A periodic function is a function that repeats its value on regular intervals or periods. A body is said to be in periodic motion, if the motion it’s executing is repeated after equal intervals of time, like a rocking chair, a swing in motion. Though periodic motion and oscillatory motion sound the same, not all periodic motions will be oscillatory motion.
What is a Periodic Function?
A function which repeats its value after a fix interval of time is called a periodic function.
y (f) = y (t + T)
T = Period of the Function.
Periodic Function Equation: Consider an oscillating object, its displacement in periodic motion is represented by a function which is periodic in time.
f (t) = A cosωt
The cosine function repeats itself in time from trigonometry,
cos θ = cos (θ + 2π)
cos (ωt) = cos (ωt + 2π) … (1)
Suppose time period is T,
f (t) = f (t + T)
A cos ωt = A cos ω (t + T)
A cos (ωt) = A cos (ωt + 2π) … (2)
From equations (1) and (2)
ωT = 2π
∴ T = ω/ 2π
The frequency of this periodic function can be given by the time period, the frequency is the number of oscillations per unit time. So, if we know the time for one oscillation then we can find the frequency by
Frequency (f) = 1/ Time Period (T).