Electromagnetic Wave in the Y-Direction

[Fascheue]

A common equation for an electromagnetic wave is Ey = Eocos(kx - wt + phi). According to this equation, wouldn’t the intensity of the electric field extend indefinitely in the y-direction? How does this make sense?

 

[Ibix]

That's a plane wave moving in the $x$ direction, and extends infinitely in the $y$ and $z$ directions, yes, assuming $E_0$ is a constant. It's an idealisation, but it's useful in the (common) circumstance where a plane wave is much, much wider than your detector and has constant amplitude across your detector.