2The Idea, Step by Step
Pluck two guitar strings that are almost — but not quite — tuned to the same note and you don't hear two notes. You hear one note that throbs: wah … wah … wah, swelling loud and soft over and over. Those pulses are beats. The closer the two strings creep toward the same pitch, the slower the throb; tune them to match exactly and the throbbing stops altogether. That single observation is the whole phenomenon.
Each string sends out a sound wave with its own frequency — the number of pressure wiggles per second, measured in hertz (Hz). Call them $f_1$ and $f_2$. When both reach your ear they simply add together. At some moments the two waves push the air the same way and reinforce (loud); a moment later one pushes while the other pulls, and they partly cancel (soft). How often does this loud-soft cycle repeat each second? Exactly the difference of the two frequencies:
Beat Frequency
$$f_{\text{beat}} = |f_1 - f_2|$$
So a 440 Hz tuning fork held next to a 444 Hz string gives $|440-444| = 4$ beats every second — four throbs you can literally count out loud.
Where does that rule come from? Add two equal-amplitude cosines and apply the sum-to-product identity:
Carrier × Envelope
$$y(t) = 2A\cos\!\big(\pi(f_1-f_2)t\big)\,\cos\!\big(\pi(f_1+f_2)t\big)$$
The second factor is a fast carrier oscillating at the average pitch $(f_1+f_2)/2$; the first is a slow envelope that swells at $(f_1-f_2)/2$. Notice the envelope itself wobbles at only half the beat rate — yet your ear hears loudness, which depends on amplitude squared, and a squared cosine peaks twice per envelope cycle. That doubling is exactly why the audible beat rate is the full $|f_1-f_2|$, not half of it. In the simulation the $f_1$ and $f_2$ sliders set the two pitches, $A_1$ and $A_2$ set how strongly each wave contributes (unequal amplitudes mean the envelope never quite reaches silence), and $\Delta\varphi$ slides the whole pattern sideways in time.
Try this in the sim above. First, drag $f_2$ until it equals $f_1$ and watch the beats slow to a standstill — a single steady tone. Then nudge $f_2$ a few hertz away and the dashed yellow envelope starts pulsing again, faster the wider you spread them. Finally, set $f_1=f_2$ but push $\Delta\varphi$ all the way to $\pi$: the two waves flatten into a near-silent line — the very trick noise-cancelling headphones use.
3Equation Derivation
Two-Source Superposition
$$y(t) = A_1\cos(2\pi f_1 t) + A_2\cos(2\pi f_2 t + \Delta\varphi)$$
Beats from Equal Amplitudes (A_1 = A_2 = A)
$$y(t) = 2A\cos\!\left(2\pi\,\frac{f_1-f_2}{2}\,t\right)\cos\!\left(2\pi\,\frac{f_1+f_2}{2}\,t\right)$$
Beat Frequency
$$f_{\text{beat}} = |f_1 - f_2|$$
Symbol Definitions
| Symbol | Meaning | SI Unit |
|---|
| $f_1, f_2$ | Frequencies of the two sources | Hz |
| $A_1, A_2$ | Amplitudes | arbitrary (e.g., Pa for sound) |
| $\Delta\varphi$ | Phase difference | rad |
| $f_{\text{beat}}$ | Beat frequency (envelope modulation rate) | Hz |
| $T_{\text{beat}}$ | Beat period $1/f_{\text{beat}}$ | s |
| $\lambda$ | Wavelength $v_{\text{sound}}/f$ | m |
1
Two waves of close frequency. Consider two pure tones $y_1=A\cos(2\pi f_1 t)$ and $y_2=A\cos(2\pi f_2 t)$. Their sum can be written using the trig identity $\cos\alpha+\cos\beta=2\cos\!\frac{\alpha-\beta}{2}\cos\!\frac{\alpha+\beta}{2}$.
2
Carrier and envelope. The result is a fast oscillation at the average frequency $\bar f=(f_1+f_2)/2$ multiplied by a slow modulation at the difference frequency $(f_1-f_2)/2$. The slow factor is the envelope; the fast factor is the carrier.
3
Why beat frequency is $|f_1-f_2|$, not $|f_1-f_2|/2$. The envelope cosine $\cos(2\pi(f_1-f_2)t/2)$ has a period $T_e=2/(f_1-f_2)$. But our ear hears loudness, which is proportional to the square of the amplitude — and the amplitude crosses zero twice per envelope cycle, so the perceived beat rate is $f_{\text{beat}}=|f_1-f_2|$.
4
Unequal amplitudes. If $A_1\neq A_2$, the envelope no longer reaches zero — there is residual amplitude $|A_1-A_2|$ at the minima. The beat is still audible as long as the difference is significant, but it is harder to perceive when $A_1\gg A_2$.
5
Mapping to the simulation. Sliders set $f_1, f_2, A_1, A_2$, and the relative phase. Three traces are drawn: the two component waves and their sum. The dashed envelope $\pm 2A|\cos(\pi(f_1-f_2)t)|$ is overlaid. Live readouts give $f_{\text{beat}}$, $\bar f$, $T_{\text{beat}}$, and $\lambda_1$ assuming $v_{\text{sound}}=343$ m s$^{-1}$.
Primary references: Halliday, Resnick & Walker 10th Ed., §17-7; Crawford — Waves (Berkeley Vol. 3), §1.5; HyperPhysics — Beats.
4Frequently Asked Questions
💡 Concept Why does the beat frequency equal $|f_1-f_2|$ instead of $|f_1-f_2|/2$?▼
The amplitude envelope oscillates at $|f_1-f_2|/2$, but the loudness — proportional to amplitude squared — peaks twice per envelope cycle (at both positive and negative maxima). Since the ear perceives loudness, the perceived beat rate is doubled to $|f_1-f_2|$. Mathematically, $|\cos\theta|$ has period $\pi$ rather than $2\pi$.
