Phy132 Experiment 1

Air Column Resonance and the Speed of Sound

Name: ______Date: ______

Lab Partners: ______

Introduction: Objects vibrate at one or more frequencies called natural, or resonance, frequencies. When a source of vibration causes the object to vibrate at one of its resonance frequencies the energy transferred from the source to the object is at a maximum for a given amplitude of the source. In other words, if the source changed its frequency of vibration to a value either higher or lower than the resonance frequency of the object while keeping its amplitude constant, the energy transferred to the object would decrease. Also, if the source increased its amplitude while vibrating at the object’s resonance frequency the energy transferred would increase.

In this lab the source of vibration will be a tuning fork and the object will be a column of air. The height of the column of air is adjustable by changing the water level in a vertical plastic pipe. As the length of the column changes so too does the column’s resonance frequencies. You will be trying to match the resonance frequencies of the column with the vibrational frequency of the tuning fork.

When a column of air vibrates at its resonance frequency a standing wave of sound is set up inside the column. Like a standing wave on a string there are displacement nodes and anti-nodes where the medium either doesn’t move, or moves with largest amplitude. The difference between the wave on a string and the air column is that the wave on a string is transverse while the wave in air (sound) is longitudinal. There is a displacement node at the bottom of the column because the water there prevents air from moving. However, there is an anti-node at the top of the column because it is there that the air can move the easiest. (The actual location of the top anti-node is slightly above the top of the plastic column.)

The shortest distance between a node and anti-node on a standing wave is one-quarter wavelength. The distance between a node and the second anti-node is three-quarters of the wavelength. Similarly, the distance to the next few anti-nodes is 5/4 , 7/4 . 9/4 , etc.

For an air column of a fixed length, L, there are several resonance frequencies, fm, with wavelengths, n, such that a node is at the bottom and an anti-node is at the top. The first wavelength is the one where only a quarter of the wave fits within the column, L = ¼ 1, or 1 = 4L. The next wavelength is the one where ¾ of the wave fits within the column so that L = ¾ 3, or 3 = 4L/3. In general, m = 4L/m where m = 1, 3, 5, 7, … Thus the resonance frequencies are

fm = mv/4L, (1)

where m = 1, 3, 5, …, and v is the speed of sound in air. If you obtained a tuning fork tuned to one of the frequencies given by equation (1) and you struck the tuning fork while holding it over the air column, you would create a standing sound wave in the air column of length L. You would hear a much louder tone than if the tuning fork’s frequency did not match equation (1).

Instead of keeping the length of the column fixed and looking at different frequencies we will pick a frequency, f, and vary the length L so that equation (1) is satisfied. If we start with the water level near the top of the plastic pipe (so that our air column’s length is short) and strike a tuning fork of frequency f above the column then we could slowly lower the water level and listen for an increase in the loudness of the tone. We could then carefully adjust the water level so that we find precisely the correct length, L1, of the air column at which we hear the first loud tone. This length, L1, would then be equal to ¼ , or in terms of equation (1) f = v/4L1. We would then lower the water level again until a second loud tone is heard. This new length, L2, would be equal to ¾ , or in terms of equation (1), f = 3v/4L2. Remember that we are keeping the frequency, and thus the wavelength, constant while we change the length.

Recall that the anti-node at the top of the pipe really is located slightly above the pipe. Therefore, all of the equations above are slightly in error. However, for all of the different resonance lengths, L1, L2, etc, the anti-node at the top is the same distance, E (called the end correction), above the pipe. Thus we should really write L1 + E = ¼ , and L2 + E = ¾ , etc. If we do not know the small distance E then we can not use the measured values of length to calculate the wavelength, . But, if we look at the difference between L1 and L2 we can calculate the wavelength.

L2 – L1 = (3/4  – E) – (1/4  – E) = ½ 

Similarly, if we continue to lower the water level and measure new lengths, L3, L4, etc, at which resonance occurs we will always find that the difference between adjacent lengths is ½ .

