27 — Banjo Drum Physics — theoretical preliminaries

This paper opens by posing the question, “why does a timpani have a discernible pitch when other drums do not?”

It also has this immensely high quality sentence toward the bottom of the first page: “To most people, a banjo is just a banjo.”

The answer to the question above, and the haughty response to the statement above, is the same: The air-membrane interaction is a complex one to model, and different banjos must be modeled entirely differently. The depth of the “pot” (the body) can dramatically change the turbulence of the air under the membrane.

Politzer addresses this by reducing the dimensions of the problem and modeling a series of simplifications that help explain the way a banjo sounds by deconstructing the sound into a series of oscillators. The first proposed model is a $0-0D$ banjo: Its membrane is represented as a damped oscillator, and the air is modeled as another damped oscillator tethered to the first one (essentially, two springs). Physicist readers should already be picturing a 1D double-pendulum on springs, and so it should be pretty obvious why this calculation is only going to get much more complex as we add dimensions.

The next model is a $0-1D$ banjo, modeled as a piston acting on a compressible fluid. Then comes $1-2D$, where the string is a 1D piston compressing the air, a 2D fluid in a cylinder.

Politzer explains that a true banjo model — a $2-3D$ — is too complicated to model here by hand, but is only a computational extrapolation of the above models. That is, it wouldn’t be fun to draw out the formulas for a real banjo, but using the above systems as models, it isn’t mathematically hard… it’s just computationally challenging.

The last few sentences of this paper are maybe my favorite part:

So the mixing up of frequencies and wavelengths contributes to shaping the sound according to our pleasure. Of course, what we like is strongly influenced by what we know and what we’re used to. Musical instrument design is not, by and large, a purely engineering endeavor.