Frequency, Time Period And Angular Frequency

In wave mechanics, any given wave enfolds parameters like- frequency, time period, wavelength, amplitude etc. The time period is the time taken by a complete cycle of the wave to pass a point, Frequency is the number of complete cycle of waves passing a point in unit time. Angular frequency is angular displacement of any element of the wave per unit time.

Consider the graph shown below. It represents the displacement y of any element for a harmonic wave along a string moving in the positive x-direction with respect to time. Here, the element of the string moves up and down in simple harmonic motion.

Period And Angular Frequency

The relation describing the displacement of the element with respect to time is given as:

y (0,t) = a sin (–ωt), here we have considered the inception of from x=0

y (0,t) = -a

As we know sinusoidal or harmonic motion is periodic in nature, i.e. the nature of the graph of an element of the wave repeats itself at a fixed duration. In order to mark the duration of periodicity following terms are introduced for sinusoidal waves.

Following is the table explaining other related concepts of waves:

What is Time Period?

In sinusoidal wave motion, as shown above, the particles move about the mean equilibrium or mean position with the passage of time. The particles rise till they reach the highest point that is the crest and then continue to fall till they reach the lowest point that is the trough. The cycle repeats itself in a uniform pattern. The time period of oscillation of a wave is defined as the time taken by any element of the string to complete one such oscillation. For a sine wave represented by the equation:
y (0, t) = -a sin(ωt)

The time period formula is given as:

\(T=\frac{2\pi }{\omega }\)

What is Frequency?

We can define frequency of a sinusoidal wave as the number of complete oscillations made by any element of the wave per unit time. For a sinusoidal wave represented by the equation:
y (0,t) = -a

The formula of the frequency with the SI unit is given as:

Formula \(f=\frac{1}{T}=\frac{\omega }{2\pi }\)
SI unit Hertz

One Hertz is equal to one complete oscillation taking place per second.

What is Angular Frequency?

For a sinusoidal wave, the angular frequency refers to the angular displacement of any element of the wave per unit time or the rate of change of the phase of the waveform. It is represented by ω. Angular frequency formula and SI unit are given as:

Formula \(\omega=\frac{2\pi }{T}=2\pi f\)
SI unit rads-1

Where,

  • ω = angular frequency of the wave.
  • T = time period of the wave.
  • f = ordinary frequency of the wave.

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Practise This Question

If the power of two lenses and P1 and P2; when they are combined; the effective power is: