The wavenumber is a quantity associated with a periodic wave that is usually denoted k and has units of the inverse of length. The wave number is related to the wavelength λ by the relation k=2π/λ. The relationship between wavenumber and wavelength for the spatial dependence of a wave is analogous to the relationship between angular frequency and period in the time dependence. The wavenumber tells you how much of an angle (measured in radians) that the argument of the wave goes through as you move a unit of distance through space. A large wavenumber means that the wave changes significantly even over small distances, while a small wavenumber means that the wave varies only gradually with distance. If a traveling wave oscillates in time with an angular frequency ω, then the speed of the wave will be ω/k.
The wavenumber is commonly used because it comes into mathematical formulas for waves in a simpler way than wavelength. The spacial dependence of a sinusoidal wave is
Acos(kx + φ)
in terms of the wavenumber. For a certain wave equation, the wavenumber and angular frequency of waves will be related by the dispersion relation ω(k).
For waves in multidimensional space, the concept of wavenumber becomes even more useful, because we can think of the wavenumber as a vector k, sometimes called the wavevector. The magnitude has the same meaning as in one dimension, and the vector points in the direction of the wave (the direction along which the wave's value varies). The equation for the wave then becomes
Acos(k⋅x + φ)
This formula would not be as cleanly formulated in terms of the wavelength, showing the advantage of wavenumber for mathematical description. In special relativity, this concept is used and the wavenumber becomes a four-vector k=(ω/c,k) and the equation for a wave becomes
Acos(kμxμ + φ)
where x is the four-vector position in spacetime. The formula then contains both time and space dependance, and the argument the wave is now Lorentz invariant.
The wavenumber comes up naturally in quantum mechanics, where particles travel as waves. The momentum of particle is directly proportional to the wavenumber of the particle's wavefunction, according to p = h/(2π)*k, a form of de Broglie's relation. In units where h bar equals 1, then the relation is simply p = k.