# Wavenumber

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.

## Approximations

### 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:

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

and

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

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.