Frans Pretorius
University of Alberta
동일 진동수 빛의 중첩
E ( x, t )  Eo sin(t  (kx   ))  Eo sin(t   )
두 빛의 중첩( 더하기)
E1  Eo1 sin(t  1 )  Eo1ei (t 1 )
E2  Eo 2 sin(t   2 )  Eo 2ei (t 2 )
E  E1  E2
 Eo1ei (t 1 )  Eo 2ei (t 2 )
I  EE*   Eo1ei (t 1 )  Eo 2 ei (t  2 )  Eo1e  i (t 1 )  Eo 2e  i (t  2 ) 
 E012  E022  E01 E02 ei (1  2 )  e  i (1  2 ) 
 E012  E022  2 E01 E02 cos(1   2 )
 E012  E022  2 E01 E02 cos( k ( x1  x2 ))
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
2
I  E012  E022  2 E01 E02 cos(k ( x1  x2 ))
두 빛이 더해진 밝기의 표현 식
계산을 간단히 하기 위해서 E01=E02라고 가정하자. 그리고 E012=E022=Io라 하자
I  I 0  I 0  2 I 0 cos( k ( x1  x2 ))
 2 I 0 (1  cos(k ( x1  x2 ))
 k ( x1  x2 ) 
 4 I 0 cos 2 

2


PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
3
Standing waves
 Standing waves are waves that look stationary, but have an amplitude that
changes with time. Several situations can produce standing waves, including
 the superposition of left and right moving waves on a string
 the “natural” modes of vibration of a string fixed at both ends (stringed instruments
work like this)
 sound waves in a tube open at one or both ends (wind instruments work likes this)
 sustained 40mph winds set up standing waves in the Tacoma Narrows Bridge in 1940,
causing it to collapse:
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
4
Standing waves on a string fixed at both
ends
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
5
정상파 (Standing
서로 반대 방향으로 진행하는, 같은 진동수의 빛이
만나면 제자리에서 진동하는 파동이 생긴다.
마디와 마시 사이 거리는 파장의 절반이다.
antinode
공명 (resonance)
node
광물리학
빛의 중첩, 회절 간섭
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Beats
 When two tones of similar frequency f1 and f2 are
added together, interference will create what is
called a beat frequency at the difference between
the two frequencies : fb=f1-f2
Example:
A 200hz tone:
A 200hz + 201hz tone:
(beat frequency is 1 hz … this is
a 5 second sample, so we should
hear ~5 beat cycles)
A 200hz + 210hz tone:
(beat frequency is 10 hz …should
hear ~50 beat cycles)
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
7
Interference of two waves sources
vibrating in phase
S1
S2
 Two wave sources, S1 and S2, are
emitting waves in phase, and of exactly
the same frequency and amplitude.
Consider a point p that is a distance d1
from source 1, and a distance d2 from
source 2.
 If
p
| d1  d2 | nl
where n is a non-negative integer and l
is the wavelength, then p will be a point
of complete constructive interference
 If
1

| d1  d 2 |  n  l
2

then p will be a point of complete
destructive interference
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
8
Example A
 Consider the
configuration of
loudspeakers and
listener shown to the
right. Assume both
loudspeakers are
playing the exact
same music. What set
of frequencies will
the listener not be
able to hear at all?
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
Image
courtesy
John Wiley
& Sons, Inc.
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Diffraction
diffraction is the bending of a wave as
it moves past edges or obstacles
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
10
Single Slit Diffraction
 With single slit diffraction, we have a
sound wave of wavelength l passing
through an opening of width D. On the
other side of the opening there will be
interference between parts of the
wave, and at an angle given by
sin  
l
D
there will be complete destructive
interference (the so-called firstminimum)
 Note: the above formula only works if
D>> l
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
11
Standing waves on a string fixed at both
ends
 Since both ends of the string are fixed, the only
possible set of wavelengths are
ln  2 L / n
 n=1 gives the first or fundamental harmonic
 n=2 gives the second harmonic or first overtone, n=3 the
third harmonic or second overtone, etc.
 Given the relationship lf=v, the set of frequencies
corresponding to these wavelengths are
 v 
f n  n 
 2L 
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
12
Standing waves in a tube
 A resonance can be used to set up standing
sound waves in a tube
 this is a longitudinal standing wave (compared to
the transverse standing wave on a string)
 If both ends are open, the possible set of
natural frequencies are (as with the string) :
 v 
f n  n 
 2L 
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
13
Standing waves in a tube
If only one end is open, the following
set of resonant frequencies are possible:
 v 
f n  n 
 4L 
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
14
Example B: (ch 17, prob. 44)
A tube, open at one end, is cut into two
shorter, unequal length pieces. The
piece that is open at one end has a
fundamental frequency of 675hz, while
the piece that is open at both ends has
a fundamental frequency of 425hz.
What was the fundamental frequency of
the original tube?
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
15
Example B
 Answer: 162Hz
PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition
and Interference
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