The angular, or circular, wave number $k$, often misleadingly abbreviated as wave number, is the modulus of the wave vector. It is inversely proportional to the wave length $\lambda$.
The wave number is the spatial analogue of the angular frequency:
application of a Fourier transformation on data in the time domain
yields a frequency spectrum,
applied on data in the spatial domain (data as a function of position)
yields a spectrum as a function of the wave number.
Note that the exact definition of the wave number depends on the
physics field studied.
In mechanics, the angular wavenumber is defined as:
where $f$ is the frequency (in Hertz), $\omega$ is the angular frequency (in rad.s$^{-1}$) and $v_{\phi}$ is the phase velocity of the wave (in m.s$^{-1}$).
Complex wave numbers
Complex wave numbers are a commonly way to account for dissipations.
For a harmonic time dependence of the form $e^{j\omega t}$, a pressure wave propagating toward negative $x$ writes:
If the wavenumber $k$ is a complex quantity of the form $k=\alpha + j\beta$ with $\alpha$ and $\beta$ are defined as real positive quantities, the pressure wave can be re-written as:
\[ p(x,t) = A e^{j(\omega t + \alpha x)} e^{-\beta x} \]$e^{-\beta x}$ is an exponential attenuation (or decay) with $\beta$ as the characteristic dimension.
Complex and frequency dependent wave number
The dissipation mechanisms (e.g. visco-inertial effects, thermal effects...) can usually not be considered as constant with the frequency (see e.g. the page about the viscous boundary layer). Thus, the complex wave number is also frequency dependent (as is the velocity $v_{\phi}$).
This frequency dependence implies, at least in a given frequency range, a dispersion of the wave following Kramers-Kronig relations.