Metallic Waveguide Boundary
Maxwell's Equation: \$\mathbf{\nabla} \times \mathbf{H} = \epsilon \frac{\partial \mathbf{E}}{\partial t}\$

Expanding and using \$E_z=0\$ for TE, we get: \$\frac{\partial H_z}{\partial y}\$ gives \$E_x\$, and \$\frac{\partial H_z}{\partial x}\$ gives \$E_y\$.

Boundary conditions require: electric field cannot be parallel to metal surface.

\$\mathbf{E_x}\$ is parallel to top &amp; bottom walls, i.e. y=0 and y=b.
\$\mathbf{E_y}\$ is parallel to left &amp; right walls, i.e. x=0 and x=a.

So \$E_x\$, hence \$\frac{\partial H_z}{\partial y}\$, should be
0 at y=0 and y=b.
And, \$E_y\$, hence \$\frac{\partial H_z}{\partial x}\$, should be 0 at x=0 and x=a.

"why... not Hz itself null?"

TE, by definition, demands that \$E_x\$ and \$E_y\$ may be nonzero, \$E_z\$ is zero, and \$H_z\$ is nonzero.