Periodic functions

A function that repeats itself after a fixed period ( regular intervals) is called a periodic function. They behave cyclically over a specific time interval. The most straightforward example is trigonometric functions that repeat after the completion of 2π radians; they are periodic functions. This concept of periodic functions is used in physics while studying oscillation, waves and other phenomenons that exhibit periodicity. Any function that doesn’t repeat after a fixed time interval is called a non-periodic function.

Definition of a periodic function : 

For a function to be periodic, having a non-zero constant p, it should satisfy its definition :

             f(x+p) = f(x)

This is the periodic function formula

If this property is satisfied by any positive constant p, it is referred to as the fundamental period / primitive period. Hence, a function with a period p will repeat on intervals having length p. These intervals are referred to as periods of the function.

The graph of a periodic function exhibits a translational symmetry which implies that a graph of a function f is periodic with period p 

if the graph of a function f is invariant under translation in the x-direction by a p distance.

This definition of a periodic function can be generalized and molded to different shapes, patterns and dimensions, such as periodic tessellations of the plane.

Examples of Periodic Functions:

• Sine functions are examples of periodic functions with a period of 2π.

• Periodic Behavior is shown by the hands of the clock and phases of the moon.

• EM Waves are also a periodic function.

• Trigonometric functions like sine, cosine etc., behave as periodic functions.

• Euler’s formula is made up of cosine and sine, which are periodic functions and hence, Euler’s formula will also be a periodic function with a period of 2π/k.

• The Fourier series is a superimposition of periodic wave functions and forms a complex periodic function. This series is used in heatwaves, vibrational analysis, quantum mechanics, electrical engineering, signal or image processing.

• The study of Electromagnetic radiations, including diverse energy forms like cosmic rays, X-Rays, ultraviolet rays, infrared radiation, visible light, radar, radio waves, and microwaves, have one property in common: they are all periodic functions. All their properties can be explained while keeping in mind the properties of periodic functions.

• High tides and low tides are also predicted using periodic functions so that scientists can record the height of water at different times of the day.

• Moving constantly in a circle is a kind of simple Periodic Behavior.

 ( P = 360°)

Period of a periodic function:

The time taken by another function for completing one cycle is called the period. The period of any periodic function is :

                       T = 2πω

  where ω is the angular velocity of the 

  oscillating object.

The SI unit for the period in seconds.

Jean Baptiste Joseph Fourier showed that any periodic function could be expressed as a superposition of sine and cosine functions of different periods with sustainable coefficients.

Frequency of a periodic function:

Frequency is the number of cycles per unit time.

 Frequency = Total No.of Oscillations

Time

               F = 1/T

               F = 1/2πω

• The SI unit for frequency is Hertz (Hz).
• Properties of a periodic function :
• The graph made by a periodic function is symmetrical and repeats itself on the horizontal axis.
• Real number values are included in the domain of periodic functions, whereas the range for the periodic functions is described for a fixed interval.
• P will be the period of the function for the entire range of the function.
• If the period of f(x) is P, then the period of 1/f(x) will also be P
• If the period of f(x) is P, then the period of f(ax+b) will be P/ |a|
• If the period of f(x) is P, then af(x)+b will also be P.

• Simple Harmonic Motion :  

• Simple harmonic motion involves a simple repetitive motion of back and forth, about a central position, and the furthermost displacement on either side of the central position is the same. Each vibration is completed in the same period. The force of motion is directed towards the central position at all times, and the force is directly proportional to the distance from the central position.
• F = -kx 
• Where F is the force, x is the displacement, and k is the restoring/spring constant. This is known as Hooke’s law.
• Linear simple harmonic motion: When a mass moves back and forth in a straight line around an equilibrium position, it is a linear simple harmonic motion. An essential condition followed by this type of motion is that the force is indirectly proportional to the displacement represented by:
• F ∝ – X
• Where F is the force and X is the displacement from the equilibrium position.

1. Angular simple harmonic motion: A body that demonstrates simple harmonic motion along with a change in its angular position is said to be performing the angular simple harmonic motion. This type of motion happens when the angular acceleration or the torque is directly proportional to the displacement of the body from its position of equilibrium. That is:

• T ∝ -θ
• Phase in Simple Harmonic Motion
• The phase of a vibrating particle is its position about its displacement and direction of displacement. It can be described using the following formula:
• x = A sin (ωt + Φ)
• This formula (ωt + Φ) is the phase of the oscillating particle. At t = 0, the phase is called the initial phase. 

Conclusion : 

Periodic function repeats itself after a fixed period. An example of this could be our heartbeat. The graph of such patterns looks like a single pattern that is being repeated over and over again. This concept applies to many fields and processes like trigonometric functions, waves, oscillations, phases of a moon, etc. The period is the distance in the x-direction that a function travels before repeating its pattern.