Frequency Formula

The frequency of a wave is generally referred to as how regularly the particles of the medium fluctuate or vibrate when a stream of disturbance is encountered through the medium. Frequency is a part of our usual, everyday language. Frequency, in aspects of physical sciences, is the quantity that determines the number of ripples that pass a settled point in unit time and it can be found using frequency formula. In other words, it is also the number of vibrations or cycles encountered throughout one unit of time by a body or an object subjected to a periodic motion.

In this article, we will have a deep insight into the what is frequency, frequency symbol and frequency formula along with a few solved examples.

Heinrich Hertz

Heinrich Rudolf Hertz (February 22, 1857 – January 1, 1894) was a German scientist who confirmed the existence of electromagnetic waves predicted by James Clerk Maxwell's electromagnetism equations for the first time. In his honor, the frequency unit cycle per second was given the name "hertz."

In his work "On Electromagnetic Effects Produced by Electrical Disturbances in Insulators," which he sent to the Berlin Academy in late 1887, Hertz presented his original finding. Following a series of experiments he conducted in 1888, he later published additional details. The waves he found were originally known as Hertzian waves, but they are now known as radio waves.

What is Frequency?

Now, what is frequency? The frequency of an object subjected to periodic motion determines the number of cycles per unit time. The object or a body considered to be executing the periodic motion will undergo one complete cycle or one vibration after crossing through a series of events or positions and returning to its original state.

The quantity frequency is usually confused with the term period. The period or time period of a wave corresponds to the total time consumed to accomplish one complete wave cycle. When an event takes place frequently, then it is said to be that the particular event is periodic and refers to the time for the event to repeat itself as the period. In spectroscopy, another unit of frequency, the wavenumber, the number of waves in a unit of distance, is sometimes used.

Frequency Symbol:

The physical quantity frequency is represented by the greek letter nu(). In other words, the frequency symbol is nu () or in general, it is denoted by f. The frequency formula is given the following expression.

We know that, according to the definition of frequency, it is the total number of cycles in a unit of time. Therefore, the frequency formula or frequency equation is given by:

$\Rightarrow f=\dfrac{1}{\text{Time}}=\dfrac{\text{Number of cycles}}{\text{Time}}$

$f=\dfrac{1}{T}=\dfrac{n}{t}$…….(1)

Where,

T- The total time period for one full cycle

t- the total amount of time

n- the total number of cycles

Equation (1) is known as the frequency formula or frequency equation. The SI unit of frequency is Hertz and it is abbreviated as Hz. The SI unit of the frequency is Named after Heinrich Hertz. One hertz is equal to one cycle per one second.

Now let us understand what is frequency and frequency formula. Let us assume that, an object took x seconds to finish one complete cycle, then the object would complete $\dfrac{1}{x}$ cycle in one second. Then, the frequency of the object f is $\dfrac{1}{y}~s^{-1}$.

The frequency term plays an important role while studying waves and oscillations. With the help of the frequency of the wave, one can determine many physical aspects of the wave under consideration. With the help of frequency, we can determine the wavelength of the wave, time period of the wave, etc.

The frequency of any wave can be determined by the following expression:

$\Rightarrow \nu=\dfrac{c}{\lambda}$

Where,

c- The speed of light

𝜆 - The total wavelength of the given wave

The frequency of a wave is inversely proportional to the wavelength, which implies that shorter wavelengths will have the longest frequencies. For example, we know that from the electromagnetic spectrum gamma rays have the shortest wavelength ranging from 10-10 - 10-16 m, thus they are having the longest frequency range among all electromagnetic radiations and the frequency of gamma rays range from ranging from $3\times10^{18}-3\times10^{24}~m$.

Interesting Facts

• In a vacuum, all electromagnetic waves travel at the speed of light, but in a medium that isn't a vacuum, they travel at a slower pace. Sound waves, for example, travel at significantly slower speeds and cannot pass into a vacuum.

• Light waves, radio waves, infrared radiation, microwaves, and gamma waves are examples of electromagnetic waves.

Examples:

Now, let us solve a few solved examples.

1. A long bar pendulum needs 8 seconds to finish one full cycle. Find out the frequency of oscillation of the given bar pendulum?

Sol:

The total time period (T) of the bar pendulum is 8 seconds. We know that we can use the frequency formula to find frequency of the bar pendulum.

The frequency formula is given by:

$\Rightarrow f=\dfrac{1}{T}$

Where,

T- The total time period to complete one full cycle

$\Rightarrow f=\dfrac{1}{8}$

$\Rightarrow f=0.125~Hz$

Therefore, the frequency of the bar pendulum is 0.125 cycles per unit time. The writing of the unit cycles is regularly considered “Hertz”, and its symbol is Hz. Hence, the frequency of this particular bar pendulum is 0.25 Hz.

2. A wave is introduced into a fragile wire held tight at each end. It has an amplitude of 5 cm, a frequency of 70 Hz and a distance from a crest to the neighbouring trough of 20 cm. Determine the period of a given wave.

Sol:

In this question, we are asked to determine the total time period of the given wave which is generated by a fragile wire held tight at each end. We know that frequency is reciprocal of the time period, so we can find the time period of the wave by the following expression:

The frequency formula is given by:

$\Rightarrow f=\dfrac{1}{T}$

$\Rightarrow T=\dfrac{1}{f}$

Where,

T- The total time period to complete one full cycle

Substituting the value of frequency in the above expression, we get:

$\Rightarrow T=\dfrac{1}{70}$

$\Rightarrow T=0.0143~s$

Therefore, the time period of the given wave which is generated by a fragile wire held tight at each end is 0.0143 seconds

3. A tennis trainer paces back and forth along the sideline 20 times in 3 minutes. Then, what is frequency of her pacing?

Sol:

We know that the frequency gives rise to the total number of oscillations of a periodic event per time and it is most frequently measured in cycles/second. In this example, there are a total of 20 cycles per 3 minutes (i.e., 20 cycles per 180 seconds). The total time period (T) of the tennis trainer is 20 times in 3 minutes (180 seconds).

We know that we can use the frequency formula to find the frequency of the pacing of the tennis trainer. Then, the frequency formula is given by:

$\Rightarrow f=\dfrac{\text{Number of cycles}}{\text{Time}}$

$\Rightarrow f=\dfrac{n}{T}$

Where,

t- the total amount of time

n- the total number of cycles

$\Rightarrow f=\dfrac{20}{180}=0.1111~Hz$

$\Rightarrow f=111.1~mHz$

Therefore, the frequency of the pacing of the tennis trainer is 111.1 mHz cycles per unit time.

Conclusion

We conclude that the frequency is defined as the number of waves passing through a given point in one unit of time, as well as the number of cycles or vibrations that a body in periodic motion experiences in one unit of time. The Greek letters nu () and omega () are the most popular symbols for frequency. The unit of frequency is hertz (Hz), which is equivalent to one occurrence per second. The period is the reciprocal of the frequency since it is the length of one cycle in a repeated occurrence.