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A Study of Wavelength, Energy, and Frequency With Their Formulas

Wavelength Formula
From UV rays to radio waves, we are surrounded by waves and have developed various ways to measure, understand, and utilize them for our benefit. Wavelength is a very significant characteristic of a wave, and outlined in this article are the various wavelength equations.
Kundan Pandey
Last Updated: Feb 21, 2018
Fast Fact:

Every type of light has a different wavelength. UV rays and X-rays have shorter wavelengths, while radio waves move faster.

A wave can be defined as a disturbance that travels through a medium, often, without the matter actually changing its position. A wave transfers energy by vibration of particles of the medium. A simple way to understand this process is by observing a toy boat in still water. When small ripples are formed on the surface of water, the boat will only bob up and down, without actually changing its location. The particles in a medium act in similar manner.

Wavelength is the distance between any two points which are at the same position on adjacent waves. For example, the distance between the highest points of two waves. The highest points on a wave are called crests and lowest are called toughs; crests are usually used for measuring wavelength, as they are easiest to measure.

Wavelength Formula
Wavelength is one the three important characteristics of a wave, the other two being amplitude and frequency. It is usually measured in units of distance such as meter, centimeter, nanometer, or Ångström; with meter (m) being its SI unit. Wavelength is denoted by the Greek alphabet λ (lambda) and the formula used for calculating wavelength is as follows -
λ = v / f

λ = wavelength in m
ν = velocity of the wave in m/s
f = frequency of wave in Hz

As the speed of electromagnetic waves remains constant, v is often replaced by c, which is the speed of light in a vacuum and is around 3x108m/s. Thus, the wavelength formula now becomes:

λ = c / f

All waves have different wavelengths, with radio waves having the longest wavelength (1m - 1km), and gamma the shortest (less than 0.01nm). The wavelength that is visible to us falls in the range of 380nm - 740nm.

Frequency and Wavelength
Frequency of a wave is the number of oscillations/vibrations produced over a period of time, say, in a second. It is calculated by measuring the number of waves passing through a point in a second. It is represented by the Greek letter nu (although f is used here to avoid confusing it with v (velocity)) and is measured in Hertz (Hz). Formula for calculating frequency is:
f = 1 / T

T = time period
Hertz, the SI unit of frequency, is defined as the number of cycles a periodic phenomena is repeated in a second i.e. number of cycles per second; and the formula for Hz using wavelength is -
Hz= v / λ
Hz= c / λ

As you can see from the above equations, wavelength and frequency are inversely proportional. Hence, shorter the wavelength, higher will be the frequency of the wave and a longer wavelength will mean low frequency. For example, microwaves with wavelength in the range of 187 mm - 1 mm have frequency from 1 to 300x106Hz; while x-rays whose wavelengths fall between 10nm to 0.01nm have frequency between 30x1015 Hz and 100x1018 Hz.

Phase Velocity
Velocity used in the wavelength formula above, is the phase velocity i.e. the speed at which a phase of the wave propagates in space. This brings us to a very important formula when it comes to study of waves, wherein phase velocity is calculated by multiplying frequency with wavelength:

v = fλ

Energy from Wavelength
Energy in a photon of light, represented by E, can be calculated using wavelength as in the following equations:

E = hc / λ
E = hv / λ

Here, h is the Planck's constant, with a value of approximately 6.63x10-34. Please note that v here is phase velocity and not frequency, though frequency can also be used to derive energy.

Electromagnetic waves are described using different characteristics, for example optical and infrared light are described by their wavelength, X-ray and gamma rays by their energy and radio waves by their frequency. As can be inferred from these equations, frequency, wavelength, and energy are mathematically inter-related, and the wavelength of a wave is inversely proportional to its frequency.
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