Periodic functions

Introduction

Periodic means something that repeatedly occurs over a fixed time interval. A common example of a body in periodic motion is a rocking chair. In this piece, you will learn about the meaning of a periodic function, its formula, equation and much more. People generally believe that periodic motion and oscillatory motion are the same things. But, they are very different from each other. A periodic function is a function that returns to the same value after a regular interval. In this article, you will learn about everything you need to know about periodic functions. 

Meaning of periodic functions

A periodic function is a function that returns to the same value after a regular interval. Periodic functions can also be defined as a movement that returns to the original position after a certain period of time. Some examples of periodic functions are a rocking chair. Any function f(x) would be a periodic function if:

If there are positive real numbers like: f (x + T) = f(x). In this example, (T) is a positive real number.

The value of T, which is the smallest, is known as the period of the function.

Understanding the difference between periodic and oscillatory motion

Students often presume that periodic motion and oscillatory motion are the same things. However, they are very different from each other. One of the main differences between the two is that periodic motion is relevant to any motion which repeats itself after a certain period; however, oscillatory motion occurs about an equilibrium point.

To understand periodic functions, let us consider a pendulum bob. A pendulum bob oscillates along with the equilibrium point periodically. While the pendulum is in motion, it moves from zero to positive and then to negative while passing through the original or initial point. Another common example of an oscillating motion is SHM. In SHM (Simple Harmonic Motion), the restoring force of periodic motion is in direct proportion to its displacement.

The formula of the periodic functions

The periodic function can easily be represented as a mathematical formula. The periodic function, i.e., f, can be destined with a non zero constant such as:

f (x+P) = f (x)

Periodic function equation

The periodic function of an oscillating object is:

f(t) = Acosωt

The cosine part repeats itself after a certain period of time. It can be represented as follows:

cosθ = cos(θ+2π)

Acos(ωt) = cos(ωt+2π) – equation (a)

Let T be the time period, then:

f(t) = f(t+T)

Acosωt = Acosω(t+T)

Acosωt = Acos(ωt + ωT) – equation (b)

Thus from equations (a) and (b) we will get the following:

ωT = 2π

T = 2π/ω

The periodic function time period

The time prediction can be given by the following:

T= 2π/ω

Here ω is used to denote the angular frequency of an oscillating object.

Periodic function frequency

Let us now learn about the frequency of the periodic function. We are all aware of the fact that frequency is the total number of oscillations per unit of time. The frequency of the period function can be represented as follows:

f = 1/T

f = 1/ (2π/ω)

Fundamental Period of a Function

If we look at the definition of the periodic function, we would be able to write the function as follows:

f(x+k) = f(x) – equation ( c)

‘k’ in equation ( c) is called the period of the function. The function f is called the periodic function.

Period of a Trigonometric Function

Let us now turn our attention to the period of a trigonometric function. When we draw trigonometric functions on graph paper, the graph shows repeating waves which occur due to trigonometric functions. Similar to any other graph, these trigonometric graphs also have recognisable characteristics like high points or peaks and low points or troughs.

Conclusion 

Periodic functions can be defined as movement that returns to the original position after a certain period of time. A periodic function is an important topic in physics. Some examples of periodic functions are a rocking chair and a pendulum. If there are positive real numbers like: f(x + T) = f(x). In this example, (f) is a positive real number. The value of T, which is the smallest, is known as the period of the function. The periodic function of an oscillating object can be mathematically denoted as follows: (f) = Acosωt