The Physics Philes, lesson 111: The Legend of Kundt’s Tube
So far in our discussion of waves, we’ve learned about nodes, antinodes, and standing waves. However, we’ve really only seen what these looks like with respect to transverse waves on a string. But sound waves are longitudinal waves. How do these concepts apply?
As it turns out, much the same way as with transverse waves. Rather than a string, let’s think about a longitudinal wave propagating through fluid in a pipe. Just like transverse waves, there is superposition of waves traveling in the opposite direction. This forms a standing wave in the same way it would do so in a transverse wave. These standing sounding waves (or normal modes) make the sound waves in the air surrounding it.
Even though we are, in essence, talking about the same thing when we talk about nodes and antinodes in transverse and longitudinal waves, we do have some different terminology. Transverse waves on a string are usually describes in terms of the displacement of the string. However, sound waves can be described in two ways: in terms of the fluid displacement or in terms of the pressure variations. So, to keep things straight, we use the terms displacement node to describe the points where particles of the fluid have zero displacement and displacement antinode to describe the points where particles of the fluid have maximum displacement.
We have a nifty little device that we can use to demonstrate standing sound waves called Kundt’s tube. It’s a horizontal tube with a bit of very light powder. One end is closed, and the other is covered with a flexible diaphragm that can transmit vibrations. We can pipe in any sound wave we want and that wave causes the diaphragm to vibrate. The sound waves in the tube are reflected by the closed end of the tube and the powder collects at the displacement nodes. A Kundt’s tube is demonstrated at various frequencies in the video below.
I don’t know about you, but I think that’s kind of cool. We can also figure out more than just what the apparatus tells us directly. Adjacent nodes are separated by a distance equal to half the wavelength. This is a distance we can measure, and we can use this to determine the wave speed by using the relationship v = λf.
If we could look at a snapshot of the sound wave flowing through the medium, we’d notice something interesting. The particles on opposite sides of a displacement node vibrate in opposite phase. When the particles approach each other, the medium is compressed and the pressure goes up. The opposite is true when the particles move away from each other. That means that at the displacement node, the medium undergoes the maximum amount of compression and expansion. In addition, the pressure and density variations are at their maximum values above and below the average. We call the point where the variations of pressure and density are the greatest the pressure antinode.
All of this is reversed when we talk about what happens around antinodes. Particles on opposite sides of antinodes vibrate in phase so the distances between the particles is nearly constant and there is no variation in pressure or density at the displacement antinodes. We call the point where the pressure and density do not vary the pressure node.
To illustrate this, think of a reflection at the end of a closed pipe. At this point, the displacement of the particles must always be zero and the pressure variation is at its maximum. The closed end of the pipe is a displacement node and a pressure antinode. At the open end of the pipe is the opposite. The pressure is constant and the particles oscillate with their maximum amplitude, so that point is a pressure node and a displacement antinode.
All of this is a little fiddly, but it really does have IRL applications. Next week we’ll look at how all of this stuff applies to musical instruments.
Featured image credit: Wikipedia