Standing Sound Waves

 Activity 1

The approximate speed of sound in dry air at temperatures near room temperature is:

\(v_{\rm air} = (331.3 + 0.606\cdot T)\ {\rm m/s}\)

where T is the temperature of the air in Celsius (en.wikipedia.org/wiki/Speed_of_sound).

In this Activity you will set up standing sound waves in a tube filled with air and determine the speed of sound.

The Apparatus

The apparatus is shown on the next page. The loudspeaker generates the sound wave. The rod inside the tube has a small microphone mounted on the end, so the sound wave inside the tube can be measured at different positions.

The figure below shows a close-up of the left side of the apparatus.

The gray box in the lower-right corner is the Sound Tube Microphone Amplifier. It is connected to the Analog Sensor A connection on the Data Acquisition Device. The connector on the top of the box labeled SPK is connected to the loudspeaker, and the connector labeled MIC is connected to the microphone.

The Software

The Speed of Sound Tube program both drives the loudspeaker and measures the output from the microphone. Here is a screen shot of the software.

To use the software for this Activity:

• Click on Acquire Data in the upper-left corner. The button will turn green as shown.

• Click on Function Generator just below the Acquire Data button. It too will turn green as shown. This causes the loudspeaker to begin generating a sound wave.

• Choose a Sine Wave as the Signal Type

• Adjust the frequency of the sound produced by the loudspeaker with the Frequency knob and the control just below the knob.

The plot on the right is what the microphone measures. It is a plot of wave displacement versus time. You will be interested in the amplitude of the wave, which is about 1.5 V in the screen shot.

 

Getting a Standing Wave in the Tube

The most challenging part of this Activity is getting a standing wave set up in the tube. The actual tube differs from the ideal case because of a number of factors:

• The cone of the loudspeaker is moving back and forth, so is only approximately a closed end.

• The sound wave will reflect off the rod, the hole in the right hand side barrier, etc. This means that you are unlikely to measure nodes that have exactly zero amplitude. Instead the amplitude at the nodes will only be close to zero but will be much less than the amplitude at the antinodes.

Here are some tips for getting a standing wave. You may wish to repeat some of the steps as you get closer and closer to a good standing wave.

• Step 1: When a standing wave is established, this is called resonance, and you will be able to hear that the sound that leaks out of the tube is louder than for a non-resonant condition.

• Step 2: Place the microphone at the closed end of the tube. Slowly adjust the frequency so that you get a maximum amplitude from the microphone.

• Step 3: Place the microphone at a node, and slowly adjust the frequency so that you get a minimum amplitude from the microphone.

For each standing wave that you study, be sure to record the range of frequencies for which you can not see any difference in the quality of the standing wave. This will determine the uncertainty in your value of the frequency.

Be careful not to push the Sound Sensor all the way into the speaker, as the speaker is made of paper!

Tube Closed on Both Ends

With the removable barrier in place so that the tube is effectively closed at both ends, get a standing wave in the tube. The stand that supports the rod straddles the metre stick. As you move the microphone from the nodes and antinodes of the standing wave, the distances between them can be determined by the position of the stand relative to the metre stick.

For a given node or antinode you will want to note how much you can vary the position of the microphone and not see any difference in the amplitude of the standing wave. This will allow you to determine the uncertainty in the position, which will allow you to determine the uncertainty in your determination of the wavelength \(\lambda\) of the sound wave.

Knowing the wavelength and frequency f of the standing wave you can calculate the speed of sound v from

\(\lambda f = v\)

In your determination of the wavelength, you should think about how to get the most precise value, i.e. how to minimize the uncertainty in your value. Should you just measure the distance from an antinode to the next node, or from an antinode to the next antinode, or from the first antinode on the right of the tube to the furthest antinode on the left or the tube or …?

Determine the speed of sound for a few different standing waves of different frequency.

1. What is your final value of the speed of sound?

2. How does your value for the speed of sound compare to the accepted value which is given near the beginning of this Activity?

Tube Open on One End

When one end of the tube is open to the air, the standing waves that are possible are the same as those for a vibrating string with one loose end. Here are some of these displacement standing waves for a tube closed on the left and open on the right:

These standing waves occur because part of the incident sound wave is reflected from the open end of the tube. However, the effective reflection point of the wave is not the exact position of the open end of the tube but is slightly beyond it, and so the effective length of the tube is greater than its real length:

\(L_{effective} = L_{real} + \Delta L\)

where:

\(\Delta L \approx 0.3 D\)

and D is the diameter of the tube. Sometimes L_effective is called the acoustic length.

Remove the barrier from the end of the tube and establish a standing wave.

3. Determine the effective length of the tube. How well do your measurements agree with the above equation?
4. If someone designs a pipe organ without being aware of the acoustic length, what will be the consequences?

 

Activity 2

If the apparatus of Activity 1 were perfect, then when the tube is closed on both ends we would not hear any sound outside the tube. Similarly, if the air inside the tube were perfect, all molecule-molecule collisions would be perfectly elastic; this means that as a sound wave travels through the air none of its energy would be converted to heat energy of the air. However, neither the apparatus nor the air is perfect, The Quality Factor Q measures the degree of “perfection” of the system.

Say we have a standing wave when the frequency is f₀. For frequencies close to the "resonant frequency" f₀ the amplitude A of the sound wave at the position where there was an maximum in the pressure wave is given by:

\(A(f)=A_0{1 \over \sqrt{1+ Q^2\left( {f \over f_0} - {f_0 \over f}\right)^2}}\)

Note in the above that the amplitude A(f) is equal to A₀ when the frequency f is equal to the resonant frequency.

 

The figure to the right shows A(f) for A₀ equal to 1, Q equal 2, and for a resonant frequency of 50 Hz. Note that we have indicated the width of the curve where the maximum amplitude is \(1/\sqrt{2}\) times the maximum amplitude A₀.

A nearly trivial amount of algebra shows that the amplitude A is \(1/\sqrt{2}\) times the maximum amplitude A₀ for positive frequencies when the frequency is:

\(f={f_0 \over 2Q}(\sqrt{1+4Q^2} \pm 1)\)

Thus, if the width of the curve is \(\Delta f\) , then Q is:

\(Q={f_0 \over \Delta f}\)

1. For a given resonant frequency f₀ how does the width of the curve of amplitude versus frequency depend on the Quality Factor Q?

2. When the Quality Factor Q is zero, the maximum amplitude A₀ is zero. When Q is infinite so is the maximum amplitude. Explain.

3. Close the tube at both ends and adjust for a standing wave in the range of 200 Hz - 1 kHz. Place the microphone at a maximum in the pressure wave and take data for the amplitude as a function of frequency for frequencies close to the resonant frequency. Calculate the Quality Factor of the tube.