Problem 5.6.3: A sine with a plus sign

Between $t=0$ and $t=T/8=31.25\mathrm{ps}$ every crest moves $\lambda /8=1.31\mathrm{mm}$ towards $-x$ (Figure 5.6.14), and $1.31\mathrm{mm}/31.25\mathrm{ps}=4.19\times 10^{7}m/s$. The direction is set by the relative sign of $\omega t$ and $\beta x$, not by the sine or the cosine. A velocity of $0.14c$ needs ${\epsilon }_r\approx 51$: microwave ceramics of this kind (${\epsilon }_r$ from $20$ to $100$) are used to shrink resonators and antennas in mobile phones.