Phase velocity
= Phase velocity
{title2=$v_p=\omega/k$}
{wiki}
The <phase velocity> is the motion of a constant-<wave phase> surface. For a one-dimensional <plane wave> $u=a\cos(kx-\omega t+\phi_0)$ with real $k\ne0$ and real $\omega$, fixing $kx-\omega t+\phi_0$ gives
$$
k\frac{dx}{dt}-\omega=0,\qquad c_p=\frac{\omega}{k}.
$$
Each <wave crest> of this monochromatic <travelling wave> advances at that signed <phase velocity>. Its magnitude is the <phase speed> $|c_p|=\lambda/T$, where $\lambda=2\pi/|k|$ is the <wavelength> and $T=2\pi/|\omega|$ is the temporal <period> when $\omega\ne0$. Choosing $\omega>0$ makes positive $k$ mean rightward <phase velocity> and negative $k$ mean leftward <phase velocity>. Reversing both signs leaves the real <plane wave> and its <phase velocity> unchanged. At $k=0$ there are no spatial crests to track, so $\omega/k$ is undefined; a finite limit as $k\to0$ is a limiting <phase speed>, not the speed of a spatially uniform oscillation.
In several dimensions, a <plane wave> has <wave phase> $\mathbf k\cdot\mathbf x-\omega t+\phi_0$. Constant-<wave phase> planes obey $\mathbf k\cdot\mathbf v=\omega$. Their normal <phase velocity> is
$$
\mathbf c_p=\frac{\omega\mathbf k}{|\mathbf k|^2}\qquad(\mathbf k\ne0).
$$
Tangential motion within a plane does not move that <wavefront>, which is why the normal direction fixes the vector convention. In an <anisotropic> medium the <group velocity> need not point along this <wavevector>. Likewise the <horizontal phase velocity> of a tilted <wavefront> is its crossing speed along a horizontal line, rather than a component of the normal <phase velocity>.
For the <nondispersive wave> branch $\omega=ck$, a <wave crest> and a <wave envelope> both move at $c$. For deep-water <surface gravity waves>, $\omega=\sqrt{gk}$ gives $c_p=\sqrt{g/k}$ and <group velocity> $c_g=c_p/2$. With the illustrative choices $g=9.81\,\mathrm{m\,s^{-2}}$ and $\lambda=1\,\mathrm m$, the model predicts $c_p\simeq1.25\,\mathrm{m\,s^{-1}}$ and $c_g\simeq0.625\,\mathrm{m\,s^{-1}}$. These are calculated values for negligible <surface tension> and sufficiently large $kh$, not measurements. A <wave crest> moves forward through the <wave packet>, so timing a crest does not time the packet's journey.
There is a direct laboratory meaning to this distinction. MIT's 2005 surface-wave lab uses a long water tank, a controllable hydraulic paddle, and an absorbing beach. Resistance probes convert submerged length into voltage; calibration converts their readouts into <free surface> elevation versus time. Two probes a known distance apart, backed by a ruler and stopwatch, let students time a <wave crest> and separately time a feature of the <wave packet>. The lab records depth and varies <frequency> and <wave amplitude>.<Surface Waves Kinematics (MIT 2.016 Hydrodynamics) by Massachusetts Institute of Technology, 2.016 course staff>{ref}
For a clean measurement of <phase velocity>, fit the periodic portion of both records or follow one identifiable <wave crest>. A phase difference is only known modulo $2\pi$: probe spacing below half the shortest relevant <wavelength>, or continuity while sweeping <frequency>, avoids assigning the wrong whole-cycle delay. Then compare the timed <phase velocity> with $\lambda/T$. Tracking pepper grains instead measures fluid-particle motion, which has a different meaning from either <phase velocity> or <group velocity>.
The <Klein-Gordon packet velocities> example makes the distinction especially sharp: its <phase velocity> exceeds $c$, yet its narrow <wave envelope> moves below $c$. A <carrier wave> labels successive oscillations; its speed alone does not establish <information velocity>. The <front velocity> depends on the full causal model, including <angular frequencies> far outside the carrier's narrow range.