Problem 5.5.3: A field between two walls

$E$ has only a $y$ component, and it does not depend on $y$: $\nabla \cdot E=\partial E_y/\partial y=0$.

The derivatives are

Substituting in ${\nabla }^2{E}_y-{\mu }_0{\epsilon }_0{\partial }^2{E}_y/\partial t^2=0$ gives $[-(\pi /a{)}^2-{\beta }^2+{\mu }_0{\epsilon }_0{\omega }^2]E_y=0$ for all $x$, $z$, $t$, so the bracket must vanish:

The pattern moves at

$\beta$ is real only if $\omega /c>\pi /a$, that is above the frequency

The three numbers form a right triangle, $(50\pi {)}^2=(40\pi {)}^2+(30\pi {)}^2$, the $3$:$4$:$5$ triangle. Along $z$ the field repeats every ${\lambda }_z=2\pi /\beta =6.67\mathrm{cm}$, longer than the free-space wavelength $\lambda =c/f=4\mathrm{cm}$ because $\beta <\omega /c$ (Figure 5.5.15).