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Beat Frequencies in Sound

The A beat frequency or beat wave is a sound of fluctuating volume caused when you add two sound waves of slightly different frequencies. The sound of a single pitch or tone consists of only one specific wavelength or frequency, represented in the form of a sine wave. The volume you hear depends on the amplitude of the wave. When you add another sound wave of the same pitch, the new volume will be the sum of the two amplitudes, provided the waves are in unison or have the same phase. But if you add two sound waves of slightly different frequencies, the sound you hear will fluctuate in volume according to the difference in their frequencies. These are called beat frequencies or beat waves.

Questions you may have include:

  • What is a sine wave?

  • What does a beat wave look like?

  • What are some applications of beat frequencies?


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.


Obviously, the sound represented by the sine wave is moving at some velocity v. The number of peaks or crests that pass by a given point is the frequency f of the sound wave. The wavelength L is the distance between crests. (Usually, the symbol for wavelength is the Greek letter lambda and the symbol for frequency is nu, but since we don't have those letters available, we'll use L and f in this lesson.)

The relationship between velocity, wavelength and frequency is:

v = L x f

The pitch or tone of a sound is usually represented by its frequency.


The unit of measurement for velocity is in distance/time, such as meters/second. The unit for wavelength is distance, such as meters. The unit for frequency relates to 1/time.

NOTE: Frequency used to be designated as cycles per second (cps). But in the 1960s they changed the name to Hertz (Hz) to honor scientist Heinrich Hertz. This change from a logical to something obscure also confused millions of students.

A good test is to check the units in the equation:

v meters/second = L meters x f 1/second = meters/second

This is to assure you aren't mixing your units of measurement.


The amplitude of the sine wave is its height. Some measure the amplitude from peak to peak, while others measure it from the center to the peak. The amplitude represents the volume of the sound.


If you add two waves of the same wavelength or same frequency, and they are in unison or the same phase, the amplitudes will add. But it 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 sine wave envelope around the original sine wave.

Sum of two waves creates beat frequency


For example, if you add a wave oscillating at 440 Hz with one that is at 445 Hz, the resulting frequency will be an average of the two, and its volume will oscillate at the beat frequency of 5 Hz.

If you add 440 Hz and 500 Hz notes, the resulting waveform will not be a pure sine wave. Instead, it will be a complex version of a sine wave and will sound like a blurred average of the two tones. Also, its beat frequency will be 60 Hz, which would sound like a very low-pitched hum instead of a fluctuating volume.

You have assumed that the beat frequency is given by A-B, whereas surely it's actually (A-B)/2, as in ...

Sin(A) + Sin(B) = 2.Cos[(A-B)/2].Sin[(A+B)/2]

In support of this, you can roughly time the period between low points with the green button of your demo, it's nearer to 2 secs (0.5 Hz) than 1 sec (1 Hz).

So I think the beat frequencies printed in your text should be halved.


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.

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

Often you will hear the voice of a singer fluctuate with a vibrato. Singing two notes very close to each other cause this beat frequency.

In conclusion

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. This can be graphically shown as a sine wave that has an envelope consisting of another sine wave at a larger wavelength. You may hear beat frequencies when objects vibrate.

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