Vibrational Modes of a Circular Membrane --- (DRUMS!)

NOTE: in the following descriptions of the mode shapes of a circular membrane, the nomenclature for labelling the modes is (d,c) where d is the number of nodal diameters and c is the number of nodal circles.

The (0,1) Mode The animated gif at left (also available as a MPEG movie ) shows the fundamental mode shape of a vibrating circular membrane. The mode number is designated as (0,1) since there are no nodal diameters, but one circlular node (the outside edge). Remember that a node is a point (or line) on a structure that does not move while the rest of the structure is vibrating. The (0,1) mode of a drum, such as a tympani, is excited when the drum head is struck at its center. When vibrating in this mode the membrane acts much like a monopole source, which radiates sound very effectively. Since it radiates sound so well when vibrating in this manner, the membrane quickly transfers its vibrational energy into radiated sound energy and the vibration dies away. The short duration (fraction of a second) of the (0,1) mode means that this mode does not contribute greatly to the musical tone quality of a drum. In fact, when struck at the center a tympani, or other large drum, produces a "thump" which decays quickly and with no definite pitch.

The (1,1) Mode The next mode is the (1,1) mode (MPEG movie ) with one nodal diameter and one circlular 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 therfore, the (1,1) mode takes longer to decay. Because the (1,1) mode "rings" for a while, it contributes to the musical sound or pitch of a drum. When a tympani, or other large drum, is struck somewhere between the center and outer edge, the sound has a definite pitch which lingers for a several seconds.

The (2,1) Mode The third mode of a circular membrane is the (2,1) mode (MPEG movie ), 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 sound than the (1,1) dipole mode and much less effective at radiating sound 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 contributes to the musical pitch of a drum. In fact, the modes which most significantly determine the tone quality of a tymani drum are the (1,1), (2,1), (3,1), (4,1), and (5,1) modes.

The (0,2) Mode The (0,2) mode, shown at right (MPEG movie ) does not have any diameter nodes, but has two circular nodes - one at the outside edge and one at a distance of 0.436 a (a is the radius of the circular membrane) from the outer edge. The frequency of the (0,2) mode is 2.295 times the frequency of the (0,1) mode. Like the (0,1) mode, the (0,2) mode is excited when the membrane is struck at the center. The sound radiation characteristics of the (0,2) mode are more complicated than the first three modes -- it appears to be a mix between a monopole and a dipole. Its decay time is longer than the (0,1) mode, but shorter than the (1,1) mode. As a result, it contributes to the "thump" sound when a drum is hit at the center, but does not contribute much to the musical pitch of a drum when hit off center.

The (1,2) Mode The (1,2) mode (MPEG movie ) has one nodal diameter and two nodal circles. The frequency of the (1,2) mode is 2.917 times the frequency of the (0,1) mode. As you might expect after looking at the first several modes of the circular membrane, the (1,2) mode does not radiate sound very effectively. It has somewhat of a quadrupole type behavior. Thus, the (1,2) mode takes a relatively long time to decay. However, this mode doesn't seem to play a dominant role in the musical tone quality of a drum.

The (0,3) Mode The (0,3) mode, shown at right (MPEG movie ) has three circular nodes, but no diameter nodes. The frequency of the (0,3) mode is 3.598 times the frequency of the (0,1) mode. Like the (0,1) and (0,2) modes, the (0,3) mode is excited when the membrane is struck at the center. The sound radiation characteristics of the (0,3) mode rather complicated. This mode is excited when the membrane is struck at the center, and it dies away fairly quickly. As a result, it contributes to the "thump" sound when a drum is hit at the center, but does not contribute much to the musical pitch of a drum when hit off center.