Transcript Physics 102 - Qatar University

Physics 102
Superposition
Lecture 7
Interference
Moza M. Al-Rabban
Professor of Physics
[email protected]
Interference
The combination, or superposition, of waves is
often called interference.
A standing wave is the interference pattern
produced when two waves of equal frequency travel
in opposite directions.
Now, we will look at the interference of two waves
traveling in the same direction
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Interference in One Dimension
Overlapping light waves and overlapping sound waves both obey the
principle of superposition, and so both show the effects of interference.
We will assume that the waves are sinusoidal, have the same frequency
and amplitude, and travel to the right along the x-axis.
D1 ( x, t )  a sin(kx1  t  10 )  a sin 1
D2 ( x, t )  a sin(kx2  t  20 )  a sin 2
source
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4
Note > Perfect destructive interference occurs
only if the two waves have equal amplitudes.
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The Phase Difference
1  kx1  t  10
2  kx2  t  20
  1  2  (kx1  t  10 )  (kx2  t  20 )
 k ( x1  x2 )  (10  20 )  2
Path-length difference
x

 0
Inherent phase difference
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The Phase Difference
The condition of being in phase, where crests are
aligned with crests and troughs with troughs, is that
 = 0, 2, 4, or any integer multiple of 2.
For constructive interference:
  2
x

 0  2m or
 x 0


m
2
 2
For identical sources, 0 = 0 rad , maximum constructive
interference occurs when x = m ,
Two identical sources produce maximum constructive
interference when the path-length difference is an
integer number of wavelengths.
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Two identical sources produce maximum constructive interference
when the path-length difference is an integer number of
wavelengths.
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The Phase Difference
The condition of being out of phase, where crests are
aligned with troughs of other, that is,
 =, 3, 5 or any odd multiple of .
For destructive interference:
x
  2
 0  2(m  12 ) or

 x 0


 m  12
2
 2
For identical sources, 0 = 0 rad , maximum constructive
interference occurs when x = (m+ ½ ) ,
Two identical sources produce perfect destructive
interference when the path-length difference is
half-integer number of wavelengths.
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Constructive and
Destructive Interference
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Destructive Interference
Three Ways
For destructive interference:
  2
x

 0  2(m  12 ) or
 x 0


 m  12
2
 2
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Example 8: Interference Between Two Sound Waves
You are standing in front of two side-by-side
loudspeakers playing sounds at the same frequency.
Initially there is almost no sound at all. Then speaker 2
is moved slowly back, and the intensity increases until it
is moved back 0.75 m. As speaker 2 continues to move
back, the sound begins to decrease.
What is the distance at which the sound intensity is
again a minimum?
SOLVE:
A minimum sound intensity implies that the two
sound waves are interfering destructively.
Initially the loudspeakers are side-by-side, so
x  0 and
   rad
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Moving one of the speakers does not change

But it does change the path-length difference.
Constructive interference condition is,
  2
x
   2

 x   / 2
x

   2 rad
   1.50 m
The next point of destructive
interference, with m=1,
occurs when
  2
x2

 0  2
x2

   2(m  12 )  3
x2    1.50 m
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Clicker Question 1
Two loudspeakers emit sound waves with =2.0 m with the phases
indicated in the figure. Speaker 2 is 1.0 m in front of Speaker 1.
What, if anything, should be done to arrange for constructive
interference between the two waves?
(a) Move Speaker 1 forward 1.0 m;
(b) Move Speaker 1 forward 0.5 m;
(c) Move Speaker 1 back 1.0 m;
(d) Move Speaker 1 back 0.5 m;
(e) Nothing. Constructive interference is already present.
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The Mathematics of Interference
sin   sin   2 cos  12 (   )  sin  12 (   ) 
D  D1  D2
 a sin  kx1  t  10   a  sin kx2  t  20 
 a sin 1  a sin 2  a  sin 1  sin 2 

 

D  2a cos  sin kxavg  t   avg
2 

A  2a cos


 2a
2

A  2a if cos
 1, i.e.,   2m
2
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Example: More Interference
of Sound Waves
Two loudspeakers emit 500 Hz sound waves with amplitudes of
a=0.10 mm. Speaker 2 is 1.00 m behind Speaker 1, and the phase
difference between the two speakers is 900.
What is the amplitude of sound waves at a point 2.00 m in front of
Speaker 1?

