KAHIBARO
Table of Contents
Two Nearly Equal Frequencies
Beats occur when two sound waves with almost the same frequency reach the ear at the same time. Instead of hearing two clearly separate tones, we often hear a single tone whose loudness rises and falls repeatedly. This regular increase and decrease in loudness is called a beat.
A simple example is produced by two tuning forks, one vibrating at $f_1 = 256 \,\text{Hz}$ and the other at $f_2 = 260 \,\text{Hz}$. The listener hears a sound that becomes louder and softer again and again.
Why Beats Happen
The effect comes from superposition. Sometimes the two waves are in step, so their displacements add strongly and the sound is loud. At other times they are out of step, so they partially cancel and the sound is weak. As the relative phase between the two waves slowly changes, the loudness changes with time.
If the two waves have equal amplitude and frequencies $f_1$ and $f_2$, we can write them as
$$
y_1 = A \sin(2\pi f_1 t), \qquad y_2 = A \sin(2\pi f_2 t)
$$
Their sum is
$$
y = y_1 + y_2
$$
Using a trigonometric identity,
$$
y = 2A \cos\!\left[ \pi(f_1 - f_2)t \right] \sin\!\left[ 2\pi \frac{f_1 + f_2}{2} t \right]
$$
This result shows two parts. The sine term gives the rapid vibration that determines the pitch, which is close to the average frequency. The cosine term changes more slowly and controls the amplitude, which causes the beats.
Beat Frequency
The number of beats heard per second is called the beat frequency. It is equal to the absolute difference between the two frequencies:
$$
f_{\text{beat}} = |f_1 - f_2|
$$
Important rule:
$$
f_{\text{beat}} = |f_1 - f_2|
$$
This formula is valid when the two frequencies are close enough that distinct beats are heard.
For the example $256 \,\text{Hz}$ and $260 \,\text{Hz}$,
$$
f_{\text{beat}} = |260 - 256| = 4 \,\text{Hz}
$$
So the loudness rises and falls 4 times each second.
What the Listener Hears
The pitch that the listener notices is approximately the average of the two frequencies:
$$
f_{\text{pitch}} \approx \frac{f_1 + f_2}{2}
$$
For the same example,
$$
f_{\text{pitch}} \approx \frac{256 + 260}{2} = 258 \,\text{Hz}
$$
So the ear hears a tone near $258 \,\text{Hz}$ whose intensity varies at $4 \,\text{Hz}$.
Visual Picture of Beats
The fast oscillation is wrapped inside a slow amplitude envelope. The envelope represents the loud and soft pattern.
Example Calculation
Suppose two instruments produce frequencies of $440 \,\text{Hz}$ and $444 \,\text{Hz}$.
The beat frequency is
$$
f_{\text{beat}} = |444 - 440| = 4 \,\text{Hz}
$$
This means 4 beats are heard each second. The apparent pitch is about
$$
\frac{440 + 444}{2} = 442 \,\text{Hz}
$$
Beats and Tuning
Beats are very useful in tuning musical instruments. If a musician compares a note from an instrument to a reference note, beats will be heard if the frequencies are not equal. As the instrument is adjusted, the beats become slower. When the beats disappear, the two frequencies are equal and the instrument is in tune.
Tuning rule:
Fast beats mean the two frequencies differ more.
Slow beats mean the frequencies are getting closer.
No beats means the frequencies are equal.
Conditions for Hearing Beats
Beats are heard clearly only when the two frequencies are close together. If the difference becomes too large, the ear no longer hears a smooth rise and fall in loudness. Instead, the sounds may be heard as two separate tones.
The amplitudes also matter. Beats are most distinct when the two waves have similar amplitudes.
Summary Table
| Quantity | Expression | Meaning | ||
|---|---|---|---|---|
| Beat frequency | $f_{\text{beat}} = | f_1 - f_2 | $ | Number of loudness variations per second |
| Approximate pitch | $\dfrac{f_1 + f_2}{2}$ | Tone heard by the listener | ||
| Cause of beats | Superposition of close frequencies | Alternating constructive and destructive interference |
Final Idea
Beats are a direct and easy-to-hear result of interference between two sound waves of nearly equal frequency. The pitch stays near the average frequency, while the loudness changes at a rate equal to the difference in frequency.
Core idea:
Two close frequencies produce a sound whose amplitude varies in time.
$$
f_{\text{beat}} = |f_1 - f_2|
$$
This is why beats are heard as periodic loud and soft sound.