In Search of the Missing Fundamental: by Richard K. Jones
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Membrane Modes

The (2,1) mode
The third mode of a circular membrane is the (2,1) mode, which has two nodal diameters (at right angles to each other) and one nodal circle (the outside edge). The exact locations of the nodal diameters depend on the homogeneity of the membrane and the initial conditions when the vibration starts. The frequency of the (2,1) mode is 2.135 times the frequency of the (0,1) mode. When vibrating in the (2,1) mode a circular membrane acts much like a quadrupole source, which is worse at radiating energy than the (1,1) dipole mode and much less effective at radiating energy than the (0,1) monopole mode. This means that the (2,1) transfers its vibrational energy into radiated sound energy much more slowly than the (1,1) and (0,1) modes and therefore takes longer to decay, and thus contributes to the harmonicity of the timpano. Mode (2,1) is the second of the preferred modes. In fact, the modes which most significantly determine the tone quality of timpani are the (1,1), (2,1), (3,1), (4,1), and (5,1) modes with the (2,1) mode often being prominent in the spectrum.


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