# Notation of plane waves

Consider a monochromatic plane wave (I am using bold to represent vectors) $$\mathbf{E}(\mathbf{r},t) = \mathbf{E}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)},$$ $$\mathbf{B}(\mathbf{r},t) = \mathbf{B}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}.$$

There are a few ways to simplify this notation. We can use the complex field $$\tilde{\mathbf{E}}(\mathbf{r},t) = \tilde{\mathbf{E}}_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}$$ to represent both the electric and magnetic field, where the real part is the electric and the imaginary part is proportional to the magnetic. Often it is useful to just deal with the complex amplitude ($\tilde{\mathbf{E}}_0$) when adding or manipulating fields.

However, when you want to coherently add two waves with the same frequency but different propagation directions, you need to take the spatial variation into account, although you can still leave off the time variation. So you are dealing with this quantity: $$\tilde{\mathbf{E}}_0 e^{i\mathbf{k} \cdot \mathbf{r}}$$

My question is, what is this quantity called? I've been thinking time-averaged complex field, but then again, it's not really time-averaged, is it? Time-independent? Also, what is its notation? $\langle\tilde{\mathbf{E}}\rangle$?

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you should be careful with your notation. $\vec E_0$ and $\vec B_0$ could be complex themselves (e.g. for elliptic polarization), so your collection of $\vec E + i\vec B$ is in general not possible. But since $i\omega \vec B = -\textrm{curl} \vec E$ in the monochromatic case, you only need the E-field for for complete description – Tobias Kienzler Nov 10 '10 at 12:44
@Tobias Kienzler, Agreed. But the E and B-fields can still be gotten from the real and imaginary parts of $\tilde{\mathbf{E}}_0$, though. – ptomato Nov 10 '10 at 15:11

Stationary field or monochromatic field. Yes, basically that is the field including the $e^{i\omega t}$ term, but even when it is omitted one still knows what is meant.