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?