Question

I find plane waves are uncompatible with light cone.

Perhaps plane waves are "virtual" and can never be measured in that case, shouldn't we call plane waves as "virtual plane waves"? (other option could be that plane waves allows waves travel faster than c)

I would like to clarify this point through this question.

If plane waves would exist(as measurable), then higher than c speed could be reached like this:

A wave going from X to Y at a speed c, it will reach Z at higher than c speed, because it will reach at same time, traveling more distance.

(X).
|   
v   
________________________________________________
plane waves ________________________________________________
going X to Y ________________________________________________
(Y). (Z).

In a real situation the wave will be a circle (or a sphere in 3d) so it will get Z later then that's not a plane wave.

Answer 1

A "plane wave" generally refers to an infinite and perfectly flat wavefront, which cannot exist in reality, of course. However, there is nothing at all impossible about a plane wave of finite extent. Such a wave will experience diffraction at its edges, of course, but can still propagate over long distances before losing its planar nature.

The problem with your question about faster than light information transmission, is that if X were the only point emitting a wave, then it would be a spherical wave, not a plane wave. A plane wave can be thought of as being composed as a superposition of spherical waves emitted in phase from every point on an infinite plane. So in your example, information would not be arriving at Z from X, but instead from some other source point within the causal past of Z. In your diagram that point could be at the same height as X, but directly above Z.

Considering a wavefront (planar or otherwise) as a superposition of spherical waves is the central feature of the Huygens-Fresnel Principle, which would be a good reading on the topic.