: you send signals here, they receive them there. The only difference is that there is no cat."

There was a young lady named Bright,

Whose speed was far faster than light;

She set out one day,

In a relative way,

And returned home the previous night.

one dimensional wave equation

yEE

0 and 0x y z

xy zEy

01

2

2

22

tc

EE

2 2y y

2 2 2

E E10

z c t

x

y zEy

one dimensional wave equation

• MOST GENERAL SOLUTION!!

• Ey = a F(z - ct) + b G(z + ct)

• = a Fb G

• where F & G are arbitrary functions that depend on the excitation

2 2y y

2 2 2

E E10

z c t

2 2y y

2 2 2

E E10

z c t

yE dF dG

a bd z d zz

2 2 22 2y

2 2 2

E d F d Ga b

z zz d d

2 2 2

2 2y2 2 2

E d F d Ga 1 b 1

z d d

ctzψ

ctzθ

yE dF dGa b

d t d tt

2 2 22 2y

2 2 2

E d F d Ga b

t tt d d

2 2 2

2 2y2 2 2

E d F d Ga c b c

t d d

2 2y y

2 2 2

E E10

z c t

ctzψ

ctzθ

• Q. E. D.

• Ey = a F(z - ct) + b G(z + ct)

• = a Fb G

• is a solution of the wave equation!!!

Ey = aF(z - ct) + bG(z + ct) = aFbG

z

Ey

t = T

z

Ey

t = 2T

2 2y y

2 2 2

E E10

z c t

Ey = aF(z - ct) + bG(z + ct)

0

-10

10z

Ey = aF(z - ct) + bG(z + ct) = aFbGLet c = 2

z - ct = z + ct = t zz 5 t

2 2y y

2 2 2

E E10

z c t

0

-10

10z

Ey = aF(z - ct) + bG(z + ct) = aFbGLet c = 2

z - ct = z + ct = t zz 5 t

2 2y y

2 2 2

E E10

z c t

perturbation transverse to the direction of

propagation“transverse waves”

perturbation transverse to the direction of

propagation“transverse waves”“trajectory”

z

t* **

*

the flood

whips and chains

“the slinky wave”

“the slinky wave”

“the slinky wave”

perturbation is in the direction of propagation

“longitudinal wave”sound waves--compression &

rarefaction of air

ion acoustic wave in a plasma

nerve propagation test for MS

nerve propagation test for MS

Pulsegenerator

general properties of linear waves

01

2

2

22

2

tcz

ctzGctzFtz ,

wavesnonlinearccif ,,,, chaossolitonsshocks

magneticandelectric kineticandpotential

soliton - soliton collision

Twinkie shortage looms March 21, 2000PORTLAND, Maine (AP) - A Teamsters strike against a bakery company is creating shortage of Twinkies across parts of the Northeast. A Philadelphia bakery was shut down after pickets appeared Monday, and Teamsters planned to set up pickets in Buffalo, N.Y., as well. A New York City bakery also was shut down over the weekend. "It's a wave that's going to go right across the country," said Richard Volpe, director for the union in the United States and Canada.