The Principle of Linear
Superposition and
Standing Waves
The principle of linear
superposition
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Simulation #25, Ch. 17, Cutnell
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.
Superposition of 2 pulses
The drawing shows a graph of 2 pulses traveling toward each other at t =
0s. Each pulse has a constant speed of v = 1 cm/s. When t = 3.0s, what
is the amplitude of the resultant pulse at a) x = 5 cm?
a. +2 cm
b. -2cm c. -3 cm
d. 0 cm
17.2 Constructive and Destructive Interference of Sound Waves
Pilots use noise-canceling headsets like the ones
shown below. The headset “hears” the exterior
cockpit noise, which is of fairly consistent frequency,
and emits a sound wave that is exactly out of phase.
Sounds transmitted through the headset, such as
radio transmissions, or music, are not “canceled”.
Two identical waves coming from two sources
• Path length (d1) between observer and left speaker
equals path length (d2) between observer and right
speaker
• Δd = 0, so observer hears wave in phase, resulting
in constructive interference.
•
Constuctive Interference
For two wave sources
vibrating in phase, a
difference in path lengths
that is zero or an integer
number such that:
Δd = mλ (m=0,1,2….)
leads to constructive
interference, and a sound
wave will sound louder:
Two identical waves coming from two sources
• Path length (d1) and (d2) between observer and
right speaker differ by a half wavelength.
• Δd = .5λ, so observer hears wave out of phase,
resulting in destructive interference.
•
Destructive Interference
For two wave sources
vibrating in phase, a
difference in path lengths
that is a half-integer
number of wavelengths,
such that
Δd = (m +1/2)λ
(m=0,1,2….)
leads to destructive
interference.
Wave interference logic
1. Find values for Δd and λ
2. Compare Δd and λ as
follows: x = Δd /λ
3. IF x = m (integer), THEN
constructive.
4. IF x = (m + ½), THEN
destructive.
5. IF x ≠ m AND x ≠ (m + ½),
THEN neither constructive
nor destructive.
6. If the problem asks for the
smallest wavelength, set
m = 0 in either 3 or 4
above.
This logic is only for 2
sources which are
vibrating in phase.
Speakers are usually
set up to vibrate in
phase.
What Does a Listener Hear?
Two in-phase loudspeakers, A and
B, are separated by 3.20 m. A
man is listening at point C, as
shown. Both speakers are playing
identical 142.9-Hz tones. The
speed of sound is 343 m/s.
Does the listener hear
A. Constructive interference,
B. Destructive interference
C. Neither
Two overlapping waves with slightly different
frequencies gives rise to the phenomena of beats.
The beat frequency is the difference between the two
sound
frequencies:
fbeat = | f1 - f2 |
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Beats
Two pure tones are sounded together. The drawing shows
the pressure variations of the two sound waves,
measured with respect to atmospheric pressure, where
t1 = 0.0206 s and t2 = 0.0250 s. What is the beat
frequency in Hz?
A. .0044
B. 40
C. 48.54
D. 8.54
http://www.walter-fendt.de/ph14e/stwaverefl.htm
Uncheck “resultant standing wave” (3rd box)
1.How long is λ in in terms of string length L?
λ = ___ L
2.Once the standing wave is completely
generated, can you find at least one place
along the string where the cumulative wave
amplitude is always 0? Where?
http://www.walter-fendt.de/ph14e/stwaverefl.htm
Now check “resultant standing wave” (3rd
box).
4. What happens at a node? Antinode?
5. How far apart are the nodes?
6. What is the largest possible λ(in terms of
L for a standing wave to occur?
λfundamental = ___ L
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Or google Cutnell Johnson 8th edition, click
on student companion site, concept
simulations, ch. 17 # 27.
7. Wave speed is 4 m/s. String length is 1 m
8. Start at f=4 Hz and work your way down
to 1Hz. Do standing waves occur at that
frequency?
9. Re-answer #6:
λfundamental = ___ L
Standing Waves on a String
For a string of fixed length L, the conditions can be
satisfied only if the wavelength has one of the values
A standing wave can exist on the string only if its
wavelength is one of the values shown above.
Because λf = v, the frequency corresponding to
wavelength λm is
Transverse standing wave patterns
m/L
Standing Sound Waves
• A long, narrow column of air, such as the air in a
tube or pipe, can support a longitudinal standing
sound wave.
• sound is a pressure wave rather than a
displacement wave. The pressure oscillates around its
equilibrium value.
• Musical instruments use longitudinal standing waves
in tubes that are either open at both ends (e.g. flute,
organ) or open at one end (e.g. clarinet, trumpet).
•An open end acts as an antinode (maximum
vibration) while a closed end acts as a node, just like
for a wave on a string.
Tube open at both ends
This is the same as
for a transverse
standing wave on a
string, except that the
antinodes are at the
ends.
 v 
f n  m 
 2L 
m  1, 2, 3, 4, 
The length of an organ pipe
vsound = 343 m/s
An organ pipe is open at
both ends
The length of an organ pipe
Check your understanding
A tube is found to have standing waves at
390, 520 and 650 Hz and no frequencies
in between. The behavior of the tube at
frequencies higher/lower than that was
not tested. The speed of sound is 343
m/s. How long is the tube, in meters?
a. 1.76
b. 1.32
c. 0.879 d. 0.440
Tube open at one end
• The open end is an antinode and the closed end is a
node.
•The distance between node and antinode is λ/4.
• Therefore the tube length (L) must be equal to an odd
number of quarter wavelengths:
 v 
f m  m 
 4L 
m  1, 3, 5, 
The notes on a clarinet in a
tropical country
The notes on a clarinet
The notes on a clarinet