Wave number

Intermediate

Number of radians per metre.

Key relationk = 2π/λ

k = 2π/λ in rad/m is the spatial analogue of ω: it counts how many radians of phase per metre.

Worked example

λ = 0.5 m gives k = 12.6 rad/m.

Common mistake

k isn't the same as 1/λ (that's the wavenumber used in spectroscopy).

In more detail

The wave number k = 2π/λ measures how many radians of phase change occur per metre. It is the spatial analogue of angular frequency.

Wave number appears in quantum mechanics, where it relates to particle momentum.
Optics uses wave number to describe light in materials.
Optical engineers use the wave number to describe the spacing of ripples in a grating or interference pattern.

What units does k have?

Radians per metre.

How is k related to ω?

The wave speed is ω divided by k.

How is wave number different from wavelength?

Wave number is 2π divided by the wavelength, so it counts radians per metre rather than metres per cycle.