TMZ TME modes, clarification

I'm refering here to Taflove's "computational electrodynamcis, 3rd ed."

He says

Let us assume that the structure being modeled extends to infinity in the z-direction with no change in the shape or position in its transverse cross section. If the incident wave is also uniform in the z-direction, then all partial derivatives of the fields with respect to z must equal zero...

I don't quite understand why the wave should be uniform along z? How could you justify this assumption? Wave's have a finite wavelength, even within media which implies that they vary with respect to position.

Any ideas?

BTW: It seems like this paragraph is also cited in another textbook:

• Just a note: not having a variation along $z$ means that $E_{x,y,z}=f(x,y)$ not that $E_z = 0$. Feb 22 '19 at 13:20
In a translationally invariant (along z) waveguide each xy-plane is equivalent to any other such plane. You can use Helmholtz-Hodge decomposition for the wave equation and what you will obtain using Fourier's method is that you will have a simple plane wave in the z-direction. The transverse modes (i.e., x,y-modes) will be dependent on the wavenumber $$k_z$$ though.