Wavenumber Calculator

Background.

An infrared spectrum of a ketone shows a strong band near 1,715 cm⁻¹. That number is a wavenumber: the count of complete waves that fit into one centimetre. Turned round, it is light with a wavelength of 5.831 µm. A physicist describing the same light in a wave equation would call its wavenumber something else again, about 1.0776 × 10⁶ rad/m, because in that field "wavenumber" means phase per metre rather than cycles per metre.

Both meanings are in everyday use, and they differ by a factor of 2π. Spectroscopists use ν̄ = 1 ⁄ λ, almost always quoted per centimetre. Physicists and engineers use the angular wavenumber k = 2π ⁄ λ in radians per metre, the k in sin(kx − ωt), in momentum p = ħk and in band-structure diagrams. Put a spectroscopic value into a formula expecting k and the answer is out by a factor of 6.283, with nothing to warn you.

Enter one wavelength, in metres, and both are given side by side, with no guessing about which convention a source intended. The per-metre spatial frequency is the SI form; to get the cm⁻¹ that infrared and Raman tables print, divide it by 100.

Wavenumber earns its place in spectroscopy because it is proportional to photon energy, E = hcν̄. Energy levels can be added and subtracted directly as band positions, so a Raman shift or a hydrogen line is just a difference between two wavenumbers. Wavelengths do not combine that way.

What is wavenumber calculator?

Wavenumber is the spatial counterpart of frequency. Frequency counts oscillations per second; wavenumber counts them per unit length. The spectroscopic wavenumber ν̄ = 1 ⁄ λ counts whole cycles per metre, or per centimetre in laboratory practice. The angular wavenumber k = 2π ⁄ λ counts radians of phase per metre, since one full cycle is 2π radians. Both have the dimension of inverse length and describe the same wave; they differ only in whether a cycle is counted as 1 or as 2π.

How to use this calculator.

1. 01Convert the wavelength to metres before typing it: 500 nm is 5e-7, 10.6 µm is 1.06e-5, 2 mm is 0.002. The field accepts anything from 2 × 10⁻¹⁵ m upward.
2. 02Use a vacuum wavelength if you want to compare with spectroscopy tables. If you measured the wavelength inside glass or water, multiply it by the refractive index first.
3. 03Read k in rad/m for wave equations, phase velocity ω ⁄ k, or photon and electron momentum.
4. 04Read the spatial frequency in m⁻¹ for spectroscopy, and divide it by 100 to compare with tables in cm⁻¹.

The formula.

A wave of the form sin(kx − ωt) repeats when its phase kx advances by 2π, and by definition that happens over one wavelength. So kλ = 2π, which gives k = 2π ⁄ λ. Counting whole cycles instead of radians removes the 2π and leaves the spectroscopic wavenumber ν̄ = 1 ⁄ λ. The ratio k ⁄ ν̄ is always exactly 2π, about 6.2832, which is a quick check that you have the two outputs the right way round.

Because both are reciprocals of wavelength, halving the wavelength doubles each of them, and a very short wavelength gives a very large wavenumber. Several familiar relations hang off them. Photon energy is E = hcν̄, so 1 eV corresponds to 8,065.54 cm⁻¹. Momentum is p = ħk = h ⁄ λ, for light and for matter waves alike. Frequency needs one more ingredient, the wave's speed: f = vν̄, or ω = vk for the angular pair.

The cm⁻¹ conversion trips people because inverse units run backwards. A metre holds 100 centimetres, so one cycle per metre is only 0.01 cycles per centimetre; per-metre values are divided by 100, not multiplied.

A worked example.

Frequently asked questions.

References& sources.

1. [1]ISO 80000-3:2019, Quantities and units — Space and time (BIBLIOGRAPHIC/PAYWALL).
2. [2]OpenStax, University Physics Volume 1, section 16.1, Traveling Waves.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.