Ch16Lectures
Thursday, April 16, 2009
12:22 PM
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The following animation illustrates the interference of
two wave pulses travelling in opposite directions:
http://physics.info/interference/
The following animation illustrates the interference of
two waves travelling in opposite directions to produce
a standing wave:
http://www.physicsclassroom.com/mmedia/waves/swf.cfm
The following animation illustrates more complex
examples of interference for transverse waves:
http://serc.carleton.edu/NAGTWorkshops/deepearth/
activities/40826.html
The following animation contrasts travelling waves
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The following animation contrasts travelling waves
and standing waves:
http://physics.info/waves-standing/
The following animation illustrates interference of
waves travelling in two dimensions (such as waves on
the surface of water):
http://phet.colorado.edu/en/simulation/wave-interference
The following pages illustrate reflection of a wave from a
boundary and the creation of standing waves:
http://www.acs.psu.edu/drussell/Demos/reflect/reflect.ht
ml
http://www.animations.physics.unsw.edu.au/jw/waves_s
uperposition_reflection.htm#reflections
http://www.animations.physics.unsw.edu.au/jw/waves_s
uperposition_reflection.htm#superposition
The following page illustrates standing waves:
http://en.wikipedia.org/wiki/Standing_wave
Video clip of standing waves on a string (captions in French):
http://www.youtube.com/watch?v=4BoeATJk7dg
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Transmission and reflection of waves at boundaries
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Fundamental frequency and higher harmonics
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Standing sound waves in tubes
The previous discussion was about standing waves on
strings, which are relevant for musical instruments
such as guitars or pianos. However, wind instruments
make sounds via standing sound waves, and for them
there are more possibilities. For simplicity, model a
wind instrument (flute, trumpet, etc.) by a straight
tube filled with air. The tube could be open at both
ends (which approximates a flute), open at one end
and closed at one end (which approximates a
trumpet), or closed at both ends (no musical
instrument is like this, but the textbook includes this
case for completeness). The standing waves on such
simple tubes can be modeled as follows, where the
amplitude of the wave represents pressure:
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OMIT Sections 16.5 AND 16.7
Interference of Waves from Two Sources
The basic simplifying assumption in this section is that
the two sources are in phase.
• interference at points along the same line as the line
joining the sources
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• interference at other points
Interference at points along the same line as the line
joining the sources is illustrated in the following two
figures. Note that the resulting interference pattern
depends on the distance between the two sources.
(Also note that the two sources have the same
amplitude and are in phase.)
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In two or three dimensions, the pattern is more
complex, as shown by the following figures:
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(Remember, we make the usual simplifying
assumptions that the sources are in phase and have
the same frequency.)
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Selected problems and solutions
CQ 1 Light can pass easily through air and water, but
light can reflect from the surface of a lake. What does
this tell you about the speed of light in air and in
water?
Solution: The fact that light can reflect from the
boundary of air and water shows that the speed of
light in the two media is different. (This is like the
story of a wave travelling across the boundary
between a thick string and a thin string, with some of
the wave transmitted across the boundary, and some
of the wave reflected.)
CQ 4 A guitarist finds that the frequency of one of
her strings is too low. Should she increase or decrease
the tension in the string? Explain.
Solution: The frequency is proportional to the speed
of the wave on the string, but the speed of the wave
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of the wave on the string, but the speed of the wave
is proportional to the square root of the tension.
Thus, the tension should be increased.
CP 6 You are holding one end of an elastic cord that
is fastened to a wall 3.0 m away. You begin shaking
your end of the cord at 3.5 Hz, creating a continuous
sinusoidal wave of wavelength 1.0 m. How much time
will pass until a standing wave fills the entire length of
the cord?
Solution: The standing wave will be set up once the
wave makes a round trip to the wall and back to your
hand. The time needed for the round trip is the
distance divided by the speed. The speed of the wave
is
Thus, the time needed for a round trip is
CP 8 The figure shows a standing wave oscillating at
100 Hz. Determine the wave speed.
Solution: Note that 3 half-wavelengths fit into a 60 cm
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Solution: Note that 3 half-wavelengths fit into a 60 cm
length; thus the wavelength of the wave is
The wave speed is therefore
CP 12 A 121-cm-long, 4.00 g string oscillates in its
m = 3 mode with a frequency of 180 Hz and a maximum
amplitude of 5.00 mm. Determine the (a) wavelength
and (b) the tension in the string.
Solution: (a) In the m = 3 mode, 3 half-wavelengths of
the wave fit into the 121 cm length of the string. Thus,
the wavelength of the wave is:
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(b) We know the length and mass of the string, so we
can determine its linear density. If we could
independently determine the speed of the wave, then
we could determine the tension as well, using
But we can independently determine the speed of the wave,
because we know the frequency and wavelength of the
wave:
Now we'll work with the relation between the speed of
the wave and the tension and linear density of the
string. Let's solve for the tension:
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CP 23 A drainage pipe running under a freeway is 30.0 m
long. Both ends of the pipe are open, and wind blowing
across one end causes air inside to vibrate.
(a) Determine the fundamental frequency of air vibration in
this pipe on a day when the speed of sound is 340 m/s.
(b) What is the frequency of the lowest harmonic that
would be audible to a human ear?
(c) What will happen to the fundamental frequency in the
later afternoon as the air begins to cool?
Solution: (a) The standing-wave frequencies for an openopen pipe are
Therefore, the fundamental frequency is
(b) The lowest frequency audible to a human ear is about
20 Hz. By calculating with various values of m, we can
determine the lowest mode whose frequency exceeds 20
Hz:
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Hz:
Thus, the lowest standing-wave frequency that is
audible to a human ear is the third harmonic (i.e., the m
= 4 mode), which has a frequency of about 23 Hz.
(c) As the air temperature decreases, the speed of
sound also decreases, and so all of the standing-wave
frequencies in the pipe also decrease.
CP 28 Two loud speakers in a 20 degrees Celsius room
emit 686 Hz sound waves along the x-axis. What is the
smallest distance between the speakers for which the
interference along the x-axis is destructive?
Solution: As you can see from Figure 16.26 on Page 523
of the textbook, the smallest distance between the
speakers that results in destructive interference is equal
to half of a wavelength of sound. First let's determine
the wavelength of sound, using the fact that the speed
of sound in air at the given temperature is 343 m/s:
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Thus, the smallest distance between the speakers that
results in destructive interference is
0.5/2 = 0.25 m = 25 cm.
CP 32 Two loudspeakers 2.0 m apart emit 1800 Hz,
equal-intensity sound waves in phase into a room where
the speed of sound is 340 m/s. Is the point 4.0 m directly
in front of one of the speakers, perpendicular to the
plane of the speakers, a point of maximum constructive
interference, perfect destructive interference, or
something more complex?
Solution: First determine the wavelength of the sound
waves emitted from the speakers. Then determine the
difference in the distances from each speaker to the
point of interest, and then express this distance in terms
of the wavelength of the sound.
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The path difference is therefore
In terms of wavelengths of sound, the path difference is
Because the path difference differs from a whole number
of wavelengths by half a wavelength, the point of interest
is a point of destructive interference.
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