# Counter-propagating waves

If I have a linearly polarised plane wave given by $$\textbf{E}_i (\textbf{r})=E_i e^{-ik_0 z}\hat{\textbf{x }}$$ Then is the counter-propagating wave (the wave coming in the other direction) just written as $$\textbf{E}_c (\textbf{r})=E_c e^{ik_0 z}\hat{\textbf{x }}~?$$

I am confused because of the sign of $\hat{x}$. I am given only the electric field, but if you do the cross product of $-\hat{x}$ with the same direction of magnetic field then you get the opposite direction. Which is the right answer?

First, you need to specify the time-component in order to establish the direction of propagation, which depends on the relative sign in $kz\pm\omega t$.
Next, as written, the wave propagates either along $\pm \hat z$, so it must be that, while you cannot establish if $e^{ikz}$ propagates to the left or the right, $e^{-ikz}$ will propagate in the reverse direction. The $\hat x$ direction does not enter in the direction of propagation beyond being orthogonal to the direction $\pm\hat z$ of propagation.
• Yeah I forgot to say $\textbf{E}_i (r,t) = E_i (r) e^{-i \omega t}$. So the conclusion is that $e^{ikz}$ moves in the opposite direction? – math4everyone Jan 22 '17 at 1:12