Object

Determine the velocity of sound in air by measuring the resonant frequencies of an open tube.

Measure the signal distribution in a closed tube, calculate the wavelength and speed of sound from this data.

Compare your results for the speed of sound in air with the results of students working on other experiments (Speed of Ultrasound and Lloyd's Mirror).

Introduction

Many musical instruments (e.g. organ pipe, trumpet) use vibrations in a column of air to produce musical tones. These vibrations resonate (are loudest) for certain frequencies which depend mainly on the length of the tube containing the column of air. The relative strength and frequencies of the resonant oscillations differ between instruments, so that a penny whistle does not sound the same as trombone. In this experiment a resonance tube is excited at an open end by a loudspeaker.

Standing waves are set up in the tube when the because the sound is reflected at both open and closed ends. In figure 1 the displacement amplitude distribution for resonance states are shown. (Note: by convention, an open tube has two open ends and a closed tube has one open end).
Figure 1

The region near an open end is a pressure node (minimum pressure amplitude) and a displacement antinode (maximum displacement amplitude). Conversely a closed end is a displacement node (minimum displacement amplitude) and a pressure antinode (maximum pressure amplitude). The microphone detects the pressure of the sound wave rather than the displacement.

The resonance condition for an open tube of length L and diameter d is:

where n = 1, 2, 3, 4.... and is the wavelength of the resonant sound wave. The tube is effectively lengthened by the end correction (0.4d) for each open end. The resonance condition for a closed tube of length L and diameter d is:

where n = 1, 3, 5, 7.... and is the wavelength of the resonant sound wave.

Apparatus

Resonance Tube (eg perspex tube, about 75 cm long and about 5 cm internal diameter), loudspeaker with similar diameter (0.1 W, higher impedance better for typical oscillator), oscillator (preferably matching loudspeaker power rating/impedance or low impedance audio amplifier may be necessary), tie clip microphone/preamp, oscilloscope, thermometer, two test leads (BNC - 4 mm). The schematic experimental set up is as shown in figure 2.

Figure 2

Setting up the apparatus

1. Set up the resonance tube, microphone, oscillator and oscilloscope as shown in figure 1. Remove the piston from the tube and ensure the microphone is mounted in the Resonance Tube.
2. Connect the low impedance (50 W or less) output of the oscillator to the loudspeaker (32 W) and to channel one on the oscilloscope. Connect the microphone output to channel two on the oscilloscope. Switch on the microphone amplifier (remember to switch off the microphone amplifier at the end of the laboratory period).
3. Trigger the oscilloscope on the signal going to the loudspeaker (channel one).
4. Set the oscillator to approximately 100 Hz and adjust the amplitude until the sound from the speaker can be heard.
5. Adjust the oscilloscope to show two steady signals, if you have any difficulties consult one of the staff running the laboratory.

Experiment

Part 1: Open Tube

1. Slowly increase the frequency of the oscillator, adjusting the oscillator signal amplitude as necessary. Listen and observe the oscilloscope to detect resonances.
2. Adjust the frequency to obtain the lowest resonant frequency (f_n, n = 1). Make a note of this frequency and the higher frequencies of other resonances (f_n, n = 2, 3, 4, 5, 6, 7, 8 ....). You should obtain at least seven resonances.
3. Plot a graph of resonant frequency f_n (vertical axis) against harmonic number n (horizontal axis). Calculate the gradient (G) of your graph.
4. Measure the length (L) and diameter (d) of the tube and the air temperature in the tube.
5. Calculate the speed of sound (v) using the equation v = 2(L + 0.8d)G. How does this compare with the accepted value of v = 331.5 + 0.607 T where T is the air temperature in Celsius?

Part 2: Closed Tube

1. Insert the piston into the tube so that the tube length is now between 0.50 and 0.70 m..
2. Set the oscillator frequency to about 1 kHz and then adjust the oscillator to obtain a resonance.
3. Move the microphone along the tube, make a note of the positions where the signal reaches a maximum and a minimum.
4. Sketch the wave amplitude along the tube. Mark on your sketch nodes, antinodes and their positions. Make sure you mark the open and closed ends of the tube on your sketch. Does your sketch agree with the displacement amplitude patterns for the closed tube shown in figure 1? Which overtone/harmonic does you sketch show?
5. The separation between adjacent nodes (weakest signal) and between adjacent antinodes (strongest signal) is equal to l/2 where is the wavelength. Calculate several times from your data.
6. Measure the frequency of the resonance using the oscilloscope.
7. Calculate the speed of sound using v = fl , how does this compare with your previous value?
   

How do your results for the speed of sound in air compare with the results of students working on other experiments (Speed of Ultrasound and Lloyd's Mirror)?

©  Mark Davison, 1997,  give feedback or ask questions   about this experiment.