phys 322 chapter 7 lecture 18 the superposition of waves
TRANSCRIPT
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Chapter 7The superposition of waves
Phys 322Lecture 18
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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.
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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
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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
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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,…
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Phase and path length difference
Maximum: =0, ±0, ±20, …
Minimum: =±0/2, ±30/2, …x=0, ±,…x=±/2,…
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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!
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Application
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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
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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.
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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?
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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
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Superposition: phasor method
)cos(2
)cos()cos()cos(
120201202
201
20
021
2022
1011
EEEEEtEEEE
tEEtEE
x
y
E01
E02
E0
2-1
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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.
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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)
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Standing wave tkxEE I cossin2 0
Animation courtesy of Dr. Dan Russell, Kettering University
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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.
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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
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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