As far I know, plane wave equation is given by:
$$\vec{E}(\vec{r},t)=\vec{E_0} \cdot e^{i(\vec{k}\cdot \vec{r}-\omega t)} \hspace{2cm} \tag{1}$$ In some textbook propagation direction of $(1)$ is given as $\vec{k}$;
while in some other books, i found propagation direction of $(1)$ as $-\vec{k}$
so, can anyone tell me what is the propagation direction of $(1)$ and how can we determine it from $(1)$?


1 Answer 1


$\vec E(\vec r,t)$ is at a peak when $e^{i(\vec k \cdot \vec r-\omega t)} = 1$, and a minimum when it is $-1$. Maxima happen when $i(\vec k \cdot \vec r-\omega t) = 0$ or $2 \pi i$. It is at a minimum when the exponent is $\pi i$.

One peak is found where $\vec k \cdot \vec r =\omega t$. As t gets larger, the peak will move to where $\vec k \cdot \vec r$ is larger. To make the dot product larger, choose an $\vec r$ farther in the direction that $\vec k$ points.

That is to say, the wave will propagate in the direction that $\vec k$ points.


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