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

Theoretically, an infinite number of modes can be generated by a vibrating circular membrane, but there are only five or six of these modes that actually contribute to a timpano’s sound spectrum with regard to giving the instrument its sustained sense of pitch. These modes are found in the lower diametric modes (1,1), (2,1), (3,1), (4,1), and (5,1) and are referred to as the preferred modes. All of the other audible modes contribute to the characteristic drum color of the instrument via the production of a myriad of inharmonic overtones. Some are more audible than others, especially within the first 500 milliseconds of the attack envelope. The preferred modes will be discussed in detain in the next chapter.

The following section (information courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State) describes how the first ten modes operate, and how the different modes do, and do not contribute to the instrument’s sound spectrum.

The First Ten Vibrational Modes
of a Ideal Circular Membrane

NOTE: in the following descriptions of the mode shapes of a ideal circular membrane, the nomenclature for labeling the modes is (d,c) where d is the number of nodal diameters and c is the number of nodal circles (also known as diametric and circular or concentric modes). An ideal circular membrane may defined as a absolutely round membrane, infinitely thin, perfectly flexible, completely homogeneous, evenly and uniformly tensioned where the outer circular edge of the membrane constitutes a fixed boundary condition in an in vacuo state (in a vacuum). This type of membrane exists in theory only.

The (0,1) Mode
The animated gif below 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 circular node (the outside edge). Remember that a node is a point (or line or a circle in this case) 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 timpano, 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 to the harmonicity or musical tone quality (sense of pitch) of the timpano. In fact, when struck at the center, a timpano, or other large drums, produces a thump which decays quickly and has no definite pitch.

mode01

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