Resonator or Radiator?

A fixed boundary condition exists between the bowl and the head, which causes the timpano bowl to become a sealed enclosure tightly coupling the head and the bowl at the lip (bearing edge).  As a sealed enclosure, the bowl functions as a baffled radiator (not a resonator) separating the top of the head from the bottom of the head so that the low frequencies from the top and bottom surfaces cannot interact (see Fig. 3s and 3t). The bowl turns the drum into a monopole omnidirectional source that radiates low frequencies well in all directions, which is often confused with the bowl resonating the sound; they are two completely different processes. If the head does not have a baffle, the low frequencies from the top and bottom of the head interfere, and no sound is radiated in the directions lying in the member plane. Simply put, they cancel each other limiting the sound.

Presence of the bowl creates a Monopole Omnidirectional Source ³⁹

As the membrane vibrates, the low frequency energy produced by the motion of the top of the head moves the air molecules radiating sound in an outward direction away from the head. The low frequency energy produced by the motion of the bottom of the head disturbs the volume of the internal air exciting internal air modes as the air tries to escape the enclosure. These internal air modes interact with the motion of the head. The air-tight seal created between the head and the lip of the bowl prevents the the low frequency energy from the top of the head to interact with the air in the cavity below the head. The top and the bottom of the head along with their respective loaded air systems work independently of each other and never coincide (see Fig. 3s and 3t); this prevents any true acoustic cavity resonance to be produced by the bowl cavity interacting with the air waves above the head where the audible acoustic energy is being generated.

Fig 3s
Acoustics and Audio Group at the University of Edinburgh, UK

Figure 3s simulates the internal and external modal patterns of four timpani in an enclosed room struck in sequence. Notice how the bowl separates the motion of the internal air modes from the external radiation patterns of the membrane modes; the two systems of air never mix, but they do interact via the membrane and the bowl acting as a baffle.

Fig 3t
Modal radiation pattern mode 2,1 with a baffle ³⁹

This baffle is also particularly important for suppressing the actual fundamental, mode 0,1, which degrades the perceived pitch of the instrument. When the head has a baffle, mode 0,1 radiates its energy very efficiently (decays quickly) leaving mode 1,1 (the actual principal tone) strong enough to produce a sense of pitch. The correct amount of air in the bowl works to lengthen the decay times of the preferred modes and shorten the duration of the inharmonic modes.

Without the bowl, the head becomes a dipole source. Dipole sources do not radiate low frequency sounds well. The low frequencies from the top of the head are destructively canceled by opposite phase sound from bottom of the head. Perhaps this is one reason the timbales chromamatiques¹⁹  created by Adolphe Sax circa 1859 or the more recent Tour Timp of Marcus De Mowbray  and DrumTone Timpani have not gained traction as serious musical instruments.

Adolphe Sax Timbale chromamatique circa 1859

As it is, the bowl isolates the low frequency energy from the underside, thereby leaving the low frequency energy from the upper side to move the air and radiate sound. Since the low frequencies from the monopole source radiate well in all directions, it is perceived that the sound is being amplified or resonated by the bowl, hence the popular misconception that the bowl is resonating the sound of the vibrating head. The video below demonstrates the process with loudspeakers.

Sound Radiation from Dipoles and Quadrupoles
^{Courtesy of Dan Russell}

It is true, however, that mechanical resonances from the bowl and frame exist, but they have been shown to not enhance the pitch of the instrument to any significant degree. If they do, it is completely serendipitous. Contrary to popular belief, the energy output from the vibrating bowl (and frame/parts) doesn’t add all that much to the actual sound output of the instrument. In the frequency range of the preferred modes (65 Hz- 520 Hz), at most, it is less than 1% in terms of actual output level. Most of the SPL (sound pressure level) energy output (<1 dB) from the bowl/frame occurs in the 1K -2K frequency range, which adds a small amount of collateral color to the sound, but it doesn’t support the preferred modes, which are the modes of vibration that give the timpano its sense of pitch. The Fleischer & Fastl studies have shown that there can be as much as 16% of the overall energy being transferred to the bowl, frame and external parts producing what they call “parasitic pitch.”  This energy can no longer be used by the vibrating head and in fact, all of these unintended resonances have been shown, more often than not, to actually detract from the sustain of the preferred modes rather than enhance or support them.

One of the important things that the bowl, and the material it is made from, can and does influence, is the amount of mechanical energy loss. Energy created by the vibrating head and the displaced internal air mass is lost through the bowl walls to a certain degree. Once this mechanical energy is lost, it is no longer available for use by the vibrating head. How the bowl reacts to this mechanical energy in turn influences how the head will vibrate contributing to its resonance and sustain, which is often misinterpreted and misunderstood as being resonance generated by the vibrations being emitted from the bowl. To date, most professional timpanists prefer hand-hammered copper artisan  bowls on their timpani. The next section “Why copper?” investigates this phenomenon.

Be that as it may, one cannot deny the timpanist the pleasure of indulging in the beauty of sound produced by the various and sundry vibrations and resonances of his or her own instruments, no matter what they hear vibrating. It all adds to the mystique of the instrument.

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