The overall equation for the beat wave is the sum of the equations for the incident waves.
MinusSinewave180x60
yR(
t) =
Y cos(2π
f1t) +
Y cos (2π
f2t)
Using trigonometric identities, it can be shown that the equation above is equivalent to the one below with two parts, one involving the sum of the frequencies of the incident waves and the other involving the difference of the frequencies.
| yR(t) = 2Ycos 2π | ( | f1 – f2 | ) | cos 2π | ( | f1 + f2 | ) |
| 2 | 2 |
The blue term of the equation is the overall wave with frequency between that of the incident waves.
| average frequency = | ( | f1 + f2 | ) |
| 2 |
The red term (which tends toward zero due to the difference in frequencies) causes the variation in amplitude known as a beat.
There are two amplitude minima (beats) per period of the modulation, and therefore the beat frequency is given by the expression below..
beat frequency = f1 – f2