≈ Beats
Two waves of close frequencies produce a pulsation of amplitude: a simulation with the envelope, sound and a beat frequency calculator.
Frequency f₁6,0
Frequency f₂6,8
Animation speed×1,0
View:
Examples:
Beat parameters
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Frequency f₁
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Frequency f₂
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Beat frequency |f₂ − f₁|
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Carrier frequency (f₁ + f₂) / 2
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Beat period 1 / |f₂ − f₁|
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Interval, cents
Sum of two waves: sin(2π f₁ t) + sin(2π f₂ t) = 2 cos(π (f₂ − f₁) t) · sin(2π · (f₁ + f₂)/2 · t)
Beat frequency: fb = |f₂ − f₁| Envelope: 2 |cos(π (f₂ − f₁) t)|
Beat frequency: fb = |f₂ − f₁| Envelope: 2 |cos(π (f₂ − f₁) t)|
- The green curve at the bottom is the sum of the two waves. When the waves are in phase, the amplitude doubles; when they are in antiphase, the sum vanishes. The yellow lines are the envelope, and the yellow dots mark the beat nodes, where the sum equals zero.
- The “In space” view repeats the original: frequencies mean the number of periods across the width of the field (relative units), and the waves travel to the right at the same speed of 0.9 cycles/s, so the envelope stands still. The “In time” view shows the signal at a single point over the last second: here the frequencies are real (Hz), the envelope moves, and the “loudness” indicator pulsates at the beat frequency.
- The “Sound” button plays a 440 Hz tone and a second tone of 440 + (f₂ − f₁) Hz, so you hear beats at exactly the frequency |f₂ − f₁|. At Δf ≈ 1 Hz this is a slow pulsation; at Δf above 15 Hz it turns into roughness.
- The interval in cents shows how close the frequencies are: 100 cents is a semitone. Beats are audible as long as the difference is less than ~20 % of the frequency.
Enter the frequencies of two sound sources in hertz. You will get the beat frequency, their period, the mean frequency and the number of beats in the chosen time. Handy for tuning strings and tuning forks.
Frequency of the first source f₁, Hz
Frequency of the second source f₂, Hz
Observation time t, s
Examples:
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Beat frequency, Hz
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Beat period, s
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Carrier frequency, Hz
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Beats in time t
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Interval between frequencies, cents
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How it sounds
fb = |f₂ − f₁|, Tb = 1 / fb, favg = (f₁ + f₂) / 2
Interval: 1200 · log₂(f₂ / f₁) cents
Interval: 1200 · log₂(f₂ / f₁) cents
Beat frequency and what we hear (approximate)
| Δf, Hz | Beat period, s | How it is perceived |
|---|---|---|
| 0,5 | 2 | very slow pulsation |
| 1 | 1 | clearly audible beats |
| 2 | 0,5 | clearly audible beats |
| 4 | 0,25 | fast tremolo |
| 8 | 0,125 | fast tremolo |
| 15 | 0,067 | border between beats and roughness |
| 30 | 0,033 | roughness, dissonance |
| 60 | 0,017 | two separate tones |
Wave Beats Online: Simulation and Beat Frequency Calculator
A free interactive tool that shows how two waves of close frequencies produce beats: the amplitude of the sum alternately grows and almost vanishes. Change the frequencies f₁ and f₂, watch the individual waves, their sum and the envelope, switch between the space and time views, and turn on the sound to hear the beats. The calculator finds the beat frequency and period, the number of beats in a chosen time, and the interval in cents.
Key concepts
Beat frequency
Equal to the absolute value of the difference between the frequencies of the two waves. The closer the frequencies, the slower the pulsation.
fb = |f₂ − f₁|
Envelope
A slow curve inside which the fast carrier wave oscillates. Its shape is 2|cos(π Δf t)|, and its zeros are the beat nodes.
A(t) = 2 |cos(π Δf t)|
Superposition
The sum of two waves at each point. In phase the waves reinforce each other, in antiphase they cancel.
y = y₁ + y₂
Where it is used
- Tuning guitars, violins, pianos and tuning forks “by beats”
- Choir and orchestra: why identical notes sound slightly “livelier”
- Radio engineering: heterodyne reception and frequency mixing
- Measuring a small frequency difference between lasers and light waves
- Doppler speed measurements from the beats of the reflected and reference signals
- Teaching wave physics and acoustics
How to use
- Move the f₁ and f₂ sliders and watch how the envelope changes
- Make the frequencies equal: the beats vanish and the sum becomes a pure sine wave of double amplitude
- Switch the view to “In time” to see the moving envelope and the pulsation of loudness
- Turn on the sound and count the beats by ear
- In the “Calculator” tab, enter the real frequencies of your strings or tuning forks
Frequently asked questions
What are beats?
They are periodic changes in the amplitude of an oscillation that arise when two waves of close frequencies are added. The sum looks like a fast wave whose amplitude slowly grows and falls. For sound this is felt as a pulsation of loudness.
What formula gives the beat frequency?
The beat frequency equals the absolute value of the frequency difference: f_b = |f₂ − f₁|. For example, tones of 440 and 443 Hz give 3 beats per second, that is, a period of about 0.33 s.
Why does the amplitude of the sum pulsate?
Because of the trigonometric identity sin(a) + sin(b) = 2 cos((a − b)/2) · sin((a + b)/2). The fast sine factor gives a carrier wave with the mean frequency (f₁ + f₂)/2, while the slow cosine factor forms the envelope, which changes at the frequency |f₂ − f₁|/2, giving |f₂ − f₁| maxima per second.
How are instruments tuned by beats?
The musician plays a reference tone and the note on the instrument at the same time and listens to the pulsation. The slower the beats, the more precisely the frequencies match. When the beats vanish, the string is tuned to unison with the reference.
Why can’t beats be heard at a large frequency difference?
When the difference exceeds about 15–20 Hz, the ear can no longer follow the pulsation: the sound becomes rough and then splits into two separate tones. Besides, beats are noticeable only while the frequencies lie within one critical band of hearing, usually about 20 % of the frequency.