Key: Beat frequency is doubled because amplitude squared (loudness) peaks twice per envelope cycle.
🎯 Simulation What exactly does the simulation show?▼
Three traces in real time: two pure-tone components in red and blue, and their superposition in yellow. A dashed envelope shows the slowly varying amplitude $\pm 2A\cos(\pi\Delta f\,t)$. As you tune $f_2$ closer to $f_1$, the beat slows — when $f_1=f_2$, the waves combine into a single steady tone (or cancel completely if out of phase). FFT mode shows the two frequency peaks directly.
Key: Live superposition with envelope overlay; FFT shows the two source frequencies as discrete peaks.
🌍 Real Life How are beats used in real-world tuning?▼
Piano tuners listen for beats between a tuning fork and a string: when no beat is heard, the string is in tune. Guitar players match strings by listening for the beat between two adjacent strings — slowing it to silence indicates unison. Aircraft engines synchronise propellers by minimising beat-induced cabin noise. In LASER physics, optical beat frequencies are used to precisely measure laser frequencies.
Key: Tuning instruments, syncing aircraft engines, and laser frequency metrology.
🔬 Non-Obvious Can you have beats from waves of very different frequencies, like 200 Hz and 800 Hz?▼
Mathematically you can superpose any two frequencies, but for the ear to perceive 'beats' as a slow loudness pulsation, $|f_1-f_2|$ must be below about 20 Hz. Above that, the difference frequency itself becomes audible as a third tone (a 'difference tone' or 'Tartini tone'), and the perception changes from beats to a chord or roughness.
Key: Beats are perceived only when $|f_1-f_2|\lesssim 20$ Hz; above that, a difference tone is heard.
📐 Mathematical What if the two waves have a phase difference $\Delta\varphi$?▼
The phase shift moves the envelope and carrier in time but doesn't change the beat frequency. The combined wave becomes $y(t)=2A\cos[\pi(f_1-f_2)t-\Delta\varphi/2]\cos[\pi(f_1+f_2)t+\Delta\varphi/2]$. The beat frequency $|f_1-f_2|$ is unchanged. Two waves of identical frequency but $\Delta\varphi=\pi$ cancel completely (destructive interference) — used in noise-cancelling headphones.
Key: Phase shifts the envelope but not the beat frequency; equal frequencies + $\pi$ phase = silence.
🧠 Deep What is the connection between beats and the uncertainty principle?▼
A pure single-frequency wave extends infinitely in time. To 'see' two close frequencies you need a long observation time $T\gg 1/|f_1-f_2|$ — otherwise the spectral peaks blur into one. This is the time-frequency uncertainty $\Delta f\,\Delta t\gtrsim 1$, the classical analogue of quantum $\Delta E\,\Delta t\gtrsim\hbar$. Short pulses contain many frequencies; pure tones require long pulses.
Key: Time-frequency uncertainty: distinguishing close frequencies requires long observation time.
Explanatory resources: Halliday, Resnick & Walker 10th Ed., §17-7; Crawford — Waves (Berkeley Vol. 3), §1.5; HyperPhysics — Beats.
5Common Misconceptions
❌ Misconception: The beat frequency $f_{\text{beat}}$ equals the average $(f_1+f_2)/2$.
✅ Correction: Beat frequency is the difference $|f_1-f_2|$, not the average. The average $(f_1+f_2)/2$ is the carrier frequency — the fast oscillation underneath the envelope. Slow envelope = beat; fast oscillation = carrier. They are independent quantities.
📖 Reference: HRW 10th Ed., §17-7 'Beats'; Serway & Jewett 8th Ed., §18.7.
❌ Misconception: Beats are caused by waves crashing into each other, like ocean waves.
✅ Correction: Beats result from linear superposition, which simply adds the two wave amplitudes at every point in space and time. There is no collision, scattering, or transfer of energy between the waves — they pass through each other unaffected. The 'beat' is purely an envelope effect of the sum.
📖 Reference: Halliday, Resnick & Walker 10th Ed., §17-7; Crawford — Waves (Berkeley Physics 3), §1.5.
❌ Misconception: If the two waves have very different amplitudes, you can't hear beats at all.
✅ Correction: Beats are still mathematically present; the envelope just doesn't reach zero. The amplitude oscillates between $A_1+A_2$ and $|A_1-A_2|$ at the beat rate. Perceptually, beats become harder to hear if one wave is much louder than the other, but they don't vanish.
📖 Reference: Pierce, J. R. — The Science of Musical Sound (rev. ed., 1992), Ch. 6.
❌ Misconception: Two perfectly identical sound sources will always sound twice as loud.
✅ Correction: Only if they are in phase. With $\Delta\varphi=\pi$, equal-amplitude waves cancel completely (silence), the foundation of active noise cancellation. With $\Delta\varphi=0$, the amplitude doubles, so intensity quadruples (+6 dB), not 'twice as loud' (which is +10 dB perceptually).
📖 Reference: HRW 10th Ed., §17-5 & §17-6; HyperPhysics — Sound Interference.
❌ Misconception: Beats only happen with sound waves.
✅ Correction: Beats occur for any superposition of close-frequency oscillations: light beats (heterodyne detection in radio and laser metrology), water waves, mechanical vibrations of coupled pendulums, and quantum probability amplitudes. The math is the same; only the medium changes.
📖 Reference: Crawford — Waves (Berkeley Physics 3), §1.5; HyperPhysics — Heterodyne Beats.
Misconception research: Linder & Erickson (1989), Sci. Educ. 73, 421; Wittmann, Steinberg & Redish (2003), Phys. Teach. 41, 158; Arons — A Guide to Introductory Physics Teaching, Wiley 1990.