Speed of sound from the measured wavelengths

Speed of SoundResonance
o Determine the effective length of a closed tube at which resonance occurs for several tuning forks.
o Determine the wavelength of the standing wave from the effective length of the resonance tube
for each tuning fork.
o Determine the speed of sound from the measured wavelengths and known tuning fork frequencies
and compare with the accepted value.
. Resonance tubes (with length scale marked on the tube)
. Tuning forks (range 500 to 1040 Hz) and rubber hammer
. Thermometer (one for the class)
Traveling waves of speed V, frequency f, and wavelength l are described by
V fl Eq: 1
We can determine the speed of a traveling wave for known frequency and wavelength from Equation 1.
It is difficult to measure the properties of a traveling wave directly. When two waves of exactly the same
speed, frequency, and wavelength travel in opposite directions in the same region, they produce standing
waves. These standing waves can be measured easily.
This laboratory uses a device called a resonance tube to produce standing waves from the sound
waves emitted from a tuning fork. The can shown in Figure 22-1 contains water, and the level of the water
in the tube can be varied as the can is moved up and down. The water acts as the closed end of the tube,
and changing the water level changes the effective length of the resonance tube.
A tuning fork, clamped just above the open end of the tube, is struck with a rubber hammer.
Sound waves travel down the tube and are reflected when they strike the water. Standing waves are
produced by these traveling waves going in both directions inside the tube. The waves reflected from the
Physics Laboratory Manual n Loyd LABORATORY 22
COPYRIGHT 2008 Thomson Brooks/Cole
2008 Thomson Brooks/Cole, a part of TheThomson Corporation.Thomson,the Star logo, and Brooks/Cole are trademarks used herein under license. ALL RIGHTSRESERVED.No part of this work
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closed end of the tube undergo a phase change of 1808, and are completely out of phase with the incident
waves. Therefore, the combined amplitude of the incident and reflected waves must be zero at the closed
end of the tube. A point in space with wave amplitude zero at all times is called a node N. From similar
considerations of the relative phase between the incident and reflected waves, at the open end of the tube
the wave amplitude must be a maximum at all times. Such a point is called an antinode A. The speed of
sound is fixed, and, for a given tuning fork, the frequency is fixed. Therefore, the resonance conditions can
be satisfied for only certain specific lengths of the tube.
Figure 22-2 illustrates the necessary relationship between the length of the tube and the wavelength of
the wave for the first four resonances of the tube. Sound waves are a type of wave known as longitudinal.
The amplitude of a sound wave is determined by pressure variations in the air along the direction of wave
motion. The sound waves in the figure are pictured as if they were transverse waves for ease of
representation. The resonances are pictured from left to right as they are encountered when the level
of the water in the tube is lowered, increasing the effective length of the tube. The distances L1, L2, L3, and
L4 refer to the distance from the top of the tube to the water level for the first four resonances. The locations
of the nodes N and antinodes A are shown for each of these resonances. In the first resonance there is a
single node and antinode. Each successive resonance adds an additional node and antinode. The distance
between a node and the next antinode is one-fourth wavelength (1/4 l). The distance between nodes is
one-half wavelength (1/2 l).
The location of several of the resonances for each tuning fork will be determined experimentally.
If the situation were ideal, the following relationships would be implied by Figure 22-2 for the first four
resonances shown.
L1 1=4 l L2 3=4 l L3 5=4 l L4 7=4 l Eq: 2
226 Physics Laboratory Manual n Loyd
Image not available due to copyright restrictions
Examine Figure 22-2 carefully to be sure that you understand how the relationships given in Equations 2
are implied by the figure.
The relationships given in Equations 2 are not valid for a real resonance tube because the point at which
the upper antinode actually occurs is just outside the end of the tube. The exact location depends upon the
diameter of the tube. Equations 2 are not directly useful to determine the wavelength l of the wave.
The end effect is the same for each of the resonances and will cancel if differences between the
locations of the individual resonances are considered. Considering the differences between adjacent
resonances gives the following
L2 $ L1 L3 $ L2 L4 $ L3 l=2 Eq: 3
Equations 3 determine the wavelength, and the frequency of the tuning fork is known. Equation 1 then
allows determination of the speed of sound.
If Equations 3 are used and the results are then averaged, it would amount to taking the sum of twice
the three differences and then dividing by three. In that process, all but the first and last resonance
positions cancel from the calculation. In effect, one might as well have not measured the middle two
resonances. There is nothing incorrect about such a procedure, but it loses some of the information
contained in the data. This shows that there is often more than one way to analyze data, but often one
technique gives more information than the others.
All the data contribute to the result if each wavelength is computed, not from the adjacent differences,
but from the differences between each resonance and the first resonance. The resulting equations for the
wavelength are given below. A subscript has been placed on the wavelength, but it is still understood
that each of the wavelengths, l1, l2, and l3, refer to the same wavelength calculated from three different
sets of resonances. The equations are
l1 2L2 $ L1 l2 L3 $ L1 l3 2=3L4 $ L1 Eq: 4
The speed of sound in air has a slight linear dependence on the air temperature for a limited range
of temperature. The speed of sound VT at a temperature of T8 C will be determined from
VT 331:5 0:607 T m=s Eq: 5
where T is the temperature in 8C.
COPYRIGHT 2008 Thomson Brooks/Cole
Figure 22-2 Nodes and antinodes of first four resonances of a tube closed at one end.
Laboratory 22 n Speed of SoundResonance Tube 227
Note carefully that tuning forks should be struck only with the rubber hammer. Take care to ensure that neither the
hammer nor a vibrating tuning fork comes into contact with the tube.
