Circular Membrane Modes

Circular Membrane Modes

The four lowest vibrational modes for a circular membrane.

Mode 11 is the lowest of the preferred modes for the timpani.

Preferred Timpani Modes

Assuming that these selected modes are excited, the relative frequencies and intervals in cents are given compared to the 11-mode. The preferred vibrational modes for a timpani are a subset of the modes of a circular membrane.

The interval values in cents here are calculated from the mode frequencies given by Berg & Stork. They can be compared to equal tempered intervals. The actual sounding frequencies are affected by air damping.

Circular Membrane

The vibrational modes of a circular membrane are very important musically because of drums, and in particular the timpani. The expression for the fundamental frequency of a circular membrane has some similarity to that for a stretched string, in that it depends on tension and density. The fundamental or 01 mode of an ideal circular membrane is given by:

A timpani head, made of mylar of thickness about 0.2 mm might have the following values for those parameters:

Starting with the values at left as the default values, parameters can be changed below to explore the effects on the membrane resonance. T = N/m σ = kg/m2 f01= Hz D = m

The nature of vibrational modes in membranes is calculated from the wave equation in two dimensions.