Problem 5.6.13: A phase in two coordinates

The wave travels along $a_k=0.6a_y+0.8a_z$, at $53.1^{\circ }$ from the $y$ axis, with $\lambda =2\pi /10=0.628\mathrm{m}$ and $v_p=\omega /|\nabla \varphi |=6.28\times 10^{7}m/s$. In free space $k_0=\omega /c=2.09rad/m$, so a lossless medium would have ${\mu }_r{\epsilon }_r=(10/2.09{)}^2=22.8$. Figure 5.6.25 shows the planes of constant phase at $t=0$, where $E_x=16\sin (6y+8z)$. Walking along $y$ alone the crests repeat every $2\pi /6=1.05\mathrm{m}$, along $z$ every $2\pi /8=0.785\mathrm{m}$. Both are longer than $\lambda$: the true wavelength is measured along $a_k$ only. As time goes on the whole set of lines slides along $a_k$ at $6.28\times 10^{7}m/s$.