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What is Wave Number?

emmission spectrum of iron

In physics, the wave number is a quantity that is inversely related to wavelength, having inverse meters as its SI unit (/m or m-1). Wave number is a measurement of a certain number of wavelengths for some given distance. In a sense, the wave number is like a spatial analogue of frequency. Typically, wave number is taken to be 2π times the number of wavelengths per unit of distance, which is the number of radians for each unit of distance as well. There are many different wave number equations that relate to many different fields of physics. We have listed a few of them here:

k = 2π/λ

where k is the wave number and λ is the wavelength. This will work for many situations in physics involving a frequency (an LC circuit in electronics, for example).

The wave number for electromagnetic waves is described with the following set of equations:

k = 2π/λ = 2πv/vp = ω/vp = E/ħc

In the second equation, v is taken to be the frequency of the wave, and vp is the phase velocity of the wave. The phase velocity is the speed at which one phase of the wave propagates in space. The variable ω appears in the next equation and is known as the angular frequency, or rotation rate (cycle rate) of the electromagnetic wave. ω can also be written as:

ω = 2πv

notice the relationship between the angular frequency and frequency. As frequency increases, so does the angular frequency, which is to say there are more cycles when we have a higher frequency for the same measured wavelength.

The last equation assumes that we are working with an electromagnetic wave propagating through a vacuum. In such a case, our phase velocity is equal to the speed of light constant, c. Angular frequency, then, is taken to be:

ω = E/ħ

where E is the energy of the wave and ħ (h bar) is the reduced Planck constant (equal to the regular Planck constant, h, divided by 2π). It is common to use the reduced Planck constant when working with angular frequencies. The above equation is a rewritten version of one of the Planck energy-frequency equivalency equations:

E = ħ ω

Matter waves are mathematically expressed as:

k = 2π/λ = p/ħ = √(2mE)/ħ

In the above set of equations, p is the momentum of the matter wave, m is the mass of the particle, and E is the kinetic energy of the particle. The above equation is a non-relativistic version that approximates the value for the wave number for most situations in which quantum factors are negligible. The relativistic equation takes account of the fact that there is a difference between reference frames and therefore relates the speed of the particle to the speed of light to determine its wave number.

In spectroscopy it has historically been convenient to use a "wavenumber", which refers to a frequency which has been divided by the speed of light in a vacuum.

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