# 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.