Frequency is the number of cycles in a given unit of time. The “cycles” can be movements of anything with periodic motion, like a spring, a pendulum, something spinning, or a wave. The frequency formula is used to calculate a wave’s frequency. For example, the sound of the fan is a sound wave, while the light is an electromagnetic wave.

The SI Unit of Frequency is measured in Hertz (Hz). In this article, let us understand the formula of frequency and how to calculate with solved examples. 

What is Frequency?

In physics, the frequency can be described as the number of times a repeating event occurs in a specified time frame. It is also equal to the number of cycles or vibrations a body undergoes in periodic motion during a unit of time. A body undergoes periodic motion when it covers one cycle or a single vibration as it passes through a series of events or points and returns to its original or initial position. In wave motion, we use the word frequency to determine how often the medium particles vibrate when a wave passes through the medium.

If a body requires \(1/2\) second to undergo one vibration or cycle, its frequency will come out to be \(2\) per second. Similarly, the frequency of an object, taking \(1/50\) of an hour to complete a vibration or cycle, is \(50\) per hour. For example, the frequency of revolution of the moon around Earth is slightly over \(12\) cycles per annum, while the vibration frequency of guitar strings is around \(400\) cycles per second.

Various symbols are used to describe frequency. The most commonly used symbol is \(f\) and, other popular symbols are the Greek letters nu \((ν)\) and Omega \((ω)\). 

We often use the symbol nu to specify the frequency of electromagnetic waves, like light, \(X-\)rays, and gamma rays. 

The rotational frequency, described by Omega, is used to define the angular frequency, i.e. the number of revolutions of rotations of a body in radians per second.

Frequency Formula

Mathematically, the frequency is defined as the reciprocal amount of time period, i.e.,

\({\rm{frequency = }}\frac{{\rm{1}}}{{{\rm{ period }}}}{\rm{ = }}\frac{{\rm{1}}}{{{\rm{ time\, interval }}}}\)

or \(f = \frac{1}{T}\)

Since the given periodic motion frequency is the inverse of the time taken to complete one oscillation thus, the unit of frequency will be:

SI unit of frequency \({\rm{ = }}\frac{{\rm{1}}}{{{\rm{SI\, Unit\, of\, time }}}}{\rm{ = }}\frac{{\rm{1}}}{{{\rm{ second }}}}{\rm{ = Hertz}}\)

One hertz can be defined as one cycle per second. The unit ‘hertz’ can be used to describe any periodic movement. Usually, frequency is expressed within the hertz unit, named in honour of the \(19^{\rm{th}}\)-century German physicist Heinrich Rudolf Hertz. One hertz is adequate to one cycle per second, abbreviated \({\rm{Hz}}.\) One kilohertz \({\rm{(kHz)}}\) is \({1,000\;\rm{Hz}}\) and one megahertz \({\rm{(MHz)}}\) is \(1,000,000\;{\rm{Hz}}.\) The frequency of a ticking clock is \({1\;\rm{Hz}}\) while the frequency of a beating heart is almost \({\rm{1.2}}\;{\rm{Hz}}.\) Wavenumber, another unit of frequency generally used in spectroscopy, gives the value of the total number of waves travelling in a unit distance.

Frequency of a Wave

The frequency of a wave is equal to the number of waves passing a specific point in the given amount of time. It is measured in hertz, where one hertz is equal to the passing of one wave in one second. The frequency of a wave is the same as the frequency of vibrations producing it. 

For a given wave of wavelength travelling at a speed \(v\) in a medium, the frequency can be given as:

\(f = \frac{y}{\lambda }\)

For a light wave travelling in a vacuum, \(f = \frac{c}{\lambda }\)

Where \(c\) is the speed of light.

Thus, frequency is inversely proportional to the wave’s wavelength; thus, higher frequency waves have crests closer together; therefore, these waves have shorter wavelengths. Also, a higher frequency wave has more energy than a smaller frequency wave with the same amplitude.

