Beats

Beats are a very annoying effect which, for sound, is often audible.

Examples

• twin engined aircraft are starting their engines, one followed by the other.

• ill tuned musical instruments playing the "same" note

The equivalents are just as prevalent in every other type of wave.

It is a interference effect, ie two waves adding together giving an unusual effect.

The condition for beating is simple, the two waves must have similar, but different, frequencies (wavelengths). The condition is then best heard if the amplitudes of the waves are close so one wave doesn't drown out the other.

What happens is that the waves go in and then out of phase with each other. When in phase, constructive interference creates "loud" - strong - signal, when out of phase a weak signal. The result is

"strong, weak, strong, weak,..."

TURN UP THE SOUND ON YOUR COMPUTER

When you have had enough, turn the sound back down.

If the repetion is quick enough, it is heard as a very nasty "buzz", otherwise as simply extremely annoying.

Well tuned instruments beat at a very slow rate, slower than the notes that are generally played so the mismatch is not noticable.

The "beat" is from "weak to strong to weak".

The number of these per second is the beat frequency, f_b.

f_b = (f₁ - f₂) f₁, f₂ are the two similar frequencies of the waves.

The basic frequency is the average of f₁ and f_2.

f_av = (f₁ + f₂)/2

Mathematically it can be derived quite neatly by using the dreadful "sums to products" for sine terms.

sin f₁t + sin f₂t = 2 sin ( f₁ + f₂ )t /2 . cos ( f₁ - f₂ )t /2

• the first "sin" term describes the averaged background frequency,

• the "2" is the maximum amplitude - it can get twice as strong as one wave alone,

• the "cos" term is the "beats" term. A full wavelength is two beats as it multiplies by the sin term to give two regions of maximum strength. Thus the beat frequency is 2 x ( f₁ - f₂ ) / 2 = ( f₁ - f₂ )