Spherical Vibrations – Vibrating Soap Bubble

The number of nodes is a key feature for the classification of spectra in one dimension. We have seen that in the case of the vibrating string and the cup rim. Let us now apply this concept to two-dimensional vibrations in order to investigate vibrations on a spherical surface.

It is easy to create vibrating spheres. For example, in the form of soap bubbles. In order to make the vibration visible, let us consider a hemisphere. We can excite this hemisphere using a vibration generator. The vibration looks usually quite wobbly and irregular.

If we change the frequency with which we provoke the vibration, the Fourier transform will show specific points where the so-called resonant frequencies are formed. What does that mean?

Let’s look at a specific resonant frequency. If the bubble is excited with this frequency, the result is a very regular, periodic vibration. We can superimpose this vibration upon itself. If we do this with a time delay of half a period, we will obtain the following picture. The nodes remain constant, and each antinode is maximally deflected. By rotating the nodes around the vertical axis, we can form nodal lines on the two-dimensional soap surface. We obtain five nodal lines in total. The following applies: l=5.

What does the subsequent resonant frequency look like? If we adjust the frequency appropriately, we’ll obtain this regular vibration. Again, we can superimpose the vibration upon itself, with a time delay of half a period. We can thus see the positions of the nodes and the antinodes again. By rotating the nodes, we can form nodal lines on the spherical surface. There are seven nodal lines in total.

It is possible to describe the entire spectrum of resonant frequencies using the number of nodal lines. For reasons of symmetry, there can only be an odd number of nodal lines, since the vibration must always be its own mirror image by design.