Period, Angular frequency and Frequency
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.
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 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.
In a 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 is given mathematically as T = 2π/ω.
The frequency of a sinusoidal wave can be defined 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, frequency of the wave is given by,
Its unit is Hertz abbreviated as Hz. One Hertz is equal to one complete oscillation taking place per second.
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 and its SI unit is rads-1.
ω = angular frequency of the wave.
T = time period of the wave.
f = ordinary frequency of the wave.
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