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# Beat Frequencies in Sound

by Ron Kurtus (revised 23 October 2013)

The sound of a beat frequency or beat wave is a fluctuating volume caused when you add two sound waves of slightly different frequencies together.

If the frequencies of the sound waves are close enough together, you can hear a relatively slow variation in the volume of the sound. A good example of this can be heard using two tuning forks that are a few frequencies apart.

A sound wave can be represented as a sine waves, and you can add sine waves of different frequencies to get a graphical representation of the waveform. When the frequencies are close together, they are enclosed in a beat envelope that modulates the amplitude or loudness of the sound. The frequency of this beat is the absolute difference of the two original frequencies.

Questions you may have include:

• What are some examples of beat frequencies?
• How can you add sine waves?
• What is the beat envelope?

This lesson will answer those questions. Useful tool: Units Conversion

## Examples and applications of beat frequencies

A good demonstration of beat frequencies can be heard in the animation below. A pure sound of 330 Hz is combined with 331 Hz to give a rather slow beat frequency of 1 Hz or 1 fluctuation in amplitude per second. When the 330 Hz sound is combined with a 340 Hz sound, you can hear the more rapid fluctuation at 10 Hz.

### Example

Click on the buttons below to hear the pure tone and the different beat frequencies.

### Another example

When you fly in a passenger plane, you may often hear a fluctuating droning sound. That is a beat frequency caused by engine vibrations at two close frequencies.

### Application of beats

A piano tuner will strike a key and then compare the note with a tuning fork. If the piano is slightly out of tune, he will be able to hear the beat frequency and then adjust the piano wire until it is at the same frequency as the tuning fork. If the piano is severely out of tune, it makes the job more difficult, because the beat frequency may be too fast to readily hear.

Although sound is a compression wave that travels through matter, it is more convenient to illustrate the sound wave as a transverse wave, similar to how a guitar string vibrates or how a water wave appears. The shape of such a wave for a single frequency is called a sine wave.

Sine wave representing a single frequency of sound with constant amplitude

When you add sound waves traveling in the same direction together, elements of the sine wave add or subtract, according to where they are in the waveform. You add the amplitude of each wave, point-by-point. Making a graphical representation of the sum of two waves can be done by hand, but that can be tedious. Usually, it is done mathematically.

## Beat envelope

If you add two waves of slightly different frequencies, the resulting amplitude will vary or oscillate at a rate that is the difference between the frequencies. That beat frequency will create a beat envelope around the original sine wave.

Beat envelope modulates the amplitude of the sound

Since the frequencies of the two sounds are so close, you would hear a sound that is an average of the two. But you would also hear the modulation of the amplitude as a beat frequency, which is the difference between the initial frequencies:

fb = | f1 − f2 |

where

• fbis the beat frequency
• f1 and f2 are the two sound frequencies
• | f1 − f2 | is the absolute value or positive (+) value of the difference

### Examples

For example, if you add a wave oscillating at 445 Hz with one that is at 450 Hz, the resulting frequency will be an average of the sum of the two waves: (445 Hz + 450 Hz)/2 = 447.5 Hz. This waveform is close to a sine wave, since the frequencies are almost the same.

The amplitude of volume of this combination will oscillate at the beat frequency of the difference between the two: (450 Hz - 445 Hz) = 5 Hz.

Now, if you add 440 Hz and 500 Hz notes, the resulting waveform will be a complex version of a sine wave and will sound like a blurred or fuzzy average of the two tones. The average frequency of this complex wave will be (440 Hz + 500 Hz)/2 = 470 Hz.

Also, its beat frequency will be 60 Hz, which would sound like a very low-pitched hum instead of a fluctuating volume.

## Summary

A beat frequency is the combination of two frequencies that are very close to each other. The sound you hear will fluctuate in volume according to the difference in their frequencies. You may often hear beat frequencies when objects vibrate. Beat frequencies can be graphically shown by adding two sine waves of different frequencies. The resulting waveform is a sine wave that has an envelope of modulating amplitude.

## Resources and references

Ron Kurtus' Credentials

### Websites

Beats - Hyperphysics

Physics Resources

### Books

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sound_beat_frequencies.htm

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