Waveguide conditions Waves in a ideal waveguide have to satisfy that $E_{\parallel}=0$ and $B_{\perp}=0$. Because of the continuity conditions I would think that this relates to the surface. But the solution says that $E_{\perp}=E_{0,x} \vec{e_x}+ E_{0,y} \vec{e_y}$ for a waveguide in $z$-direction with $\vec{E}= \vec{E}_0(x,y)\mathrm{e}^{\mathrm{i}(k_z-\omega t)}$ and $\vec{B}= \vec{B}_0(x,y)\mathrm{e}^{\mathrm{i}(k_z-\omega t)}$ as an approach. So it seems that they refer to the direction of propagation. Intuitively I would say that we have no $z$-component of the electric/magnetic field because they must be perpendicular to the direction of propagation. But this follows not from Maxwell‘s equations so can we have also $z$-components of the electric/magnetic field because the fields depend on $x$,$y$?
 A: Based on the expressions you have for $\vec E$ and $\vec B$, the electromagnetic (EM) wave is propagating in the $z$ direction while the vectors $\vec E$ and (possibly) $\vec B$ are in the $xy$ plane. You can imagine the 3 vectors $\vec E$, $\vec B$, and $\vec k=k_{z}\vec e_{z}$. $\vec k$ denotes the direction of propagation of the wave and is perpendicular to the $xy$ plane which the $\vec E$ and (possibly) $\vec B$ are in.
As for the boundary conditions for $\vec E$ and $\vec B$ that you provided, assume that you have a rectangular waveguide that is infinitely long in the $z$ direction such that the EM wave with properties discussed in the previous paragraph can propagate through. The boundaries of the rectangle may be assumed as perfect electric conductors (PEC)s. Now on the boundary (the 4 sides of the rectangular cross section of the waveguide), the tangential component of the electric field and the normal component of the magnetic field should be zero since you don't have fields in the conductor.
Some further comments on your question:
$\vec E$ and $\vec B$ are vectors. Thus, the expressions $\vec E=E_{0}(x,y)e^{i(k_{z}z-\omega t)}$ and $\vec B=B_{0}(x,y)e^{i(k_{z}z-\omega t)}$ should be corrected. In particular components of $\vec E$ and $\vec B$ should be specified. Also $E_{0}(x,y)$ seems a bit confusing notation-wise as the subscript 0 suggest the quantity may be constant while the $(x,y)$ implies that the quantity changes through the $xy$ plane.
I suspect $E_{\parallel}$ in your question is with respect to each side of the rectangular waveguide. This connects to the boundary condition that the electric field tangential (i.e., parallel) to a conductor should be 0. On the other hand, I think $E_{\perp}$ is with respect to the direction of wave propagation which in your problem is the $z$ direction. In your problem you can write $\vec E=E_{\perp}\vec e_{\perp}$ since the electric field is in the $xy$ plane.
The $B_{\perp}$ argument is similar to the previous paragraph. Please let me know if you have any other questions or there are errors in my analysis.
