AcademySound
Academy
Beats
Level 1 - Physics topic page in Sound.
Principle
Beats are slow amplitude variations from superposing nearby frequencies.
Notation
\(f_1,f_2\)
two sound frequencies
\(\mathrm{Hz}\)
\(f_{\mathrm{beat}}\)
beat frequency
\(\mathrm{Hz}\)
\(f_{\mathrm{avg}}\)
average tone frequency
\(\mathrm{Hz}\)
\(\omega_1,\omega_2\)
angular frequencies
\(\mathrm{rad\,s^{-1}}\)
\(A\)
individual amplitude
varies
Method
Add two equal-amplitude waves at one position, with slightly different angular frequencies.
Two tones
\[s(t)=A\cos(\omega_1t)+A\cos(\omega_2t)\]
Trig identity
\[s(t)=2A\cos\left(\frac{\omega_1-\omega_2}{2}t\right)\cos\left(\frac{\omega_1+\omega_2}{2}t\right)\]
Beat frequency
\[f_{\mathrm{beat}}=|f_1-f_2|\]
Average tone
\[f_{\mathrm{avg}}=\frac{f_1+f_2}{2}\]
The ear hears a tone near \(f_\{\\mathrm\{avg\}}\) whose loudness rises and falls at \(f_\{\\mathrm\{beat\}}\).
Rules
These are the compact beat relations.
Beat frequency
\[f_{\mathrm{beat}}=|f_1-f_2|\]
Average tone
\[f_{\mathrm{avg}}=\frac{f_1+f_2}{2}\]
Two-tone sum
\[s(t)=2A\cos\left(\frac{\omega_1-\omega_2}{2}t\right)\cos\left(\frac{\omega_1+\omega_2}{2}t\right)\]
Examples
Question
Two tuning forks produce
\[440\,\mathrm{Hz}\]
and \[446\,\mathrm{Hz}\]
Find the beat frequency.Answer
\[f_{\mathrm{beat}}=|446-440|=6\,\mathrm{Hz}\]
The loudness rises and falls six times per second.Checks
- Beats require nearby frequencies; identical frequencies give no beat envelope.
- The beat frequency is a difference, not an average.
- The heard pitch is usually near the average frequency.
- Beat loudness varies even when each source has steady amplitude.