Oscillations of plane waves I am reading this notes on electromagnetism.
On page 25 it reads the following, about planes waves:

A plane wave propagating in the direction of z has no oscillations in
  the transverse plane $(k_x^2 + k_y^2 = 0)$, whereas, in the other limit, a plane wave propagating
  at a right angle to $z$ shows the highest spatial oscillations in the transverse
  plane $(k_x^2 + k_y^2 = k^2)$.

But as far as I know, plane EM waves oscillate in a plane which is orthogonal to the direction of propagation, so if the direction of propagation was $z$, then the oscillations of E and H should actually take place in the $xy$ plane. Is this a mistake in the text, or am I getting something wrong?
 A: Your definition of oscillation in a direction is different to the one in the book.
In the book: A plane wave propagating in the z-direction is of the form: $A(z,t)=f(k_z z - \omega t)$. Specifically it does not depend on $x,y$, expressing the fact that the wave has the same value anywhere in the x,y-plane (for fixed $z$). If you look at the plane wave at fixed time, you see an oscillation/periodic function along the z-direction, which is what the book means by "oscillation along z". Note that this has nothing to do with electromagnetic waves, but would equally be true for other plane waves.
Your definition: You are considering the direction in which the electric and magnetic field are oriented, which indeed is perpendicular to the wave propagation. 
A: Your text is completely correct (though it arguably uses language that's more baroque than it really needs to, depending on whether the context around that passage warrants that construction or not). The passage is not talking about the direction of polarization at all; instead, it's talking about the spatial dependence of the wave on the position. 


*

*If a plane wave is propagating along $z$, then there is no dependence on the $x$ or $y$ coordinates.

*If a plane wave is propagating at a right angle to $z$, then its propagation direction is in the $x,y$ plane, and the direction of highest spatial oscillations (i.e. the propagation direction, along $\vec k$), is in the $x,y$ plane.

