In Search of the Missing Fundamental: by Richard K. Jones
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The Bearing Edge

Oscillation of Mode 0,1 (Ideal Circular Membrane)

The amplitude of sound will diminish over time due to resistance. The decay of amplitude over time is called damping. For timpani, there are five possible reasons for energy loss resulting in the damping of the sound emitted from the instrument. 47 48 50

1) radiation of sound

2) mechanical loss in the membrane

3) viscothermal loss in the confined  air [inside of the bowl]

4) mechanical loss in the kettle [bowl] walls

5) mechanical loss in the frame and external parts

The lip of the timpano bowl, also known as the bearing edge, dictates the boundary conditions for the vibrating membrane. For timpani, this is a fixed or rigid boundary condition. This means that energy from the part of the membrane that creates the perceived pitch does not continue past this point. This boundary condition also defines the first nodal point (a point of minimal vibration) of the membrane, and it is the only point of physical contact between the head and the bowl. It is at this point where the vibrating portion of the head has the least amount of physical motion. A bowl with a wider, more rounded lip will have much more of the vibrating head in contact with the bowl for a longer period of time. This contact produces a stronger coupling with the bowl, potentially increasing the transmission of energy to the bowl and external parts and decreasing the amount of energy that can be used by the head to produce sound. A sharper lip, or stronger point on which the head can rest, will cause less coupling adding the potential for less damping because more energy can be used by the vibrating head. The strong pointed lip enforces the required boundary conditions for an ideal circular vibrating membrane, where the rounded lip does not hold to them as rigidly (see Fleisher & Fastl).

One must bear in mind, however, that this energy is minimal compared to the energy created by the motion of the mass of air trapped inside of the bowl as the head vibrates (concentric modes). The motion of the trapped concentric air modes has the most impact on how the bowl will vibrate; not how much of the lip is in contact with the head. However, how much of the lip that comes in contact with the head does contribute substantially to the mechanical energy loss through the membrane. (see #2 above)

pressure field copyJAES Volume 61 Issue 10 pp. 737-748; October 2013

Snapshots of the cross section of the pressure field for a timpano, at times as indicated. The “resonant” qualities (collateral color) of the bowl are a result of the pressure of the internal air mass coupling with the mass of the bowl, in as much as the bowl is allowed to vibrate.

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