Chapter 7The superposition of waves

Phys 322Lecture 18

Principle of superposition

2

2

22

2

2

2

2

2 1tzyx

vWave equation:

n

iii trCtr

1

,,

If i are solutions of the wave equation, then their linear combination is also a solution

Principle of superposition: resultant disturbance at any point is is the sum of the separate constituent waves.

For E and B fields it stems from definition: these are forces, and the resultant force is a vector sum of individual forces.

Adding waves of the same frequency kxtEtxE sin, 0Consider a plane wave:

kxx,The case of two waves coexisting in space:

11011011 sincoscossinsin ttEtEE

Resulting wave: 21 EEE

tEEtEEE cossinsinsincoscos 202101202101

22012022 sincoscossinsin ttEtEE

Can simplify: tEE sin0

202101

202101

coscossinsintan

EEEE

210201202

201

20 cos2 EEEEE

If the waves are harmonic, superposition is also harmonic

Interference term

Adding waves of the same frequency tEE sin0

202101

202101

coscossinsintan

EEEE

210201202

201

20 cos2 EEEEE

Crucial factor: phase difference

12

Maximum: =0, ±2, ±4, …in-phaseConstructive interference

Minimum: =±, ±3, …out-of-phaseDestructive interference

Phase and path length difference 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,…

Phase and path length difference

Maximum: =0, ±0, ±20, …

Minimum: =±0/2, ±30/2, …x=0, ±,…x=±/2,…

Analogy: sound waves

- At any time the two waves have the same magnitude but are 180o

out of phase: complete destructive interference

speakers

‘Noise canceling earphonesuse interference principle

No sound!

Application

Superposition of many waves

Superposition of any number of harmonic waves having given frequency and traveling in the same direction leads to a harmonic wave of that frequency

N

iii tEE

10 cos

N

iii

N

iii

E

E

10

10

cos

sintan

N

i

N

ijjiji

n

ii EEEE

1 100

1

20

20 cos2

tEE cos0

Non-coherent sources

N

i

N

ijjiji

n

ii EEEE

1 100

1

20

20 cos2

tEE cos0

what is the resulting wave amplitude?

Atoms spontaneously emit light that changes phase randomly every ~10 ns.

n

iiEE

1

20

20 For non-coherent sources intensity of the

resulting wave is equal to intensities of constituent waves.

Coherent sources in phase

N

i

N

ijjiji

n

ii EEEE

1 100

1

20

20 cos2

tEE cos0

0 ji

2

10

1 100

1

20

20 2

n

ii

N

i

N

ijji

n

ii EEEEE

For simplicity assume all sources have the same amplitude E01

20

220 iENE

Is the energy conservation law violated?

Superposition: complex representation ti

jjjeEE 0 jjj kx

N

j

tij

jeEE1

0 ti

N

j

ij eeE j

10

tii eeE 0

Can compute irradiance: *0020

ii eEeEE

Complex amplitude

N

j

ij

i jeEeE1

00

Case N=2: 212102010201

20

iiii eEeEeEeEE

tieE 0

Superposition: phasor method

)cos(2

)cos()cos()cos(

120201202

201

20

021

2022

1011

EEEEEtEEEE

tEEtEE

x

y

E01

E02

E0

2-1

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