Spherical Vibrations – Chladni Sound Figures

The vibrating string of the monochord is an example of a one-dimensional standing wave. The potential basic modes on the string can be classified according to the number of nodes. As early as in the 19th century, Chladni investigated potential vibrational modes in two dimensions. Two-dimensional nodes form nodal lines. Chladni had a simple and ingenious idea of how to make those lines visible. For example, we can use a violin bow to make a round glass plate vibrate. Let’s sprinkle the plate with fine powder. If we hold the plate at the appropriate point, we will obtain a so-called Chladni figure. How can this be explained?

The stationary granules reveal the places where the plate does not move, that is, where nodal lines are formed. If we cut the glass plate in the middle, we will see a one-dimensional standing wave with two nodes. Antinodes are at the edges, and in the middle of the glass plate. The granules do not accumulate there. They only stay where the nodes are. As a result of rotation, the nodes form a nodal line. This vibrational mode of the glass plate creates a so-called radial nodal line with a radius r₁.

If we hold the plate at the appropriate point at the edge, we get a pattern. Which vibration mode does this figure correspond to? Now, let us consider the vibration on the outer semicircle, with the corresponding nodes and anti-nodes. A standing wave with four antinodes is created at the edge of the glass plate. This pattern continues throughout the entire glass plate, that is, for all radii. The nodal lines are therefore located at fixed angles; that’s why they are called azimuthal nodal lines. In this case, the angle Φ between the two azimuthal nodal lines is equal to 90°.

Nodal lines with a fixed angle are called azimuthal lines. Nodal lines with a fixed radius are radial lines. Of course, there are also vibration modes that combine both types. To demonstrate this, we shall use a glass plate with a slightly greater radius. With fingers positioned appropriately, we obtain a vibration mode with both a radial and an azimuthal nodal line.

The larger the glass plate, the more place for nodal lines, and the more complex the vibrational patterns. If we draw the nodal lines, we obtain this figure. This is just one example of hundreds of complex vibration modes that can be produced on a glass plate. Chladni analysed many of them.

All these intricate patterns can be attributed to superposition of simple basic modes, in line with the Fourier transform. These basic modes can be described using the number l of azimuthal nodal lines, and the number r of radial nodal lines. That is: l = 0, 1, 2, 3, and r = 0, 1, 2, 3, etc., and combinations of r and l.

On the other hand, it is possible to describe any potential vibration on the glass plate by superimposing these basic modes. For example, here we can see the combination of the basic mode r=2, l=2 and the basic mode r=3. In this way, the Chladni figures come back as a superposition of basic modes.