Chapter 7The superposition of waves

Phys 322Lecture 19

Superposition of waves of the same frequencyStanding wavesSuperposition of waves of different frequencyPhase vs. group velocity

Superposition of waves of same frequency tEE sin0

202101

202101

coscossinsintan

EEEE

210201202

201

20 cos2 EEEEE

12

Maximum: =0, ±0, ±20, …

Minimum: =±0/2, ±30/2, …

kxx,

2211 kxkx

21212 xx

If the coherent waves are initially in phase (1- 2=0), then:

210

2122 xxnxx

Optical path difference: 21 xxn

Coherent waves: 1- 2=constant

x=0, ±,…x=±/2,…

Example: problem 7.6

Superposition of many waves:

N

iii

N

iii

N

ij

N

ijiji

N

ii

N

iii

E

E

EEEE

tEtEE

10

10

100

1

20

20

01

0

cos

sintan

)cos(2

)cos()cos(

1) Random phase (incoherent light)

) (suppose 010201

1

20

20 EENEEE i

N

ii

2) Uniform phase (coherent and in-phase)

) (suppose

2

010

201

2

100

1

20

20

EE

ENEEEE

i

N

ij

N

iji

N

ii

The interference of coherent waves only redistribute the energy in space, it cannot change the total amount of energy.

Standing wave

In general: txgCtxfCtx vv 21,two waves traveling in opposite direction

Consider 2 waves, incident and reflected:

III tkxEE sin0

RRR tkxEE sin0

RII tkxtkxEE sinsin0

2

cos2

sin2

sinsin

02 sin cos2 2

I R R IIE E kx t

Can select x origin and t=0 so that: tkxEE I cossin2 0

(Typically E=0 on the surface of a metal mirror)

Standing wave tkxEE I cossin2 0

Animation courtesy of Dr. Dan Russell, Kettering University

Standing wave and resonance

If the number of /2 is integer in example above, the string can oscillate forever (if there are no losses) - resonance.

Standing electromagnetic wave1890 - Otto Wiener experiment

Where is the energy when E is zero?

tkxEE cossin2 0

tkxBB sincos2 0 tB

xE

see problem 7.11

Standing wave: microwave

f=2.5 GHz=12 cm

Why is microwave dish designed to spin?How could you measure the wavelength of microwaves?

1946-invention of microwave oven

Superposition of waves of different frequency

2 txkEE 11011 cos

txkEE 22012 cos

assume the same amplitude and phase:

+

= txktxkEE 221101 coscos

2cos

2cos2

coscos

2

cos2

cos2 2121212101

txkktxkkEE

221

221

m

average angular frequency

modulation angular frequency

221 kkk

average propagation number

221 kkkm

modulation propagation number

txktxkEE mm coscos2 01

Superposition of waves of different frequency

txktxkEE mm coscos2 01 221

221

m

221 kkk

221 kkkm

If 1 2, then m

slowly changing amplitude, E0

The irradiance:

2 2 20 01( , ) 4 cos m mE x t E k x t

2 20 01( , ) 2 1 cos 2 2m mE x t E k x t

txktxkEE mm coscos2 01

2 20 01( , ) 2 1 cos 2 2m mE x t E k x t

Superposition of waves of different frequency

Animation courtesy of Dr. Dan Russell, Kettering University

beat frequency: 2m=1 - 2

Group velocity

txktxkEE mm coscos2 01

Amplitude modulation (envelope)

carrier wave

Animation courtesy of Dr. Dan Russell, Kettering University

Group velocity

Phase velocity in non-dispersive medium: v=/kconstant

dispersive medium: v=v()

txktxkEE mm coscos2 01

The speed at which the modulation envelope moves differs from that of individual waves

carrier wave

t

x

xt

//

v

txk carrier:

txk mm modulation:

k/v

mmg k/v

group velocity

Phase (red dot) vs. group (green dot) velocity

Group velocity

carrier: k/v

mmg k/vgroup velocity21

21

kk

For dispersive media: = (), or = (k) (k=2/)

If frequencies are close:

kgv

The modulation signal propagates at speed vg that may be greater than, equal to, or less than v, the velocity of the carrier

dispersion relationship

Any wave is finite in time and space - superposition of many waves is needed to create such a pulsePropagation of these ‘pulses’ is described by vg.

Group index of refraction

kgv

knkkc

nkkc

knkc

g/11)/(v

Non-dispersive media: n = const

=kv =kc/n(k)

kn

nkc

nc 2

11

kn

nkc

2v

kn

nk

g 1vv

c/ng vv

Group index of refraction: gg cn v/

SuperPos-GroupVelocity-no-dispersion.py

Near absorption band:anomalous dispersion

Group velocity and dispersion

kn

nk

g 1vv

gg cn v/ Normal dispersion

0/ knvv g

k=2/

vv gnng

SuperPos-GroupVelocity-anomalous_despersion.py

SuperPos-GroupVelocity-normal_despersion.pynng

0/ kn k=2/v