# 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:

## 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:

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:

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:

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.