Electromagnetic plane waves in vacuo

[madness]

Consider a plane wave f = cos(kz - wt). Applying Maxwell's equation (divE=0 in vacuo) gives
kcos(kz - wt) = 0 which means that k = 0. This surely doesn't make sense?

 

[whybother]

Well, mathematically,

$$\bigtriangledown \cdot E_0 \cos(kz - wt) = E_0 \sin(kz - wt) = 0$$

$$\Longrightarrow k = 0$$ or $$sin(kz - wt) = 0$$ $$\forall{ z, t }$$

EDIT: But yes, addressed below is the conceptual issue here...

 

[nicksauce]

Well the reason you get that is that EM waves are transverse waves, and you have oversimplified the situation. The more general result is:
Consider a plane wave
$$\vec{E} = \vec{E_0}cos(\vec{k}\cdot\vec{r} - \omega t})$$
Then,
$$\nabla\cdot\vec{E} = \vec{k}\cdot\vec{E} = 0$$

Which shows that the wave vector is perpendicular to the field.

 

[madness]

I think I see my mistake. I thought all i was doing was choosing my coordinates such that my z axis coincided with the wave vector k. The way i did it missed out the fact that k and E point in different directions.