A complex wavenumber $k=\beta-i\alpha$ can be defined, that when substituted into a time-harmonic solution $e^{i(\omega t - kx)}$ yields $$e^{-\alpha x}e^{i(\omega t - \beta x)}$$ The first negative exponential indicates a reduction (or growth depending on the sign of the exponent) of the oscillation with respect to distance, and the second exponential represents the phase change of the wave with time and distance. We define $\alpha$ in this case to be the spatial absorption coefficient. The algebra checks out when I expand the brackets for a complex-valued $k$:
$e^{i(\omega t -(\beta -i \alpha)x)}$
$e^{i \omega t}e^{-(i \beta x - i^2 \alpha x)}$
$e^{i \omega t}e^{-(i \beta x+\alpha x)}=e^{-\alpha x}e^{i(\omega t - \beta x)}$
Apparently it's also possible to derive a time-dependent absorption coefficient starting from $e^{ik(x-ct)}$ where $c = \omega/ \beta$ and $k$ is here taken to be real-valued $\beta = k +i \alpha$
The resulting expression is meant to be: $$e^{-\alpha t}e^{i(\omega t - \beta x)}$$
But I can't figure out the steps in between. Can anyone help with this?
