A horn equation quantifies how bell flare shapes resonance frequencies.

Evidence

The rate of increase (flare) in the bell diameter determines where along the bell the various resonances have their antinodes, and thus controls the frequencies of the partials. The flare has to be just right to bring these into harmonic alignment. There is a way to quantify these effects that goes back to the 1760s with the work of Leonhard Euler, Joseph Lagrange, and Daniel Bernoulli. Their results have been forgotten and rediscovered several times since. The modern horn equation is associated with the work of A. G. Webster in 1919. A horn function U(x) was defined as the inverse of the product of the two radii of curvature, R1 and R2, defined and illustrated in figure 16.16: where x is the place along the axis of the instrument where the curvatures are measured. — Eric J. Heller, Why You Hear What You Hear- An…, 1507

Categorization

• DEFINE: Defines the horn function and its origins with Euler, Lagrange, Bernoulli (1760s) and Webster (1919).
• ADDENDUM: The horn-function mechanism of impedance reflection parallels how I described trees/sentient structures shaping their environments — here waves are reflected before escaping the bell.