Virtual laboratory/Physics/Oscillations/Superposition of oscillations — beats

Superposition of oscillations — beats

Superpose two oscillations of nearly equal frequencies. Their sum pulsates — these are beats, heard as the sound growing and fading. The beat frequency is |f₁−f₂|. Bring the frequencies closer and watch the envelope.

Superposition of oscillations — beats
Physics · Oscillations and waves · Superposition of oscillations (FIZ.DRGANIA.PP.05)
Frequency f1f_16.0 Hz
Frequency f2f_25.0 Hz
Two oscillations of nearly equal frequencies, superposed, give:
frequency f₁
6.0 Hz
frequency f₂
5.0 Hz
beats f_d
1.0 Hz
beat period T_d
1.00 s
🎯 Challenge: set the beats
streak: 0

I'll give you a target beat frequency — detune f₁ and f₂ to reach it. f_d = |f₁ − f₂|.

What just happened?

Two oscillations of almost equal frequencies were superposed. At times they are in phase (reinforcing each other), at times in antiphase (cancelling out) — that is why the loudness rises and falls in the rhythm of beats.

Superposition of oscillations:

y(t)=cos⁡(2πf1t)+cos⁡(2πf2t)y(t) = \cos(2\pi f_1 t) + \cos(2\pi f_2 t)

Using the sum-to-product formula for cosines, this splits into a slow envelope and a fast carrier oscillation:

y(t)=2 cos⁡ ⁣(2π f1−f22 t) cos⁡ ⁣(2π f1+f22 t)y(t) = 2\,\cos\!\Big(2\pi\,\tfrac{f_1-f_2}{2}\,t\Big)\,\cos\!\Big(2\pi\,\tfrac{f_1+f_2}{2}\,t\Big)

Beat frequency:

fd=∣f1−f2∣f_d = |f_1 - f_2|

The amplitude pulsates with frequency fdf_d — the beat period is Td=1/fdT_d = 1/f_d. The closer f1f_1 and f2f_2 lie, the slower the beats; when they become equal, the pulsation disappears.

Where will you hear it?

Tuning a guitar "to zero beats", the drone of two slightly detuned engines, measuring close acoustic frequencies — the same principle works everywhere.

Why you need it for the exam:

Beat problems boil down to a single formula: fd=∣f1−f2∣f_d = |f_1 - f_2|. The trick: compute the frequency difference and its reciprocal to get the beat period — the envelope does the rest.

Remember: two nearby oscillations give beats, and their frequency is simply the difference of the component frequencies — f_d = |f₁ − f₂|.

Illustrative model · NexTutor Virtual laboratory

Enjoyed the experiment? Share it.

It's free — send it to friends from school.

Theory & formulas

Beats

Superposing two oscillations of nearly equal frequencies produces a pulsation of the amplitude (an envelope). We hear it as the sound rising and falling.

Beat frequency

fd=∣f1−f2∣f_d = \lvert f_1 - f_2\rvert The closer f1f_1 and f2f_2 are, the slower the beats. When they vanish, the frequencies have become equal (tuning instruments).

Symbols: fdf_d — beat frequency, f1,f2f_1,f_2 — component frequencies.

Frequently asked questions

›

What are beats?

Periodic amplification and cancellation of the amplitude when oscillations of nearly equal frequencies are superposed; we hear it as a pulsation of loudness.

›

What is the beat frequency?

Equal to the difference of the component frequencies: f_d=|f₁−f₂|.

›

What are beats used for?

For tuning instruments — when the beats vanish, the frequencies have become equal.

Teaching? Embed this experiment on your site.

Free widget — paste it on your school site, blog or notes. It links to the full version.

Teaching a class or tutoring? Guide students in NexTutor →

Related experiments

Wave interference — amplification and cancellation →Resonance →The simple pendulum — what determines its period? →