Question: Why are the electric and magnetic field components of guided EM waves independent of the direction of propagation?
Details: I'll go on to paraphrase Griffiths' section on guided waves here-
For a waveguide that is a perfect conductor, the boundary conditions at the inner wall are $E^{||}=0$ and $B^\perp=0$
For monochromatic waves that propagate down such a waveguide along the $z$ direction-
$\boldsymbol{\tilde{E}}(x,y,z,t) = \boldsymbol{\tilde{E_0}}(x,y)e^{i(kz-\omega t)}$
$\boldsymbol{\tilde{B}}(x,y,z,t) = \boldsymbol{\tilde{B_0}}(x,y)e^{i(kz-\omega t)}$
Now, the book says that confined waves are not (in general) transverse; in order to fit the boundary conditions, we shall have to include longitudinal components $E_z$ and $B_z$:
$\boldsymbol{\tilde{E_0}} = E_x\boldsymbol{\hat{x}} + E_y\boldsymbol{\hat{y}} + E_z\boldsymbol{\hat{z}}$
and
$\boldsymbol{\tilde{B_0}} = B_x\boldsymbol{\hat{x}} + B_y\boldsymbol{\hat{y}} + B_z\boldsymbol{\hat{z}}$
$\textbf{where each of the components is a function of x and y}$.
I don't understand why each of the components has to be a function of $x$ and $y$ but not $z$.
