= Wavenumber
{title2=$k$}
{wiki}
= Wave number
{synonym}
A <wavenumber> measures how rapidly a <wave phase> changes with distance. Two conventions coexist: angular <wavenumber> counts <radians> per unit length, while ordinary <spatial frequency> counts cycles per unit length. For a <monochromatic wave> with <wavelength> $\lambda>0$,
$$
K=\frac{2\pi}{\lambda},\qquad \nu_x=\frac1\lambda,\qquad K=2\pi\nu_x.
$$
Here $K\geq0$ is the magnitude of the angular <wavenumber>. The symbol $k$ can instead denote a signed one-dimensional angular <wavenumber>: a factor $e^{ikx}$ has <wavelength> $2\pi/|k|$ for real $k\ne0$. Both conventions have dimensions of inverse length; writing $\mathrm{rad\,m^{-1}}$ or “cycles per metre” exposes the convention. A quoted value in $\mathrm{m^{-1}}$ alone does not resolve the factor of $2\pi$. <Spectroscopic wavenumber> usually uses the cycles-per-length convention.<University Physics Volume 1>[section 16.2, equations 16.2–16.5: angular convention and sinusoidal wave phase]{ref}
For a real <monochromatic> <plane wave>, fix the convention
$$
u(\mathbf x,t)=A\cos\phi(\mathbf x,t),\qquad \phi=\mathbf k\cdot\mathbf x-\omega t+\phi_0,\qquad K=|\mathbf k|.
$$
The <wavevector> $\mathbf k$ supplies both the spatial <wave phase> slope and the normal to the <wavefronts>; its <dot product> with position makes the <wave phase> constant on parallel planes. The <angular frequency> is $\omega=2\pi f$, with $f>0$ the ordinary <frequency>. Moving a distance $\lambda$ along $\mathbf k$ changes the <wave phase> by $2\pi$. Holding a <wave crest>'s <wave phase> fixed gives its <phase velocity> $\omega\mathbf k/K^2$ and <phase speed>
$$
v_p=\frac{\omega}{K}=f\lambda.
$$
In one dimension the signed <phase velocity> is $\omega/k$. This identifies a moving pattern; it does not identify the motion of the medium's particles or a <wave packet>'s <group velocity>.
For the synthetic <plane wave> below, choose $\lambda=0.40\,\mathrm m$ and $f=5\,\mathrm{Hz}$. Then $K=5\pi\,\mathrm{rad\,m^{-1}}$, $\nu_x=2.5$ cycles per metre, $\omega=10\pi\,\mathrm{rad\,s^{-1}}$, and $v_p=2\,\mathrm{m\,s^{-1}}$. The <wave crests> are $0.40\,\mathrm m$ apart at one instant; after a quarter <oscillation period>, each has advanced $0.10\,\mathrm m$. The lower panel retains whole turns of <wave phase>, making its slope visible instead of reducing it modulo $2\pi$.
\Image[/wavenumber-phase-crest-distance.png]
{title=Synthetic plane-wave snapshots: crest spacing, phase slope, and a quarter-period advance}
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