Periodic Function

A periodic function is a mathematical function that repeats its values at regular intervals or periods. These functions are essential in mathematics and physics, especially in modeling cyclical phenomena such as sound waves, seasons, or electrical currents. Common examples include sine, cosine, and tangent functions, which play a vital role in trigonometry, Fourier analysis, and signal processing.

1.0Periodic Motion Definition

A periodic function in physics is a function that repeats its values at regular intervals or periods. It is commonly used to describe oscillations, waves, and other repetitive physical phenomena.

Key Points of Periodic Motion:

• Many common periodic functions (like sine and cosine) are continuous, meaning they have no breaks or jumps.
• Periodic functions repeat their pattern every interval of length. This means their graphs look identical over each interval of one period.
• The shape and frequency of a periodic function remain unchanged over time, unless influenced by an external force or damping.
• Many physical systems exhibit linear response to periodic inputs — meaning the output is also periodic with the same frequency, though amplitude and phase may vary.
• Periodic functions can be amplified or attenuated depending on the system’s resonant frequency.

2.0Examples of Periodic Motion

3.0Mathematical Definition of Periodic Motion

• Periodic functions are mathematical functions used to describe motions or phenomena that repeat at regular intervals.
• As sine and cosine functions exemplify periodic behavior, a particle undergoing periodic motion returns to its original position after each complete cycle. If Trepresents the period, then all physical quantities associated with the motion repeat after every interval of T.

$$\begin{gathered} (y=a\sin \omega t=a\sin \omega (t+T)) \\ (x=a\cos \omega t=a\cos \omega (t+T)) \\ \text{We know that value of sine or cosine function repeats after a period of 2π radian} \\ \therefore (\omega (t+T)=\omega t+2\pi ) \\ (\omega T=2\pi ) \\ \omega =\frac{2\pi }{T}=2\pi \nu [\because (1/T=\nu )] \\ \text{where ω is an angular frequency} \end{gathered}$$

The linear combination of sine and cosine functions is itself a periodic function, as explained below. Consider a function defined as a linear combination of sine and cosine terms 

$$\begin{gathered} x=f(t)=a\sin \omega t+b\cos \omega t \\ a=R\cos \varphi \dots \dots (1) \\ b=R\sin \varphi \dots \dots (2) \\ x=R\cos \varphi \sin \omega t+R\sin \varphi \cos \omega t=R\sin (\omega t+\varphi ) \\ \text{This function describes a periodic behavior characterized by a time period T and an amplitude R.} \\ \text{On squaring and adding equation (1) and (2) we get} \\ {a}^2+b^2=R^2{\cos }^2\varphi +R^2{\sin }^2\varphi =R^2({\cos }^2\varphi +{\sin }^2\varphi )=R^2 \\ {R}^2=a^2+b^2 \\ R=\sqrt{a^2+b^2} \\ \text{And on dividing (2) by (1)}\frac{b}{a}=\frac{R\sin \varphi }{R\cos \varphi } \\ \Rightarrow \tan \varphi =\frac{b}{a} \\ \varphi ={\tan }^{-1}\left(\frac{b}{a}\right) \end{gathered}$$

The combination of multiple periodic functions is also periodic, with the overall time period being the smallest common multiple of the individual periods of the functions involved.

4.0Analysis of Periodic Function

$$\begin{gathered} \text{Trigonometric functions such as sin θ and cos θ are periodic, repeating their values at regular intervals. Specifically, they have a period of 2π radian.} \\ \sin (\theta +2\pi )=\sin \theta \\ \cos (\theta +2\pi )=\cos \theta \\ \text{If the independent variable represents a physical quantity like time, we can define periodic functions with a specific time period denoted by T.} \\ f(t)=\sin \frac{2\pi t}{T} \\ \text{Checking the given function is periodic by replacing t by t + T} \\ f(t+T)=\sin \frac{2\pi }{T}(t+T)=\sin \left(\frac{2\pi t}{T}+2\pi \right)=\sin \left(\frac{2\pi t}{T}\right)=f(t) \\ \text{Hence Function is Periodic.} \end{gathered}$$

Illustration-1.A particle executes SHM described by

$x(t)=3\cos \left(4t+\frac{\pi }{6}\right)$

(1) Find the time period of motion.

(2) What is the displacement after a time equal to one period?

Solution:

$$\begin{gathered} x(t)=A\cos (\omega t+\varphi ),\omega =4,\text{rad/s} \\ \text{(1) Time Period} \\ T=\frac{2\pi }{\omega }=\frac{2\pi }{4}=\frac{\pi }{2};\text{seconds} \\ \text{(2) Displacement after time (T)} \\ x(t+T)=3\cos \left[4(t+T)+\frac{\pi }{6}\right]=3\cos \left[4t+4T+\frac{\pi }{6}\right] \\ x(t+T)=3\cos \left[4t+4\cdot \frac{\pi }{2}+\frac{\pi }{6}\right]=3\cos \left(4t+\frac{\pi }{6}+2\pi \right) \\ x(t+T)=3\cos \left(4t+\frac{\pi }{6}\right)=x(t)\text{Hence Displacement remains the same.} \end{gathered}$$

5.0Graph of Periodic Function

Sin x Graph

6.0General Periodic Function