Experiment 08 · Theory
Kundt's Tube
Kundt's tube turns something invisible into something you can measure with a ruler. Sound sent into a closed column of air comes back off the far end, meets itself on the way out, and settles into a pattern that does not travel. Sawdust lying in the tube is shaken off the places where the air moves most and left in ridges where it does not move at all, and the spacing of those ridges is half a wavelength.
What a sound wave is
Sound is energy carried as a mechanical wave. A vibrating object disturbs the material around it, and that disturbance passes on from particle to particle through collisions. Because it needs particles to carry it, sound cannot cross a vacuum, unlike light.
Sound waves are longitudinal: the particles of the medium vibrate back and forth along the same line the wave travels, not across it. The result is a train of compressions, where the particles crowd together and the local pressure rises, separated by rarefactions, where they spread apart and the pressure falls.
Properties worth naming
| Quantity | What it measures | SI unit |
|---|---|---|
| Wavelength (λ) | Distance between two consecutive compressions, or two consecutive rarefactions | metre |
| Frequency (n) | Number of complete oscillations produced per second. It fixes the pitch | hertz |
| Amplitude (A) | Greatest displacement of a particle from its mean position. It fixes the loudness | metre |
| Velocity (v) | Distance the wave travels per second, related to the other two by v = nλ | metre per second |
| Time period (T) | Time taken for one full oscillation, T = 1/n | second |
How fast sound goes depends on the medium rather than on the source. It is governed by the elasticity of the medium and its density, so sound moves fastest through solids, more slowly through liquids and slowest through gases. The human ear responds to frequencies between roughly 20 Hz and 20 000 Hz. Anything below that is infrasonic and anything above it is ultrasonic, and neither is heard, though both are put to work in industry and medicine.
Stationary waves
Send a wave down a tube and let it reflect off a closed end. The reflected wave has the same frequency and nearly the same amplitude as the wave still arriving, and the two travel in opposite directions through the same air. They superpose. At some points along the tube they always cancel, and at others they always reinforce, so the pattern of loud and quiet stops moving. That is a stationary wave, and its fixed points are called nodes, where the air never moves, and antinodes, halfway between them, where it moves most.
Consecutive nodes are half a wavelength apart, which is the single fact the whole experiment rests on. One loop of the pattern, measured node to node, is therefore:
Why the dust shows you where they are
Fine sawdust spread along the floor of the tube is light enough to be moved by the air itself. At an antinode the air is sweeping back and forth with the greatest speed, so the dust there is swept aside. At a node the air is still, so whatever arrives stays. Given a few seconds the powder collects into a sharp ridge at every node and the pattern becomes visible along the length of the tube.
This only happens at resonance. The reflector at the end of the tube forces a node at its own face, and the speaker at the other end drives the column at whatever frequency the generator is set to. Only when the length of the column is a whole number of half wavelengths do the two agree, the pattern builds up, and the heaps appear. Move the reflector a few millimetres either way and the ridges slump back into a flat bed of dust.
Getting the velocity out of it
The frequency is known, because you set it on the generator. The wavelength comes from the dust. Multiply them:
In practice you measure the length L of an air column holding N whole loops, rather than trusting one loop on its own, since a millimetre of error spread over ten loops matters ten times less. One loop is then l = L/N, the wavelength is twice that, and the velocity follows.
The answer is the velocity in the laboratory, at whatever the room happens to be that day. Warm air carries sound faster, because the molecules are moving faster to begin with, and the accepted figure is quoted at 0 °C. The correction is small and linear over the range a room ever covers:
Rearranged, that gives v0 from the value you measured. Near 27 °C the correction is about 5 %, which is the difference between a result that looks wrong against the textbook and one that agrees with it.
Where the method is used
- Velocity of sound in a solid. The classic form of the experiment replaces the speaker with a metal rod clamped at its centre and rubbed with a rosined cloth. The rod and the air column share a frequency, so comparing the node spacing in the rod with the node spacing in the gas gives the velocity in the rod from the velocity in the gas.
- Velocity of sound in other gases. Keep the source fixed and fill the tube with carbon dioxide or hydrogen instead of air, and the heaps move to a new spacing that gives the velocity in that gas.
- Young's modulus. The velocity in a solid is √(E/ρ), so a rod of known density yields its own elastic modulus from an acoustic measurement.
- Acoustic testing. The same idea, grown up, is the impedance tube used in engineering to measure how much sound a material absorbs, which is how insulation and cabin panels are graded.
- Teaching. Interference of sound is otherwise something you take on trust. Here it is a row of ridges you can point at.