In order to show a smooth motion, these animated gifs are fairly large. They demonstrate the oscillations of the surface of a circular vibrating drumhead.

The frequency is labeled in each animated gif, given in arbitrary units so that the lowest frequency is one. The indices listed in each case counts the number of nodes of the oscillation, with "m" representing the number of angular nodes and "k" representing the number of radial nodes (counting the edge).

The functions describing these surfaces are products of Bessel functions of r and trigonometric functions of θ. In a couple of cases you see simple approximate forms for the functions, and you can use these approximate forms to get good estimates of the oscillation frequencies. There is a little on this in chapter eight of the text "Mathematical Tools." A series solution for the Bessel function appears in chapter four.

 Animations Equation for the mode m = 0, k = 1 ω = 1 m = 0, k = 2 ω = 2.3 m = 0, k = 3 ω = 3.6 m = 1, k = 1 ω = 1.6 m = 1, k = 2 ω = 2.9 m = 2, k = 1 ω = 2.1 m = 3, k = 3 ω = 5.4