# Beats

When two sound waves of different frequency approach your ear, the alternating constructive and destructive interference causes the sound to be alternatively soft and loud - a phenomenon which is called "beating" or producing beats.a beat is an interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies.

### Beat Frequency

There are two wave of frequency ω_{1} and ω_{2}.Then the beat frequency will be ω_{1} - ω_{2}.

The wave of frequency is denoted by ω

Also ω = 2πν

Therefore ν_{beat} = ν_{1} – ν_{2}

Mathematically beats can be understood from a mathematical identity for sine waves. For sounds with (angular) frequencies of w_{1} and w_{2} added together the identity gives

sin(w_{1}t) + sin(w_{2}t) = 2 sin(w_{3}t)cos( w_{4}t)

where w_{3} is the average of w_{1} and w_{2}

and w_{4} is one half of their difference.

If the difference frequency is small, the sound is the sound of a single tone at the average frequency, but with a volume which varies in time at the difference frequency. This is illustrated on the graphic to the right. The sum is a sine wave with a variable amplitude and the pattern repeats every 0.05s. That is, there are 1/0.05 = 20 repeats per second.