A  2a cos
 2a
2
v 343 m/s
 
 0.686 m
f
500 Hz
(1.00 m) 
  10.73 rad

(0.686 m) 2
10.73 rad
A  2(0.100 mm) cos
 0.121 mm
2
  2
x
 0  2
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Interference in 2D and 3D
A  2a cos
  2
r


2
 0
Constructive Interference:
r
  2
D(r, t )  a sin  kr  t  0 
D  D1  D2
 a sin  kr1  t  10 
 a sin  kr2  t  20 
The motion of the waves does not affect the points
of constructive and destructive interference.
r  m

 0  2m
Destructive Interference:
r
  2
 0  2 m  12 





r  m  12 
m  0,1, 2,
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Example11: 2D Interference between Loudspeakers
Two loudspeakers are 2.0 m apart and
produce 700 Hz sound waves in phase, in a
room where the speed of sound is 341 m/s.
A listener stands 5.0 m from one
loudspeaker and 2.0 m to one side of
center.
Is the interference there constructive,
destructive, or something in between?
How will this result differ if the
speakers 1800 are out of phase?
v (341 m/s)
 
 0.487 m
f
(700 Hz)
r

r1  (5.0 m) 2 +(1.0 m) 2  5.10 m
r2  (5.0 m) 2 +(3.0 m) 2  5.83 m
r  (5.83 m)  (5.10 m)  0.73 m

(0.73 m)
 1.5
(0.487 m)
Destructive interference if 0=0;
Constructive interference if 0=.
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Visualizing Interference
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Example: Intensity of Two
Interfering Loudspeakers
Two loudspeakers are 6.0 m apart and produce
in-phase equal amplitude sound waves with a
wavelength of 1.0 m. Each speaker alone
produces an intensity I0. An observer at point A
is 10.0 m in front of the plane containing the
speakers on their line of symmetry. A second
observer at point B is 10.o m directly in front of
one of the speakers.
In terms of I0, what is the intensity IA at
point A and IB at point B?
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2
rB  (10.0 m) +(6.0 m)  10.0 m  1.662 m
rA  0 by symmetry, so A =0.

AA  2cos A  2a
2
I A  cAA2  c(2a)2  4ca2  4I0
B  2
rB
 2
(1.662 m)
 10.44 rad
(1.00 m)


AB  2cos B  2a cos(5.22 rad)  0.972a
2
I B  cAB2  c(0.972a)2  0.95ca2  0.95I0
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Clicker Question 3
Two loudspeakers emit equal-amplitude
sound waves in phase with a wavelength of
1.0 m.
At the point indicated, the
interference is:
(a) Constructive;
(b) Perfect destructive;
(c) Something in between;
(d) Cannot tell without knowing
the speaker separation.
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Beats and Modulation
If you listen to two sounds with very different frequencies, you
hear two distinct tones.
But if the frequency difference is very small, just
one or two Hz, then you hear a single tone whose
intensity is modulated once or twice every second.
That is, the sound goes up and down in volume, loud,
soft, loud, soft, ……, making a distinctive sound
pattern called beats.
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Beats and Modulation
The periodically varying amplitude is
called a modulation of the wave.
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For
1. The two waves have the same amplitude a,
2. A detector is located at the origin (x=0),
3. The two sources are in phase,
4. The source phase happen to be
1  2   rad
The modulation frequency is:
mod 
1
1  2 
2
And the beat frequency is,
f beat  2 f mod
mod
1  1 2 
2
 2. 

  f1  f 2
2
2  2 2 
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Example 13: Listening to Beats
One flutist plays a note of 510 Hz while a second plays a note of 512
Hz. What frequency do you hear? What is the beat frequency?
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Graphical Beats
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Clicker Question 1
You hear three beats per second when two sound tones
are generated. The frequency of one tone is 610 Hz.
The frequency of the other tone is:
(a)
(b)
(c)
(d)
(e)
(f)
604 Hz;
607 Hz;
613 Hz;
616 Hz;
Either (a) or (d)
Either (b) or (c)
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Chapter 21 - Summary (1)
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Chapter 21 - Summary (2)
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Chapter 21 - Summary (3)
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End of Lecture 7