Standing Sound Waves
Introduction
Standing sound waves can be produced in a tube that is closed on one end and open to the air on the other end. According to theory, the wavelength, λ, of these standing waves should be:
[1] λ = 4L/n
L = length of standing wave
n = 1, 3, 5, …
There is a subtle difference between the length of the standing waves and the length of the tube, so equation [1] should not be used to directly calculate the wavelength. If we measure the lengths of the tube for multiple standing waves, then we can calculate the wavelength with the following trick. Equation [1] for an ordered pair of standing waves can be written as follows:
λ = 4Ln/n
λ = 4Ln+2/(n + 2)
The above two equations can be solved for wavelength in terms of the difference in length of the standing waves.
Ln = nλ/4
Ln+2 = (n + 2)λ/4
Ln+2 – Ln = λ/2
[2] λ = 2(Ln+2 – Ln)
The difference in length will not depend on the error mentioned above, so we will use equation [2] to calculate the wavelength of standing waves. We will introduce sound waves of a known frequency into the tube using tuning forks. We will use the wavelength and frequency to calculate the experimental speed of sound using the following equation:
[3] v = λf
The speed of sound strongly depends on temperature and subtly depends on humidity and other variations in gas composition. There are a number of formulas for the theoretical speed of sound, but the simplest is the following:
[4] v = a + bTc
v = speed of sound in m/s
a = 331.5 ± 1.0 m/s
b = 0.6 ± 0.1 m/s/°C
Tc = temperature in °C
The error in the formula comes from the failure to adjust for the effect of unknown humidity and gas composition in addition to any error resulting from inaccuracy in temperature measurement.
Experimental Procedures
1) Grab a towel and a water-tube apparatus. Set the reservoir can at its lowest position and slowly fill it with water. Select a tuning fork from the cart with a frequency of at least 320 Hz.
2) Measure the temperature of the air with multiple thermometers and calculate the theoretical speed of sound using equation [4].
3) Move the reservoir can to its highest position. Using a rubber hammer (never strike a tuning fork on a hard surface!), strike the tuning fork and hold it, tongs vertical, over the open end of the tube. Slowly adjust the water level until the sound volume is a maximum. Record the exact location of the resonance level. If you are unsure if you found a resonance level, you probably did not find it. The resonance should be very obvious.
4) Increase the height of the air to the next resonance level. Repeat the above procedure for the next level. You should find 2 to 4 resonance levels.
5) Rank your resonance levels from smallest to largest length of air in the tube. The depth of the water is not relevant.
6) Calculate the wavelength for each pair of adjacent resonant levels using equation [2].
7) Calculate the experimental value for the speed of sound using equation [3]. You may assume that the tuning forks are accurate to at most 1 Hz.
8) Compare the theoretical and experimental values for the speed of sound. Do they agree within experimental error?
9) Repeat the experiment with a tuning fork of a different frequency.