Speed of Sound

Objective: To measure the speed of sound in air at room temperature using resonance in air columns.

Equipment: Resonance column apparatus, a set of tuning forks.

Reference: R.D. Knight, Physics for Scientists and Engineers, Ch.21 Superposition

Theory:

You will determine the speed of sound in air by measuring the wavelength of a standing wave for a sound of known frequency. A standing wave is what you get when two or more traveling waves combine in such a way that there are some places where there is no motion at all, and those places are called nodes. For any wave with wavelength λ (in m) and frequency f (in vibrations/s, or 1/s, or Hz), the speed of the wave (v, in m/s) is:

λf = v

A sound wave is a traveling variation in air pressure, the air itself is not transported from one side of the room to the other. The speed it travels depends on the pressure, humidity and temperature of the air. High humidity, high temperature and high pressure all lead to a higher speed v. In a tube, which is closed at one end and open at the other, you can get a standing sound wave set up in the tube with a displacement node at the closed end, and a displacement antinode at the open end. See pictures below. What that means is, the open end will be a place where the air vibrates most vigorously (a displacement antinode) and at the closed end there will be a minimum amount of vibration (a displacement node).

For the first standing wave shown, notice that the wavelength is 4 times the length of the portion of the tube containing air, so we've “fit" 1/4 of a wavelength in the tube. If we now make the air column 3 times longer, we'll be able to “fit" ¾ of a wavelength in the tube. In general, we can capture 1/4; 3/4; 5/4; … of the wavelength in the tube, by adjusting the level of water to just the right length. It all comes from insisting that there be a displacement antinode at the open end and a displacement node at the closed end.
Next, you should know about resonance. For example: play a note A (440 vibrations/s) next to a string whose length is such that one of its possible standing waves has this same frequency. Then the string will vibrate at 440 Hz, even if you don't pluck it. This is called resonance. So, say you hold a tuning fork above a tube with one end open and the other end closed. If you adjust the length of the air column in the tube, and you find the shortest length at which the tube will resonate (you will be able to hear it), you will know that the length of the column is 1/4 of the wavelength of the sound wave. Now keep making the column longer, and the next time you hear resonance, your tube will have reached 3/4 of the wavelength. The next resonance will be at a length of 5/4 the wavelength, and so on. If the frequency is stamped on the tuning fork, then you will have frequency and wavelength, and you can multiply them together and find the speed of sound in air.
A general expression for the speed of sound in a gas, from which we can derive the expression:

where γ= 1.4, R = 8314 J/(kmol·K), T = temperature of the room during the experiment in K, and M = 28:8 kg/kmol. Thus, by measuring the temperature of the room in _C and adding 273 to convert it to K, you can make an independent estimate of what the speed of sound should be.

Activity:

As explained above, for a tone of wavelength λ, there can be a standing wave in an air-filled cavity of length L closed at one end if:

In reality it is not quite true. The top antinode is located slightly above (x cm) the tube, so you have

Ln + x = n·λ/4

Using a glass tube filled to a variable height with water, you will vary L until you find the place of resonance for various tuning forks of known frequencies, and thus find λ. The easiest way to do this is to find the distance between any two neighboring resonance points (which will be 1/2 of a wavelength), and multiply that by 2 to get λ. In this case