Starting from Maxwell's equations in vacuo, show that the cartesian components of $\mathbf{E}$ and $\mathbf{B}$ each satisfy

$$\nabla^{2} f=\frac{1}{c^{2}} \frac{\partial^{2} f}{\partial t^{2}}$$

Consider now a rectangular waveguide with its axis along $z$, width $a$ along $x$ and $b$ along $y$, with $a \geqslant b$. State and explain the boundary conditions on the fields $\mathbf{E}$ and $\mathbf{B}$ at the interior waveguide surfaces.

One particular type of propagating wave has

$$\mathbf{B}(x, y, z, t)=B_{0}(x, y) \hat{\mathbf{z}} e^{i(k z-\omega t)}$$

Show that

$$B_{x}=\frac{i}{(\omega / c)^{2}-k^{2}}\left(k \frac{\partial B_{z}}{\partial x}-\frac{\omega}{c^{2}} \frac{\partial E_{z}}{\partial y}\right)$$

and derive an equivalent expression for $B_{y}$.

Assume now that $E_{z}=0$. Write down the equation satisfied by $B_{z}$, find separable solutions, and show that the above implies Neumann boundary conditions on $B_{z}$. Find the "cutoff frequency" below which travelling waves do not propagate. For higher frequencies, find the wave velocity and the group velocity and explain the significance of your results.