Give the theory of beats and hence find the expression for beta frequency.

Explanation

Let `y_(1) = A sin omega_(1) t ` and `y_(3) = A sin omega_(2) t ` represent two progressive waves travelling in the same direction and superpose with each other.
`y y_(1) + y_(2) = A ( sin omega_(1)t + sin omega_(2) t)`
i.e, `y =2A sin ""((omega_(1)+omega_(2)))/(2)t cos""((omega_(1)+omega_(2)))/(2)t`
i.e., `y = 2A cos ((omega_(1)-omega_(2))/(2)) t . sin omega t ` where`omega =( omega_(1) +omega_(2))/(2)`
and `omega_(b) = omega_(1)-omega_(2)` or `f_(b) = f_(1) -f_(2)` where `'f_(b)'` is known as the beat frequency and `t_(b) = (1)/( f_(b)) = (1)/( f_(1)-f_(2))`as beat period.