Wavenumber
k = 2π/λ, the spatial angular frequency of a wave, measured in rad/m. The wavevector k has magnitude k and direction along propagation. In spectroscopy, wavenumber often means k̃ = 1/λ in cm⁻¹.
Definition
The wavenumber k = 2π/λ is the spatial analogue of angular frequency ω = 2πf. It measures how many radians of phase the wave accumulates per unit distance along the propagation direction, with units of rad/m. The wavevector k is the vector generalisation: its magnitude is the wavenumber, and its direction is the direction of propagation. The phase of a plane wave at position r and time t is written k·r − ωt, making the wavevector a natural Fourier-space variable.
The dispersion relation ω(k) specifies how wavenumber relates to frequency in a given medium: in vacuum, ω = ck (linear, non-dispersive); in a refractive medium, ω = ck/n(ω) (dispersive, since n depends on ω); in a plasma above the cutoff, ω² = c²k² + ω_p². Group velocity v_g = dω/dk and phase velocity v_p = ω/k diverge in dispersive media, which is the origin of pulse spreading in optical fibres and of coloured fringes in prism spectrometers. A separate convention in vibrational spectroscopy uses "wavenumber" to mean k̃ = 1/λ in cm⁻¹ (without the 2π), which is why infrared absorption peaks are usually quoted in units like "1730 cm⁻¹" for a carbonyl stretch — that is the reciprocal wavelength, not the angular wavenumber.