Sound Beats
Sound beats are periodic variations in amplitude — heard as a rhythmic pulsing or "wah-wah" sound — that occur when two sound waves of slightly different frequencies interfere. The beat frequency equals the absolute difference between the two source frequencies, and the sound alternately gets louder (constructive interference) and quieter (destructive interference). Musicians use beats to tune instruments: when no beats are heard, the two sources are in tune; as beats slow to zero, the frequencies converge.
Key Formula
f_beat = |f1 - f2|
LaTeX: f_{beat} = |f_1 - f_2|
| Symbol | Meaning | Unit |
|---|---|---|
| f_{beat} | Beat frequency (number of amplitude pulses per second) | Hz |
| f_1 | Frequency of the first sound source | Hz |
| f_2 | Frequency of the second sound source | Hz |
Worked Example
Problem
A guitar string vibrates at 440 Hz. A tuning fork of 436 Hz is sounded simultaneously. How many beats per second are heard, and what does this tell the guitarist?
Solution
Step 1: Use f_beat = |f₁ − f₂|. Step 2: f_beat = |440 − 436| = 4 Hz. Step 3: The guitarist hears 4 beats per second, meaning the guitar string is 4 Hz out of tune. Step 4: The guitarist should loosen the string (lower its frequency toward 436 Hz) or tighten it until no beats are heard.
Answer
Beat frequency = 4 Hz; guitar string is 4 Hz sharp
Beat Frequency Examples for Musical Tuning
| Frequency 1 (Hz) | Frequency 2 (Hz) | Beat Frequency (Hz) | Beats per Second | Perception |
|---|---|---|---|---|
| 440 | 440 | 0 | 0 | Perfect unison |
| 440 | 438 | 2 | 2 | Slow, gentle pulsing |
| 440 | 436 | 4 | 4 | Noticeable out-of-tune |
| 440 | 430 | 10 | 10 | Rapid, unpleasant flutter |
| 440 | 420 | 20 | 20 | Perceived as rough dissonance |
| 440 | 400 | 40 | 40 | Two separate pitches heard |
Interactive Tools
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Related Terms
Constructive Interference
Constructive interference occurs when two or more waves overlap in such a way that their displacements add together, producing a resultant wave with greater amplitude than either individual wave. This phenomenon arises when the waves are in phase — that is, their crests and troughs align — leading to a net increase in energy at that point. It is fundamental to technologies such as noise-cancelling headphones (in reverse), optical coatings, and phased-array antennas.
Destructive Interference
Destructive interference occurs when two waves overlap out of phase — with the crest of one aligning with the trough of the other — causing their displacements to cancel partially or completely, reducing the resultant amplitude. When two waves of equal amplitude are exactly 180° out of phase, the resultant amplitude is zero, meaning complete cancellation. This principle underlies active noise cancellation in headphones, anti-reflective optical coatings, and acoustic dead spots in concert halls.
Acoustic Resonance
Acoustic resonance occurs when an object or air column vibrates at its natural frequency in response to an external sound source at that same frequency, resulting in a dramatic amplification of the sound. The phenomenon arises when standing waves are set up within the resonating object, with nodes and antinodes at fixed positions. Acoustic resonance is exploited in all musical instruments — strings, pipes, and percussion — as well as in architectural acoustics, industrial machinery fault detection, and medical imaging.
The word "beat" comes from Old English "beatan" (to strike repeatedly), apt since a beat in music corresponds to a repeated pulse. In physics, the term was formalised in the 18th century with the mathematical analysis of wave superposition.