Phase velocity

The phase velocity $u_p$ is the speed at which a point of constant phase on a sinusoidal wave moves through space. For a wave

$\psi (z,t)=A\cos (\omega t-\beta z),$

a point of constant phase satisfies $\omega t-\beta z=\text{const}$, so

$u_p=\frac{dz}{dt}=\frac{\omega }{\beta }\text{(m/s)}.$

Here $\omega$ is the angular frequency (rad/s) and $\beta$ is the phase constant (rad/m). The relation $u_p=\omega /\beta$ is universal — it works for transmission lines, plane waves, sound waves, water waves, anything sinusoidal.

On a transmission line

For a lossless Transmission line with distributed inductance $L^{\prime }$ and capacitance $C^{\prime }$ per unit length:

$u_p=\frac{1}{\sqrt{L^{\prime }{C}^{\prime }}}.$

Since $L^{\prime }{C}^{\prime }=\mu ϵ$ for any TEM line (a universal geometry-independent identity — see Transmission line):

$u_p=\frac{1}{\sqrt{\mu ϵ}}=\frac{c}{\sqrt{{\mu }_r{ϵ}_r}}.$

For a non-magnetic dielectric (${\mu }_r=1$):

$u_p=\frac{c}{\sqrt{{ϵ}_r}}.$

Common values: RG-58 coax has ${ϵ}_r\approx 2.3$, giving $u_p\approx 0.66c\approx 2.0\times 10^8$ m/s. Twisted-pair phone cable: $u_p\approx 0.7c$. A “5 ns/m” rule of thumb for PCB FR-4 traces corresponds to $u_p\approx 0.67c$.

The takeaway: signals on real lines travel slower than $c$, by a factor of $\sqrt{{ϵ}_r}$. Wavelength on the line is shortened by the same factor: $\lambda ={\lambda }_0/\sqrt{{ϵ}_r}$.

In a uniform dielectric

For a Plane wave in an unbounded uniform medium, the same formula:

$u_p=\frac{1}{\sqrt{\mu ϵ}}=\frac{c}{\sqrt{{\mu }_r{ϵ}_r}}.$

In free space, $u_p=c$ — exactly. (The defined speed of light, since 1983.)

Versus group velocity

Phase velocity is the speed of a single Fourier component. A real signal — a pulse, a wave packet — is built from many frequencies. If $u_p$ depends on $\omega$ (a dispersive medium), different components travel at different speeds and the pulse spreads. The pulse envelope moves at the group velocity:

$u_g=\frac{d\omega }{d\beta },$

different from $u_p=\omega /\beta$. In non-dispersive media (where $\beta \propto \omega$), the two coincide.

In a dispersive medium, $u_p$ can exceed $c$ without violating relativity — phase velocity isn’t the speed of information transport. $u_g$ is closer to the physical signal speed, and stays at or below $c$ in normal media. Electromagnetics mostly works in the non-dispersive lossless regime where the distinction doesn’t matter; group velocity is properly introduced in a later wave-propagation course.

Why “phase” velocity

A snapshot of $\cos (\omega t-\beta z)$ at $t=0$ is a sinusoid in $z$. At $t=\Delta t$, the same shape has shifted by $\Delta z=(\omega /\beta )\Delta t$. The pattern slides through space at speed $\omega /\beta$ — the speed at which “the same phase” (e.g., a particular zero crossing, or a peak) appears at successive points.

This is a kinematic quantity: it describes the geometry of the wave pattern, not the speed of energy or information.

Relation to wavelength and frequency

The standard wave-equation identities:

$u_p=f\lambda =\frac{\omega }{\beta }=\frac{\omega }{2\pi /\lambda }=\frac{\omega \lambda }{2\pi }.$

Given any two of $(f,\lambda ,u_p)$, the third is fixed. On a transmission line you typically know $f$ (the source frequency) and $u_p$ (from the line’s $L^{\prime }$, $C^{\prime }$ or equivalently $\mu$, $ϵ$); you compute $\lambda =u_p/f$.

For a 1 GHz signal on a coax with $u_p=0.66c$:

$\lambda =\frac{0.66\times 3\times 10^8}{10^9}\approx 0.2\text{ m}=20\text{ cm}.$

A quarter-wave stub on such a line is 5 cm — physically shorter than free-space ${\lambda }_0/4=7.5$ cm by the same $\sqrt{{ϵ}_r}\approx 1.5$ factor.

Why it shows up everywhere

The phase-velocity relation $u_p=\omega /\beta$ is the bridge between time-domain and space-domain wave behavior. Any time you want to translate “how fast does it oscillate?” ($\omega$) and “how packed are the wavelengths?” ($\beta$) into “how fast does it move?” ($u_p$), this is the formula. It recurs in TL theory, plane-wave optics, waveguide cutoff analysis, antenna theory, and the dispersion relations of any wave system.