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For the stipulated Klein-Gordon packet velocities, take the original normalized spectral amplitudewith $k_0>0$ and $0<\sigma \ll k_0$. Here $\sigma$ is the width of the spectral amplitude, not the standard deviation of its squared magnitude. Initially $\psi (x,0)=e^{-{\sigma }^2{x}^2/2}{e}^{i{k}_{0}x}$, a Gaussian function multiplying a carrier wave. Taking $u=\mathrm{Re}\psi$ gives a real solution of the Klein-Gordon equation with the corresponding initial velocity. The tiny negative-$k$ tail is retained in this exact integral; the statement that it is mainly right-going is a narrow-spectrum approximation, not a claim of strictly positive wavenumbers.

The dimensionless figure uses $c={\omega }_0=k_0=1$ and $\sigma =0.18$ at $t=0,4,8$. Blue curves are $\mathrm{Re}\psi$, orange bounds are $\pm |\psi |$, and the marked carrier wave reference is $x=v_{p}t$; it need not be a crest of the full wavefield. The dashed envelope-centre reference $x=v_{g}t$ is a leading approximation. These are calculated curves, not experimental records. The exact dispersion relation makes the packet slightly asymmetric; a quadratic approximation predicts symmetric broadening about $v_{g}t$.

The first-order Taylor expansion translates the initial wave envelope without changing its shape. Retaining curvature instead givesThe distinction between translation and curvature-driven spreading is the same one exposed by the Gaussian pulse calculation in Pulse Propagation by Richard Fitzpatrick (equations 9.8–9.18). Its first-order approximation is not an assertion that a dispersive packet keeps its shape indefinitely.

Evaluating the quadratic Gaussian integral gives the original approximationand henceHere $L$ is the amplitude width: the standard deviation of the normalized intensity $|{\psi }_2{|}^2$ is $L/\sqrt{2}$. The peak falls as the packet widens, while $\int |{\psi }_2{|}^{2}dx$ stays constant. This conserved integral is not the complete physical wave energy of the Klein-Gordon equation.

Rigid translation requires $|\beta |t{\sigma }^2\ll 1$. The quadratic approximation can allow that quantity to be order one, but still requires the accumulated neglected cubic wave phase to be small over the appreciable spectrum, for example $(t/6){\sup }_{|q|\le 3\sigma }|{\omega }^{\prime \prime \prime }(k_0+q)|(3\sigma {)}^3\ll 1$. This sufficient check controls only the stated three-width spectral window; the Gaussian tails must also be negligible at the required accuracy. Higher derivatives or very long times require returning to the exact spectral integral. The width $L(t)$ is therefore a controlled local approximation; it neither predicts front velocity nor establishes a universal identity between group velocity and information velocity.

The phase velocity is the motion of a constant-wave phase surface. For a one-dimensional plane wave $u=a\cos (kx-\omega t+{\varphi }_0)$ with real $k\ne 0$ and real $\omega$, fixing $kx-\omega t+{\varphi }_0$ givesEach 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 $\omega \ne 0$. 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 $k\to 0$ is a limiting phase speed, not the speed of a spatially uniform oscillation.

In several dimensions, a plane wave has wave phase $\mathbf{k}\cdot \mathbf{x}-\omega t+{\varphi }_0$. Constant-wave phase planes obey $\mathbf{k}\cdot \mathbf{v}=\omega$. Their normal phase velocity isTangential 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 $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$ and $\lambda =1\mathrm{m}$, the model predicts $c_p\simeq 1.25\mathrm{m}{\mathrm{s}}^{-1}$ and $c_g\simeq 0.625\mathrm{m}{\mathrm{s}}^{-1}$. 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.^[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.