Types of Frequency

Based on the periodic motion of the particle, there are mainly two types of frequency,

1. Angular Frequency: Angular frequency is associated with a particle undergoing rotation motion; the angular frequency is equal to the number of revolutions in a specific time. The unit of angular frequency is hertz. The frequency and angular frequency of a particle are related as follows: 

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

Where, \(ω\) – angular frequency and \(f\)- frequency of vibration

2. Spatial Frequency: Its spatial coordinate governs the spatial frequency of a particle. Spatial frequency is inversely proportional to the wavelength of the vibrating particle. The spatial frequency is the measurement of the structure that is a characteristic of its period in space.

Solved Examples

Q.1. The light wave has a wavelength of \(400\,\rm{nm}\). Compute its frequency?
Ans: We are Given: Velocity of light, \(c = 3 \times {10^8}\;\rm{m/s}\)
The wavelength of the wave, \(\lambda = 400\;{\rm{nm}} = 400 \times {10^{ – 9}}\;{\rm{m}}\)
Frequency is given by,\(f = \frac{c}{\lambda }\)
\(f = \frac{{3 \times {{10}^8}\;{\rm{m}}/{\rm{s}}}}{{400 \times {{10}^{-9} }{\rm{m}}}} = 7.5 \times {10^{14}}\;{\rm{Hz}}\)

Q.2. A long pendulum takes \(5\) seconds to complete one back-and-forth cycle. Find out the frequency of the pendulum’s motion?
Ans: We are given, The period ‘\(T\)’ of the pendulum is \(5\) seconds. 
Frequency of the pendulum \(f = \frac{1}{T}\)
\(f = \frac{1}{5}\)
\(f = 0.20\frac{{{\rm{cycles}}}}{{\rm{s}}}\) or \(\,0.20\;{\rm{Hz}}\)

Q.3. A car’s tachometer measured the number of revolutions per minute of its engine. A car travels at a constant speed, and the reading of the tachometer is \(1200\) revolutions per minute. Find out the frequency of the engine spinning. Also, find out the period in seconds.
Ans: We are given,
The number of cycles or revolutions per minute \(=N=1200\)
frequency, \(f = \frac {\rm{Number}\;\rm{of}\;\rm{cycles}}{t}\)
\({\rm{f = 2200 \,cycles\, per\, minute = }}\frac{{{\rm{1200\, cycles}}}}{{{\rm{60}\,\rm{s}}}}{\rm{ = 20\, cycles/s}}\)
To find the time period, use the relation, \(f = \frac{1}{T}\)
\(T = \frac{1}{f}\)
\(T = \frac{1}{{20}}\)
\(T = 0.05\;{\rm{s}}\)

Summary

In physics, the frequency can be described as the number of times a repeating event occurs in a specified time frame. It is also equal to the number of cycles or vibrations of a body in periodic motion during a unit period. Mathematically, the frequency is defined as the reciprocal amount of time. SI unit of frequency is hertz. One hertz can be defined as one cycle per second. The frequency of a wave is equal to the number of waves passing a specific point in the given amount of time. The wave’s frequency is obtained by dividing the speed of a wave in the given medium by its wavelength. 

FAQs

Q.1. Write the SI unit of frequency.
Ans: The SI unit of frequency is hertz.

Q.2. Define frequency?
Ans: Frequency is defined as the number of times an event takes place per unit of time.

Q.3. What is the angular frequency?
Ans: The angular frequency is defined as the angular displacement of any element of the wave per unit of time. It can be given as, \(\omega = 2\pi f.\)

Q.4. Give the formula of frequency.
Ans. The frequency is equal to the reciprocal of the time period, thus, \(f=\frac{1}{T}\).

Q.5. Write the formula for the frequency of a wave.
Ans: The frequency of a wave can be given as \(f=\frac{v}{\lambda}\)  where \(v\) is the velocity and \(\lambda\) is the wavelength of the wave.