For electromagnetic waves characterized by the Helmholtz equation, all of the corresponding wave properties can be derived from the wavenumber \(k\). The wavenumber at a particular frequency depends on the physical properties of the propagation medium and is given by:

\[k = \sqrt{\mu \epsilon \omega^2 - i \mu \sigma \omega}\]

The wavenumber has both real and imaginary components and may be decomposed as follows:

\[k = \alpha - i \beta\]

such that the general solution for EM planewaves propagating in the vertical direction becomes:

\[\mathbf{E} = \mathbf{E}_0^- \, e^{\beta z}e^{i(\alpha z-\omega t)} + \mathbf{E}_0^+ \, e^{-\beta z}e^{-i(\alpha z+\omega t)}\]

According to [Str41, WH88], \(\alpha\) and \(\beta\) are given by:

\[\alpha = \omega \left ( \frac{\mu \epsilon}{2} \left [ \left ( 1 + \frac{\sigma^2}{\epsilon^2 \omega^2} \right )^{1/2} + 1 \right ] \right )^{1/2} \geq 0\]
\[\beta = \omega \left ( \frac{\mu\epsilon}{2} \left [ \left ( 1 + \frac{\sigma^2}{\epsilon^2 \omega^2} \right)^{1/2} - 1 \right ] \right ) ^{1/2} \geq 0\]

When deriving a general solution, we stated that \(\alpha\) (the real component of the wavenumber) determines the wavelength and velocity of the planewave. Whereas \(\beta\) (the imaginary component of the wavenumber) determines the attenuation. The details of this can be learned visually through app as well as through the following material on planewave properties.


Quasi-Static Approximation

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

\[k \approx \sqrt{- i \mu \sigma \omega}\]

where it can be shown that:

\[\alpha = \beta = \left ( \frac{\omega \mu \sigma}{2} \right ) ^{1/2}\]

In this case, EM waves oscillate and decay as they propagate.

Wave Regime Approximation

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

\[k \approx \alpha = \sqrt{\mu \epsilon \omega^2} = \omega \sqrt{\mu \epsilon}\]


\[\beta \approx \frac{\sigma}{2} \sqrt{\frac{\mu}{\epsilon}} \sim 0\]

For a perfect wave equation, \(\beta = 0\) and the waves do not decay in amplitude as they propagate. In geophysical problems (ground-penetrating radar for example), signals still experience amplitude loss as they propagate through the Earth.