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What would be the theoretical characteristics of an acoustic resonator cavity which has a completely flat gain frequency response over 200Hz-3000Hz (Roughly the range of a violin)

In other words, what would a musical instrument look like if it were to have no distortion in the frequency response, only a constant gain.

My thoughts:

  • A thin membrane?
  • Right angles would probably be bad, but if you just have a circle which gets progressively larger you'd end up with a horn or a bell, which obviously has a resonant frequency
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  • $\begingroup$ If the frequency response is absolutely flat, energy is equally spread across all considered frequency, so by definition you can't really say the cavity is a resonate structure (or system). A resonator by definition traps, amplifies energy at specific frequencies. $\endgroup$ – docscience Dec 14 '18 at 1:25
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The resonator you describe would need to be either 1) broadly-tuned or 2) multiresonant in order to resonate all the way across that frequency range.

Broad tuning means its resonance peak is wide, and the tradeoff in obtaining a broad peak is that the resonant "gain" (the system amplitude at the resonant peak) is necessarily low.

Multiresonance means that as the driving frequency shifts about within its frequency range, different resonators with different sizes come into play, and this can include such effects as having the radiating surface break into smaller elements at higher frequencies, and/or having the resonating cavity driving the radiator dynamically segment itself into smaller volumes as the driving frequency increases.

All the instruments in the violin family, for example, use the effects of both radiator and cavity segmentation to get broadband performance. This is accomplished by carving and constraining the radiating top of the instrument to enable segmentation and by shaping the body cavity and the port vents (the "f" holes) to support segmentation.

A more useful way (at least from an acoustic-engineering standpoint) to approach this problem instead of framing it as a resonance issue is by considering it as an impedance-matching problem with a bandwidth requirement as a design constraint.

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  • $\begingroup$ So would a "better" question would be "How to design a cavity with constant impedance over X range"? $\endgroup$ – nanotek Dec 14 '18 at 1:42
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    $\begingroup$ Yes- at least from the standpoint of a recovering ex-engineer like me! $\endgroup$ – niels nielsen Dec 14 '18 at 1:45

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