A Little Bit About Partial Differential Equations

The Vibrating Drumhead

Consider a circular membrane with radius R, which could be a drumhead. Assume that in a state of equilibrium it lies in a plane. The forces pushing it back into equilibrium are just due to the tension of the drum; we assume that the head is not very stiff. This means that we can neglect any forces caused by the bending of the material. Let us introduce a coordinates with coordinates x and y in the plane, and let us position the center of the system in such a way that it coincides with the center of our circular membrane. If we stay close to equilibrium we can describe the position of the drum as the graph of a function

which also depends on time, and one can show that it has the properties that

Here

denotes the second derivative of the function with respect to the variable x, that is the rate of change of the rate of change if only that variable is changed, and all others kept the same. Now there are special solutions of this equations in which the drumhead vibrates with one frequency only, and in the form that a fixed form in space just oscillates back and forth, or mathematically we could say

Now in what follows we have some graphs of possible functions v, so-called eigenfunctions. They are presented as a contour plot, with lines along levels of v, just like isothermals in a weather map. The regions where v is positive are in green to red with increasing size of v, where it is negative it ranges from green to blue, with increasing negative size of v. Now here are the pictures. Mathematically all this belongs to the 19^th Century and is produced using special functions, the Bessel functions to be precise. Only the method of producing the graphics with a computer is of course of a more recent vintage.

This produces the vibration with the lowest frequency. All parts of the drumhead are at all times either above or below the equilibrium plane. Click on thumbnails to enlarge.

Now we have the oscillations with increasing frequencies, at least the frequencies increase till we reach the picture with the next explanatory text in front of it.

Now we start a second series of pictures with two rings of oscillations. I can say that the frequency of the sound is higher than that of the first one, I was too lazy to determine where exactly in the row these pictures would belong. They in themselves are again ordered according to increasing frequency. Any actual vibration would normally be a superposition of these idealized vibration types, which can in principle also occur in their pure form, but are not too likely to do so.