Spherical Vibrations – Vibrating Cup

We can use a simple coffee cup to generate interesting variants of Chladni figures. All we need is a spoon. If the cup is struck opposite the handle, it sounds like that… Now, strike slightly offset to one side, and it sounds like that… Here, the cup sounds deeper, and there, higher. Why is that? Apparently, the angle between the handle and the place where the spoon strikes is essential.

We can illustrate the phenomenon using a figure of standing waves. A standing wave forms on the cup rim. In the simplest scenario, the rim fits four antinodes, or two wavelengths. When an antinode forms at the handle, more mass is vibrating. The frequency decreases, and the sound becomes lower. When a node forms at the handle, less mass is vibrating. The frequency increases, and the sound becomes higher.

How can we prove this theory? To do that, we need to adequately visualize these standing waves.

Let us fill the cups with coffee. Let us strike the cup directly across from the position of the handle. A standing wave having four antinodes forms and spreads across the surface of the coffee. We can see that a lot of coffee is stirred at the four antinodes, and a wave is running into the centre of the cup. Indeed, the antinode is located at the handle.

Let us now strike the cup 45° from that position. What will happen on the surface of the coffee? Again, a standing wave with four antinodes forms. However, we can clearly see that the node is located at the handle. We have thus found an explanation for the mystery of the two tones emitted by the vibrating cup. Either the handle vibrates with the cup, or it stands still.

Let us now consider the entire spectrum of the vibrating cup. We have already observed that standing waves with four antinodes can be formed in two ways. Exactly two wavelengths, that is, 2λ, fit on the cup rim. In the vibration 2λ₊, the handle vibrates with the rim; in 2λ_–, it does not.

Are there more frequencies in the spectrum? If so, what do they mean? One obvious possibility would be that the rim can fit not just the double wavelength, but the triple wavelength as well. It would look like this. Again, either the node, or the antinode is positioned at the handle. To create these standing waves, we need to strike the cup at the appropriate place.

With a little practice, it is indeed possible to generate standing waves having six antinodes. Let’s start with striking the cup 60° from the handle.

Indeed, we can see six antinodes forming at the cup rim. The antinode is located at the handle.

Let us strike the cup 30° from the handle.

Again, the cup rim vibrates with six antinodes, but this time the handle is positioned precisely between two antinodes. It thus lies directly at the node. The hexagon has shifted by 30°, and the frequency is slightly increased.

We have thus explained the spectrum of the cup. We can create double peaks with two times, three times, and, with some effort, even four times the wavelength. Try it out, Bob!