WAVES
UM Physics Demo Lab 07/2013
EXPLORATION
Exploration Materials
1
1
1
2
wave-motor apparatus
battery board
card alligator leads
squeeze clamps
1.
Figure 1: Standing Wave Unit
1. Observe standing waves on a string. The standing wave apparatus has two
parts: a motor-elastic-pen unit, and a base unit with a plastic screw. Slip the bottom
of the pen into the base unit and tighten the screw. Clamp the motor mount to the
table so you can stretch the base unit away from it.
Wire a 1.5V cell with a switch and connect it across the motor. Have one group
member close the switch, and another move the base unit. Try to find the
fundamental wave which looks just like a jump rope: the ends are fixed and the
center swings up and down. Draw your observation below. Clamp the base unit
down.
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Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan 48109
2. Add another cell, increasing the voltage to 3V and close the switch. You can move
the base unit a small amount to rectify the wave if it is not smooth. What happens to
the wave? Draw your observation below.
3. What does adding cells to the battery do to the wave? Explain and predict what
will happen when you add a 3rd cell and increase the voltage to 4.5V.
Add a 3rd cell to the battery and observe what happens to the wave. Again, you can
slide the base unit slightly to clean up the wave if it is not smooth. What happens to
the wave? Does this agree with your prediction? Draw the wave you observe
below.
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Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan 48109
Challenge Work:
Slide the pen cap while the motor is running. How does this change the number of
wave maxima (antinodes)?
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Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan 48109
Everyday Applications

Musical Instruments
APPLICATION: OVERTONES IN A TUBE
Having studied standing waves on strings we will now think about which acoustic
(sound) standing waves are possible in a tube. We’ll study three cases: a tube
open at both ends, closed at both ends, and open at one end and closed at the
other.
For the following diagrams the top of the box represents the maximum
positive amplitude (positive antinode) the centerline of the box zero
amplitude (node) and the bottom of the box the maximum negative
amplitude (negative antinode) for acoustic standing waves in a tube. The
ends of the box are the ends of the tube. These features are illustrated in
the diagram below:
(+)Antinode Level
Tube End------------------Node Level (0)----------------Tube End
(-) Antinode Level
For each case that follows, the object is to draw the correct number of
wavelengths for each fundamental or overtone standing wave specified
while ensuring that open ends are antinodes (maximum positive or negative
amplitude) and closed ends are nodes (zero amplitude). For each case,
label the ends of the tube as open or closed to guide your thinking. A closed
end must be a node (zero amplitude) and an open end must be an antinode
(maximum positive or negative amplitude). Finally, you might find it helpful
to do your trial and error drawings on scrap paper until you figure out the
correct pattern for each case.
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Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan 48109
Overtone Diagrams
First we’ll represent the overtones for a tube open at both ends.
1. Draw a diagram of an open tube’s fundamental standing wave. The tube’s
length is equal to 1/2 of the fundamental wavelength.
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2. Draw a diagram of an open tube’s first overtone. The tube’s length is equal to
1 of the overtone’s wavelengths.
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3. Draw a diagram of an open tube’s second overtone. The tube’s length is
equal to 1.5 of the overtone’s wavelengths.
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Now we’ll draw overtones for a tube closed at both ends.
4. Draw a diagram of a closed tube’s fundamental standing wave. The tube’s
length is equal to 1/2 of the fundamental wavelength.
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Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan 48109
5. Draw a diagram of a closed tube’s first overtone. The tube’s length is equal
to 1 of the overtone’s wavelengths.
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6. Draw a diagram of a closed tube’s second overtone. The tube’s length is
equal to 1.5 of the overtone’s wavelengths
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Finally, we’ll consider the standing wave patterns for overtones in a tube
that is closed at one end and open at the other. This is characteristic of the
pipes in a pipe-organ.
7.
Draw a diagram of an open-closed tube’s fundamental standing wave. The
tube’s length is equal to 1/4 of the fundamental wavelength.
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8. Draw a diagram of an open-closed tube’s first overtone. The tube’s length is
equal to 3/4 of the overtone’s wavelengths.
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Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan 48109
9. Draw a diagram of an open-closed tube’s second overtone. The tube’s length
is equal to 5/4 of the overtone’s wavelength.
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Challenge Work
The lowest musical note is the low C of a pipe organ which corresponds to 32
cycles/second or 32 Hz. Assume that an organ pipe is closed at one end and open
at the other. How many meters long must the organ pipe be to produce this note
as its fundamental frequency? The speed of sound in air is 343 m/s and the speed
of sound is related to the frequency and wavelength of a sound wave by v  f 
where the speed is measured in m/s, the frequency in Hz and the wavelength
meters.
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Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan 48109

in
Summary:
1. Traveling waves are a disturbance in a medium that propagates energy
from one place to another.
2. The elastic medium oscillates about its equilibrium position as the traveling
wave passes but does not travel from one place to another with the wave.
3. The speed of propagation for an elastic wave depends only on the properties
of the medium, not on the amplitude, frequency or wavelength of the wave.
4. The distance between two successive wave crests is called the wavelength
and is denoted by the Greek letter λ. The units of wavelength are meters.
5. The number of wave crests passing a fixed point per second is called the
frequency of the wave and is denoted as f. The units for frequency are
cycles/second denoted as Hertz (Hz).
6. The amplitude of a wave is the maximum excursion from equilibrium in
the medium as the wave propagates. The amplitude is independent of
frequency or wavelength.
7. The fundamental relationship governing all traveling waves is v  f  where v
is the propagation speed of the wave, f the frequency and λ the wavelength of
the wave. Changing f or λ does not change the propagation speed.
8. For waves on a string the propagation speed is
v
T
where T is the
m
L
tension force applied to the string and m/L the mass per unit length of the
string.
9. Standing waves occur on a string when both ends of the string are fixed and
the string is made to vibrate.
10. Nodes are positions along the standing wave where the amplitude of the
oscillation is zero at all times.
11. Antinodes are positions along a vibrating string where the amplitude of the
oscillation is a maximum at all times.
12. For sound standing waves in a tube, a closed end of the tube is forced to
be a node.
13. For sound standing waves in a tube an open end of the tube is forced to be
an antinode.
14. The lowest frequency standing wave that can exist on a string or in an
acoustic tube is called the fundamental frequency.
15. Higher frequency standing waves which can also exist on a vibrating string
or in an acoustic tube are called overtones.
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Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan 48109