Wavelength

Fig. 46 A plane harmonic wave as it propagates into the earth.

Wavelength defines the physical distance a wave travels during a single oscillation. As it turns out, the wavelength for EM waves depends on the real component of the wavenumber (\(\alpha\)) and is given by:

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

As we can see from the previous equation, higher frequencies correspond waves with to shorter wavelengths.

EM Wavelengths for Various Materials

The table below shows wavelengths for EM waves travelling in 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 wavelength.

Type	\(\sigma\)	\(\mu_r\)	\(\epsilon_r\)	\(\lambda\) (1Hz)	\(\lambda\) (1kHz)	\(\lambda\) (1MHz)	\(\lambda\) (1GHz)
Air	0 S/m	1	1	299,800 km	299,800 m	299.8 m	0.2998 m
Sea Water	3.3 S/m	1	80	1.7 km	55 m	1.7 m	0.032 m
Igneous	\(10^{-4}\) S/m	1	5	316 km	10,000 m	132 m	0.13 m
Sedimentary (dry)	\(10^{-3}\) S/m	1	4	100 km	3,200 m	90 m	0.15 m
Sedimentary (wet)	\(10^{-2}\) S/m	1	25	32 km	1,000 m	30 m	0.06 m
Sulphide Skarn	\(10^{2}\) S/m	1	5	0.32 km	10 m	0.32 m	0.01 m
Magnetite Skarn	\(10^{2}\) S/m	2	5	0.22 km	7 m	0.22 m	0.007 m

Approximations

Quasi-Static Approximation

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

\[\lambda = 2\pi\sqrt{ \frac{2}{\omega \mu \sigma} } = 2\pi\delta\]

Wave Regime Approximation

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

\[\lambda = \frac{1}{\omega \sqrt{\mu \epsilon}}\]