To summarize; by measuring several lengths, and calculating their differences, we can calculate the wavelength of the standing sound wave. Combining this with the known frequency we can calculate, or measure, the speed of sound in air.

We can then compare our measure speed of sound with result of the equation:

vs = (331.5 + 0.6 T) m/s,(2)

where T is the air temperature in degrees Celsius.

Part I: Tuning Fork with Known Frequency: Measure and record the air temperature in the room.

The air column apparatus (see figure 1) consists of a tall plastic tube and an aluminum can, known as the reservoir, connected to the bottom of the plastic tube by a rubber hose. Measure the inside diameter of the plastic tube with a vernier caliper and record this in the data section.

Figure 1

Raise the reservoir as high as it will go and then fill the plastic tube with water to about an inch or so from the top. Once filled, you will notice that as you lower the reservoir the water level in the plastic tube lowers as well.

Wrap four rubber bands around the plastic tube spaced about 5 to 10 cm apart. You will use these to mark the locations where the water level is when resonance is heard.

Obtain a tuning fork whose frequency is greater than 500 Hz. Record the frequency of this fork in Table 1. Use a rubber mallet (see figure 2), or the palm of your hand to strike the tuning fork so you hear a nice clear tone. You may hear a tinny tone in addition to the expected tone. To remove this tinny tone you may touch the tuning fork about 1/3 of the way up from the base.

Figure 2

With the water level at its maximum height hold the vibrating tuning fork over the air column. (How do you orient the fork for maximum effect?) Practice lowering the reservoir and listening for increases in loudness of the tone. When you are confident that you know what you are listening for start over and carefully measure (using the rubber bands as markers) the locations of the water level when resonance is heard. Try to locate the resonance lengths L1, L2, etc, as precisely as you can. Measure the lengths L1, L2, L3, etc. and record them in Table 1. Using these lengths calculate the differences in length, L. Calculate the average L and then the average .

Using the known frequency and the average wavelength, calculate the speed of sound.

Repeat the above procedures for a second tuning fork with a significantly different frequency.

Calculate the average of the calculated speeds of sound from fork 1 and fork 2 and compare this average to that obtained from equation (2).

Part II: A Tuning Fork With Unknown Frequency: Obtain a tuning fork whose frequency marking has been covered up. No peeking! Follow the procedures in part I to determine the locations of the resonances to find the average wavelength. Use this average wavelength along with the average speed of sound from part I to determine the frequency of the tuning fork. Compare this measured frequency with the accepted value.

Data

Temperature of Air = ______(°C)

Inside Diameter of Tube = ______(cm)

Tuning Fork 1 / Tuning Fork 2
Frequency = ______/ Frequency = ______
Resonance Position / L / Resonance Position / L
L1
L2
L3
L4
L5
Average L
Average 

Table 1

Speed of sound from Fork 1 = ______

Speed of sound from Fork 2 = ______

Average speed of sound = ______

Speed of sound from equation (2) = ______

Percent Error = ______

Show Calculation Below:

Unknown Tuning Fork
Resonance Position / L
L1
L2
L3
L4
L5
Average L
Average 

Table 2

Experimental Value of Unknown Frequency = ______

Accepted Value of Frequency = ______

Percent Error = ______

Questions

1.Discuss some of the largest sources of error, and their effect on the results, and state whether each was random or systematic.

2.Suppose the water level is at length L1 for the first resonance of the first tuning fork.

a) Could another tuning fork with a frequency lower than that of the first tuning fork produce a resonance? Explain your answer and if it is yes, then calculate the frequency of the new tuning fork.

b) Could another tuning fork with a higher frequency produce a resonance? Explain your answer and if it is yes, then calculate the frequency of the new tuning fork.

3.From your data explain how you would calculate the end correction, E, for the top anti-node. Calculate E and compare to the theoretical value obtained from E = 0.4 * I.D., where I.D. is the inside diameter of the tube.

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