Fermi estimate — a whistling microphone

Problem

In an audio system the microphone produces a high-pitched whistle at $\sim 3000\;\mathrm{Hz}$. Estimate the wavelength in air and the microphone-to-speaker distance that favours triggering the whistle.

Solution

Setup. Wavelength from the relation $\lambda = v/f$, with $v \approx 343\;\mathrm{m/s}$.

$\lambda = \frac{343}{3000}$

$\ev{\lambda \approx 0{,}11\;\mathrm{m}}$

Triggering distance. The whistle (Larsen effect, feedback) is self-sustaining when the sound returning from the speaker into the microphone comes back in phase, reinforcing itself. This happens when the microphone-speaker path is an integer multiple of the wavelength:

$d \approx n\,\lambda \approx n\cdot 0{,}11\;\mathrm{m}, \qquad n = 1, 2, 3, \dots$

That is, distances of order $\sim 11\;\mathrm{cm}$ and its multiples favour triggering; slightly moving the microphone changes the phase and often kills the feedback.

Linked atoms

Topics: Onde Concepts: Periodo e frequenza Skills: Stime di Fermi