Physics · Sound

Beats

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Wave InterferenceExtend the same superposition idea to spatial interference patterns

Key takeaway

A beat is the slow envelope around a faster resultant wave, not a separate source turning volume up and down.
Beat frequency is set by the frequency difference |f2 - f1|.
The carrier frequency follows the average of the two source frequencies, so the same beat rate can sit on a higher or lower pitch.

Common misconception

If two tones create beats, each tone must be changing its own loudness.

In this bench, each source keeps a steady amplitude the whole time.

Carry beats into superposition and resonance

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Motion and pitchWaves

Doppler Effect

Watch a moving sound source compress wavefronts ahead and stretch them behind, then see how source motion and observer motion combine to change the heard pitch on one bounded classical bench.

Broader superpositionWaves

Wave Interference

Superpose two coherent sources, trace their path difference to phase difference, and watch bright and dark regions emerge on the same live screen.

Pattern from interferenceWaves

Standing Waves

Track fixed nodes, moving antinodes, and harmonic mode shapes on one live string while the same probe trace shows the underlying oscillation in time.

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Reference and support

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Equation snapshotEnvelope versus carrier snapshot · Resultant with envelope and carrierShow 2 equations

Keep the two time scales separate: the frequency difference sets the slow beat envelope, while the average frequency sets the faster carrier inside it.

Envelope versus carrier snapshot
$f_{b e a t} = ∣ f_{2} - f_{1} ∣$
The envelope rate comes from the difference between the source frequencies.
Resultant with envelope and carrier
$y = y_{1} + y_{2} = 2 A cos (π (f_{2} - f_{1}) t) sin (2 π \frac{f _{1} + f _{2}}{2} t)$
The slow cosine factor sets the envelope, while the faster sine factor keeps the average-frequency carrier visible inside it.

Why it behaves this way

Explanation

Beats happen when two sounds are almost, but not exactly, the same frequency. You do not hear the sources taking turns getting louder. You hear one rapid oscillation whose amplitude rises and falls because the two waves keep drifting between reinforcement and cancellation.

This bench shows the two source traces, their resultant, and a bounded loudness cue from that same superposition. The slow envelope rate is set by $∣ f_{2} - f_{1} ∣$ , while the faster carrier follows the average of the two frequencies. That is why you can change the beat rate without changing amplitude, or keep the same beat rate while shifting both source frequencies upward or downward together.

Key ideas

01Beat frequency is set by

∣ f_{2} - f_{1} ∣

, not by amplitude.

02The loud-soft pulse comes from changing phase alignment in the resultant, so neither source has to change its own amplitude.

03The average frequency sets the fast carrier, while the frequency difference sets the slow envelope.

Worked examples

Live beat checks

Open examples when you want to see the same idea walked through step by step.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

These checks use a selected source pair, source amplitude, and setup time from the same superposition model, so the algebra stays tied to the envelope and carrier cues.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For this beat setup with $f_{1} = 1 Hz$ and $f_{2} = 1.12 Hz$ , what beat frequency and average carrier frequency follow from that pair?

f_{1}

Source A frequency

1 Hz

f_{2}

Source B frequency

1.12 Hz

f_{b e a t}

Beat frequency

0.12 Hz

f_{a v g}

Average carrier frequency

1.06 Hz

1. Start from the nearby-frequency relations

Use

f_{b e a t} = ∣ f_{2} - f_{1} ∣

for the envelope rate and

f_{a v g} = \frac{f _{1} + f _{2}}{2}

for the faster carrier underneath it.

2. Substitute the setup source pair

f_{b e a t} = ∣1.12 - 1∣ = 0.12 Hz

and

f_{a v g} = \frac{1 + 1.12}{2} = 1.06 Hz

3. Interpret the pulse rate

So the envelope repeats at

0.12 Hz

, each loud-soft cycle takes 8.33 s, while the faster oscillation still centers on

1.06 Hz

Beat and carrier result

f_{b e a t} = 0.12 Hz, f_{a v g} = 1.06 Hz

The frequency difference is small, so the carrier oscillates many times before the loudness envelope completes one slow beat cycle.

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows two source drivers on the left, a shared time-trace region in the middle, and one listener cue on the right. The upper trace compares the two source motions, while the lower trace shows their combined resultant with an optional envelope guide around it.

Optional overlays label the envelope, the loudness cue, and the current frequency difference. In compare mode, the same compact bench appears in two rows so the learner can contrast beat rate and carrier frequency without leaving the shared layout.

Graph summary

The first graph plots Source A, Source B, and the resultant displacement against time so the faster carrier stays visible inside the superposition.

The second graph plots the normalized envelope ratio and a bounded loudness cue against time so the slow beat cycle can be read separately from the fast oscillation.

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Starter track

Step 3 of 5

Sound and Acoustics

Beats appears later in this track, so it is cleaner to start from the beginning first.

Previous step: Pitch, Frequency, and Loudness / Intensity

Also appears in Waves.

Start from track beginning