14.4 Beats

Strike two tuning forks that are very nearly, but not exactly, alike and hold them near your ear. You do not hear two notes. You hear a single note whose loudness rises and falls — wah… wah… wah… — a few times a second. Musicians use exactly this effect to tune: a guitarist plays the same note on two strings and turns a peg until the throbbing slows down and stops; a piano tuner counts these throbs to set each string of the piano to a precise fraction of a hertz. The steady drone of a twin-engine propeller aircraft pulses in the same way when its two engines run at slightly different speeds.

The rise and fall is called beats. It is the principle of superposition of Section 14.3 once more, but now applied at one place to two waves of slightly different frequency: the two sounds drift slowly into step and out of step, reinforcing each other and then cancelling. This short section finds the beat frequency, $|\nu_1-\nu_2|$, and puts it to work — finding unknown frequencies, deciding whether a fork is above or below a reference, and tuning strings and pipes with the formulas of the last two sections.

Two notes of nearly equal frequency

Two sound waves of the same amplitude $a$ and frequencies $\nu_1$ and $\nu_2$ (with $\nu_1\gt\nu_2$, and the two close together) arrive at your ear. At the ear, take their displacements as $s_1=a\cos\omega_1t$ and $s_2=a\cos\omega_2t$, with $\omega=2\pi\nu$. By the principle of superposition and the identity $\cos A+\cos B=2\cos\dfrac{A-B}2\cos\dfrac{A+B}2$:

Read this as an oscillation at the average angular frequency $\dfrac{\omega_1+\omega_2}2$ — the pitch you hear — whose amplitude, the factor in square brackets, changes slowly with time. The loudness depends on the size of that amplitude, $2a\left|\cos\pi(\nu_1-\nu_2)t\right|$, not on its sign. It is greatest ($2a$) whenever $\cos\pi(\nu_1-\nu_2)t=\pm1$, i.e. at

and zero half-way between. So the loudness reaches a maximum $\nu_1-\nu_2$ times every second. The same result in words: the phase difference between the two waves, $(\omega_1-\omega_2)t$, grows steadily; each time it passes a whole multiple of $2\pi$ the waves are in step (loud), and each time it passes an odd multiple of $\pi$ they cancel (soft).

Notice the factor of two that catches people out: the amplitude factor $\cos\pi(\nu_1-\nu_2)t$ completes one cycle in $\dfrac{2}{\nu_1-\nu_2}$, but it peaks twice in that time (once positive, once negative), and the ear hears both as loud. The ear can follow beats up to roughly $10$–$15$ per second. With a larger difference the sound becomes rough, and beyond a few tens of hertz you simply hear two separate notes.

Using beats: unknown frequencies and tuning

Beats measure a frequency difference very precisely — counting $3$ beats in a second against a $512\ \text{Hz}$ fork pins down a difference of $3\ \text{Hz}$ out of $512$, better than $1\%$, with nothing but your ears. But a beat count alone cannot say which note is higher: $512\pm3\ \text{Hz}$ give the same $3$ beats per second. To decide, change one frequency slightly in a known direction and see whether the beats speed up or slow down.

• Loading a prong of a tuning fork with a little wax (or sticky tape) adds mass and lowers its frequency — like adding mass to a spring–block oscillator, $\omega=\sqrt{k/m}$ (Section 13.3).
• Filing the ends of the prongs removes mass and raises the frequency.
• For a string, tightening it or shortening its vibrating length raises the frequency ($\nu\propto\dfrac1L\sqrt T$); for a pipe, shortening the air column, or warming the air ($v\propto\sqrt T$), raises it.

Exercises

Take the speed of sound in air as $340\ \text{m/s}$ unless told otherwise. Give answers to 2–3 significant figures.