Poe, Bessel, and Drones

What’s a connection between Edgar Allan Poe and Friedrich Wilhelm Bessel? Poe’s story The Pit and the Pendulum describes a pendulum with a semicircular blade being lowered on a cable. This is an interesting variation on the usual harmonic oscillator because the length L of the pendulum is not constant but rather an increasing function of time L(t). Using the small angle approximation sin(θ) ≈ θ, the equation of motion is L θ″ + L′ θ′ + g θ = 0. If we assume the length is increasing at a constant rate, then with a change of variables described in [1], this differential equation becomes Bessel’s equation and its solutions are Bessel functions. The authors of [1] state that their primary interest is whether a pendulum can have the motion that Poe describes. But as they indicate in their opening sentence, The study of the motion of a steadily descending pendulum has practical applications (e.g., to the oscillations of a load being lowered by a crane). A more recent application would be…