Problem — Resonance of the glass

Problem

A singer who holds a note aimed at a crystal glass at a specific frequency can make it vibrate violently until it shatters. Explain why it only works at one precise frequency and not at “similar” frequencies.

Solution

The glass has a natural frequency $f_0$ and low damping (high quality factor $Q$: the crystal dissipates little). The resonance curve $A(f)$ has a narrow peak around $f_0$: only if the driving frequency falls within the narrow band $\sim f_0/Q$ does the amplitude grow cycle after cycle, because each push arrives in phase and adds energy.

At “similar” but out-of-band frequencies the pushes gradually go out of phase: on average they do not transfer energy, and the amplitude stays small.

$A(f) \text{ maximal only if } |f - f_0| \lesssim \frac{f_0}{2Q}$

High $Q$ means the energy accumulates with every cycle until it exceeds the breaking load of the glass.

$\ev{\text{large amplitude only for } f \approx f_0,\ \text{band} \sim f_0/Q}$

Links

Topics: Oscillazioni e moto armonico Concepts: Risonanza Skills: Analisi di casi limite e fantafisica