1. Measure the room temperature of the air and record it in Data Table 1.
2. Adjust the water level until the can is essentially empty when the tube is almost full. The water level
in the tube should come to at least within 0.050 m of the open end of the tube. It may be necessary to
remove some water from the can when the water level is near the bottom of the tube.
3. Clamp a tuning fork above the top of the tube, and one partner should strike it repeatedly with the
rubber hammer. Keep the fork vibrating continuously with a large amplitude. With the tuning
fork vibrating, another partner should slowly lower the water level from the top while listening for a
resonance. The sound will be very loud when a resonance is achieved. Try to measure the position of
each resonance to the nearest millimeter. Raise and lower the water level several times to produce
three trials for the measured position of the first resonance and record the values in Data Table 2.
Record the frequency of the tuning fork in Data Table 2.
4. Repeat the procedure in Step 3 to locate as many other resonances as possible. Depending upon the
frequency of the tuning fork, either three or four resonances should be attainable. Record in Data
Table 2 the location of resonances that are attained.
5. Use a second tuning fork of different frequency and repeat Steps 1 through 4. Record in Data Table 3
the frequency of the tuning fork and the position of as many resonances as are attained.
1. Use Equation 5 to calculate the accepted value of the speed of sound from the measured room
temperature. Record it in Data Table 1.
2. Calculate the mean and standard error of the three trials for the location of each of the resonances.
Record each of the means and standard errors in Calculations Tables 2 and 3.
3. Use Equations 4 to calculate the wavelengths that are appropriate. If four resonances were found, then
all three values of l can be determined. If only the first three resonances were measured, then only
two values of l can be determined. If this is the case, just leave the at the
appropriate position. Use the mean values of the lengths to calculate the wavelengths.
4. Calculate the mean and standard error for the number of independent wavelengths measured for each
tuning fork. Record those values in the Calculations Tables as l and al.
5. From the values of l and the known values of the tuning fork frequencies, calculate the experimental
value for V, the speed of sound.
6. Calculate the percentage error of the experimental values of V compared to the accepted value of the
speed of sound in Data Table 1.
228 Physics Laboratory Manual n Loyd
Name ………………………………………………………………. Section ……………. Date …………….
22 LABORATORY 22 Speed of SoundResonance Tube
1. What is the equation that relates the speed V, the frequency f, and the wavelength l of a wave?
2. How are standing waves produced?
3. What name is given to a point in space where the wave amplitude is zero at all times?
4. What name is given to a point in space where the wave amplitude is a maximum at all times?
COPYRIGHT 2008 Thomson Brooks/Cole
5. What are the conditions that must be satisfied to produce a standing wave in a tube open at one end
and closed at the other end?
6. For an ideal resonance tube an antinode occurs at the open end of the tube. What property of real
resonance tubes slightly alters the position of this antinode?
7. A student using a tuning fork of frequency 512 Hz observes that the speed of sound is . What is
the wavelength of this sound wave? Show your work.
8. A student using a resonance tube determines that three resonances occur at distances of L1 0.172 m,
L2 0.529 m, and L3 0.884 m below the open end of the tube. The frequency of the tuning fork used is
480 Hz. What is the average speed of sound from these data? Show your work.
230 Physics Laboratory Manual n Loyd
Name ………………………………………………………………. Section ……………. Date …………….
Lab Partners ……………………………………………………………………………………………………..
22 LABORATORY 22 Speed of SoundResonance Tube
COPYRIGHT 2008 Thomson Brooks/Cole
Data Table 1
Room Temperature 8C Speed of sound m/s
Data Table 2
Frequency Fork One Hz
L1 (m) L2 (m) L3 (m) L4 (m)
Data Table 3
Frequency Fork Two Hz
L1 (m) L2 (m) L3 (m) L4 (m)
Calculations Table 2
L1 m L2 m L3 m L4 m
aL1 m aL2 m aL3 m aL4 m
l1 2(L2 $ L1) m l2 (L3 $ L1) m l3 2/3(L4 $ L1) m
l m al m V fl m=s % Err
Calculations Table 3
L1 m L2 m L3 m L4 m
aL1 m aL2 m aL3 m aL4 m
l1 2(L2 $ L1) m l2 (L3 $ L1) m l3 2/3(L4 $ L1) m
l m al= m V fl m=s % Err
1. Speed Sound 331.5 0.607 T
2. l1 2(L2 $ L1)
3. l2 (L3 $ L1)
4. l3 2/3(L4 $ L1)
5. V fl
6. % Error jE $ Kj/K (100%)
1. What is the accuracy of each of your measurements of the speed of sound? State clearly the evidence
for your answer.
2. What is the precision of each of your measurements of the speed of sound? State clearly the evidence
for your answer.
3. Equations 2 provide a means to determine the end correction for the tube. Using the value of l
for the first tuning fork, calculate values for L1 and L2 from those equations. They should be larger
than the measured values of L1 and L2 by an amount equal to the end correction. Repeat the
calculation for the second tuning fork. Compare these values for the end correction and comment on
the consistency of the results.
232 Physics Laboratory Manual n Loyd
4. Suppose that the temperature had been 10 8C higher than the value measured for the room temperature. How much would that have changed the measured value of L2 $ L1 for each tuning fork?
Would L2 $ L1 be larger or smaller at this higher temperature?
5. Draw a figure showing the fifth resonance in a tube closed at one end. Show also how the length of
the tube L5 is related to the wavelength l.
COPYRIGHT 2008 Thomson Brooks/Cole
Laboratory 22 n Speed of SoundResonance Tube 233
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