Phase Velocity

Phase velocity defines the speed at which waves oscillating at a particular frequency propagate. Like the wavelength, the phase velocity depends on the real component of the wavenumber (\(\alpha\)) and is given by:

\[v_{ph} = \frac{\omega}{\alpha} = \left ( \frac{\mu \epsilon}{2} \left [ \left ( 1 + \frac{\sigma^2}{\epsilon^2 \omega^2} \right )^{1/2} + 1 \right ] \right )^{-1/2}\]

Phase Velocity for Various Materials

The table below shows phase velocity for certain rocks at various frequencies. This is meant to serve as a general guide, as rock types are classified by a range of physical properties values which can lead to order of magnitude differences in phase velocity.

Type	\(\sigma\)	\(\mu_r\)	\(\epsilon_r\)	\(v_{ph}\) (1Hz)	\(v_{ph}\) (1kHz)	\(v_{ph}\) (1MHz)	\(v_{ph}\) (1GHz)
Air	0 S/m	1	1	299.8 m/us	299.8 m/us	299.8 m/us	299.8 m/us
Sea Water	3.3 S/m	1	80	0.0017 m/us	0.055 m/us	1.7 m/us	32 m/us
Igneous	\(10^{-4}\) S/m	1	5	0.32 m/us	10 m/us	132 m/us	134 m/us
Sedimentary (dry)	\(10^{-3}\) S/m	1	4	0.1 m/us	3.2 m/us	90 m/us	150 m/us
Sedimentary (wet)	\(10^{-2}\) S/m	1	25	0.032 m/us	1 m/us	30 m/us	60 m/us
Sulphide Skarn	\(10^{2}\) S/m	1	5	0.00032 m/us	0.01 m/us	0.32 m/us	10 m/us
Magnetite Skarn	\(10^{2}\) S/m	2	5	0.00022 m/us	0.007 m/us	0.22 m/us	7 m/us

Approximations

Quasi-Static Approximation

In the quasi-static regime (\(\epsilon\omega \ll \sigma\)), the phase velocity simplifies to:

\[v_{ph} = \sqrt{ \frac{2\omega}{\mu \sigma} }\]

Thus the phase velocity is faster to waves which oscillate at higher frequencies. EM waves also move slower in media that a conductive and highly permeable.

Wave Regime Approximation

In the wave regime ( \(\epsilon \omega \gg \sigma\) ), the phase velocity simplifies to:

(122)\[v_{ph} = \frac{1}{\sqrt{\mu \epsilon}}\]

Thus at sufficiently high frequencies, waves at all frequencies propagate as the same speed. In free space, the previous equations simplifies to \(1/ \! \sqrt{\mu_0\epsilon_0} = 3\times 10^8\) m/s, which is the speed of light.

Relating Wavelength and Phase Velocity

As we have shown, both the wavelength and phase velocity can be defined in terms of the real component of the wavenumber (\(\alpha\)). As a result, we can define a mathematical relationship when relates the wavelength and phase velocity at a given frequency. This relationship is given by:

\[\lambda = \frac{2\pi v_{ph}}{\omega} = \frac{v_{ph}}{f}\]

where \(f\) is the frequency of oscillation in Hz. From this expression, we can see that EM waves which propagate faster through the Earth correspond to longer wavelengths.