Wavenumber Calculator — Spectroscopic (cm⁻¹) & Angular (rad/m)

Formula

How this is calculated

Wavenumber is the spatial frequency of a wave — how many complete cycles or radians fit into one unit of length. Two conventions are in common use. Spectroscopic wavenumber ν̃ = 1/λ (read "nu-tilde") counts full cycles per centimetre; it is the default axis unit in infrared and Raman spectra because its range (roughly 100–4000 cm⁻¹ for IR-active vibrations) is practically convenient and linearly proportional to energy. Angular wavenumber k = 2π/λ counts radians per metre; it appears in the wave equation y = A sin(kx − ωt), the de Broglie relation p = ℏk, and quantum-mechanical dispersion relations.

The calculator converts your wavelength to metres first, then to centimetres for ν̃. A 500 nm green photon has ν̃ = 1/(500×10⁻⁷ cm) = 20000 cm⁻¹ and k = 2π/(500×10⁻⁹ m) ≈ 1.257×10⁷ rad/m. The spatial frequency in m⁻¹ (without the 2π factor) equals 1/λ in metres, another alternative that some texts use.

Both wavenumbers are exact for a monochromatic wave in vacuum. In a medium with refractive index n, the effective wavelength shortens to λ/n, so the wavenumber inside the medium is n times larger. The calculator uses the vacuum (free-space) wavelength you enter.