Explain periodic function.

Explanation

The displacement can be represented by a mathematical functions of time.
Periodic function are those functions which are used to represent periodic motion.
One of the simplest periodic function is `f(t) =" A cos " omega t`
The periodic time of this function is `T= (2pi)/(omega)` because, `omega t` is increased by an integral multiple of `2pi` radians, the value of the function remains the same.
Thus, the function f(t) is periodic with period T. `therefore f(t) = f(t+T)`
If we consider a sine function, `f(t) = Asin omega t` is a periodic function with the same period T.
If we consider a sine and cosine function with a linear combination, then
`f(t) = A sin omega t+ B cos omega t` is also a periodic function with the same period T.
If `A= D cos phi " ""......"(1)`
`B= D sin phi " ""........"(2)` then
`f(t) = D sin omega t cos phi + D cos omega t sin phi`
`= D[sin omega t cos phi + cos omega t sin phi]`
`=D[sin (omega t +phi)]`
`f(t)= D sin (omega t+phi)`
Where D and `phi` are constants. D is a resultant amplitude.
Adding and squaring equation (1) and (2)
`A^(2)+B^(2)= D^(2) cos^(2) phi + D^(2) sin^(2) phi`
`= D^(2) [cos^(2) phi + sin^(2) phi]`
`= D^(2)`
`therefore = sqrt(A^(2)+B^(2))`
and taking ratio of equation (2) and (1)
`(B)/(A)= (D sin phi )/(D cos phi)`
`(B)/(A)= tan phi`
`therefore phi = tan^(-1) (B/A)`.