Problem 5.6.12: Two snapshots on a plot

Read off the plot: the amplitude is $E_0=2V/m$, and the black crests are at $x=0$, $0.2$ and $0.4\mathrm{m}$, so $\lambda =0.2\mathrm{m}$. The form $\cos (\omega t-\beta x)$ travels to $+x$, so the crest at $x=0$ has moved to the red crest at $x=0.05\mathrm{m}$ in $\Delta t=0.5\mathrm{ns}$. There is no decay: the medium is lossless, with $\mu ={\mu }_0$.

Lossless and nonmagnetic, $v_p=c/\sqrt{{\epsilon }_r}$:

$\beta =2\pi /\lambda =10\pi rad/m$ (check: $\omega /\beta =10^{8}m/s$):

At $t=0.5\mathrm{ns}$, $\omega t=\pi /2$ and $E_y=2\sin (10\pi x)$: the red curve, with its crest at $0.05\mathrm{m}$. The interval $0.5\mathrm{ns}$ is $T/4$, and the shift is $\lambda /4$.