Leonel Schwartz

2022-09-06

What is the explicit expression of a plane wave in the frequency domain?
A plane wave in the time domain can be written (using notation for an electric field):
$\mathbit{E}\left(\mathbit{r},t\right)={\mathbit{E}}_{0}{e}^{i\left(\mathbit{k}\mathbit{r}-\omega t\right)}$
what is the corresponding expression for a plane wave, $\mathbit{E}\left(\mathbit{r},\omega \right)$, in the frequency domain?

ralharn

The key for all these formulas is
$\begin{array}{}\text{(1)}& {\int }_{{\mathbb{R}}^{d}}{e}^{ix\cdot y}\phantom{\rule{thinmathspace}{0ex}}\frac{dy}{\left(2\pi {\right)}^{\frac{d}{2}}}=\delta \left(x\right),\end{array}$
where $\delta$is the d-dimensional Dirac distribution. This question is probably concerned with d=1 only, but there is no extra difficulty in doing the general case.
To finally answer the question, we apply (1) to obtain
${\int }_{\mathbb{R}}{E}_{0}{e}^{ik\cdot x-\omega t}{e}^{-its}\phantom{\rule{thinmathspace}{0ex}}ds=\left(2\pi {\right)}^{\frac{1}{2}}{E}_{0}{e}^{ik\cdot x}\delta \left(t+\omega \right).$
This is the Fourier transform of the given function, the "plane wave", in the time variable.

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