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

The (1,1) mode
The second mode is the (1,1) mode, which has one nodal diameter and one circular node (the outside edge). The exact location of the nodal diameter depends on the homogeneity of the membrane and the initial conditions when the vibration starts. The frequency of the (1,1) mode is 1.593 times the frequency of the (0,1) mode. When vibrating in the (1,1) mode a circular membrane acts much like a dipole source; instead of pushing air away from the membrane like the (0,1) mode does, in the (1,1) mode one half of the membrane pushes air up while the other half sucks air down, resulting in air being pushed back and forth from side to side. As a result, the (1,1) mode radiates sound less effectively than the (0,1) mode, which means that it does not transfer its vibrational energy into radiated sound energy as quickly as the (0,1) mode and therefore the (1,1) mode takes longer to decay. Because the (1,1) mode “rings” for a while, it contributes to the sense of pitch generated by a timpano. When timpani, or other large drums, are struck somewhere between the center and outer edge, the sound has a definite pitch which lingers for a several seconds. This mode is what we hear as the “principal tone” of the timpano. It is the first of the preferred modes.


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