Lecture 19
Superposition, interference,
standing waves
Today’s Topics:
• Principle of superposition
• Constructive and Destructive Interference
• Diffraction
• Beats
• Standing Waves
The principle of linear
superposition
When two or more waves are present simultaneously at the same place, the
resultant disturbance is the sum of the disturbances from the individual waves.
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Constructive and Destructive
Interference
When two waves always meet
condensation-to-rarefaction,
they are said to be exactly out
of phase and to exhibit
destructive interference.
When two sound waves always meet
condensation-to-condensation and
rarefaction-to-rarefaction, they are
said to be exactly in phase and
to exhibit constructive interference.
Conditions for Constructive and
Destructive Interference
For two wave sources vibrating in phase, a difference in path lengths that
is z ero or an integer number (1, 2, 3, . . ) of wavelengths leads to constructive
interference; a difference in path lengths that is a half-integer number
(½ , 1 ½, 2 ½, . .) of wavelengths leads to destructive interference.
d1
d2
Example -- Interference
A speaker generates a continuous tone of 440 Hz. In the drawing, sound travels
into a tube that splits into two segments, one longer than the other. The sound
waves recombine before being detected by a microphone. The speed of sound in
air is 339 m/s. What is the minimum difference in the lengths of the two paths for
sound travel if the waves arrive in phase at the microphone?
(a) 0.10 m (b) 0.39 m (c) 0.77 m (d) 1.11 m (e) 1.54 m
For constructive interference,
the minimum path difference = λ
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Standing waves
In reflecting from the wall, a
forward-traveling half-cycle
becomes a backward-traveling
half-cycle that is inverted.
The incoming and reflected cycles
can enhance or diminish another,
depending on the timing.
Repeated incoming and reflected
cycles causes a large amplitude
standing wave to develop.
Standing waves on a vibrating string
occur at well-defined frequencies
Integer numbers of ½ wavelengths are allowed
String fixed at both ends
⎛ v ⎞
f n = n⎜ ⎟
⎝ 2L ⎠
n = 1, 2, 3, 4,!
λ
Longitudinal Standing Waves
The basis for “wind” instruments!
Integer numbers of ½ wavelengths are allowed
Tube open at both ends
⎛ v ⎞
f n = n⎜ ⎟
⎝ 2L ⎠
n = 1, 2, 3, 4,!
3
Or only one end open
Only odd numbers of ¼ wavelengths are allowed
Tube open at one end
⎛ v ⎞
f n = n⎜ ⎟
⎝ 4L ⎠
n = 1, 3, 5,!
Example – Standing Waves
A rope of length L is clamped at both ends. Which one of the following is not a
possible wavelength for standing waves on this rope?
(a) L/2 (b) 2L/3 (c) L (d) 2L (e) 4L
L
⇒ L = 2λ
2
2L
3λ
b )λ =
⇒L=
3
2
c )λ = L ⇒ L = λ
a )λ =
d )λ = 2 L ⇒ L =
e)λ = 4 L ⇒ L =
λ
2
λ
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Example – Standing Waves
A string with a linear density of 0.035 kg/m and a mass of 0.014 kg is clamped
at both ends. Under what tension in the string will it have a fundamental
frequency of 110 Hz?
(a) 270 N (b) 410 N (c) 550 N (d) 680 N (e) 790 N
⎛ v ⎞
f n = n⎜ ⎟
⎝ 2L ⎠
n = 1, 2, 3, 4,!
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So we can change the frequency of a standing wave
on a string by changing the tension in the string.
How do we change
the frequency (e.g.
pitch) of a sound in
a resonant cavity?
Sound travels
through gases,
liquids, and solids
at considerably
different speeds.
Beats
Let ’s move from the spatial domain to the time domain
What happens when two
tones with slightly different
frequencies interfere?
The beat frequency is the difference between the two sound
frequencies.
Example - Beats
Two timpani (tunable drums) are played at the same time. One is correctly tuned
so that when it is struck, sound is produced with wavelength of 2.20 m. The
second produces sound with a wavelength of 2.08 m. If the speed of sound is
343 m/s, what beat frequency is heard?
(a) 7 Hz
(b) 9 Hz (c) 11 Hz (d)13 Hz (e) 15 Hz
Beats = f1 − f 2
343 m/s
= 165 Hz
2.08 m
v 343 m/s
f2 = =
= 156 Hz
λ 2.20 m
Beats = 165 Hz − 156 Hz
f1 =
v
λ
=
Beats = 9 Hz
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