The Traveling Electromagnetic Wave, Quantitatively — Question 7

Question 7

A supplied dispersion law separates phase speed from packet speed

In an ideal transparent dispersive medium, a transverse mode obeys $\omega^2=\omega_p^2+c^2k^2$ for positive k. Define phase speed as omega/k and the narrow-packet group speed as $d\omega/dk$.

1. Derive both speeds and their product.

2. At $\omega=2\omega_p$ find both speeds in units of c. Explain why a phase speed above c does not by itself establish faster-than-c information transport.

Question 7 - Solution

A supplied dispersion law separates phase speed from packet speed

(a) Method and calculation

$$k=\frac{\sqrt{\omega^2-\omega_p^2}}{c}\Rightarrow\boxed{v_\phi=\frac{c}{\sqrt{1-\omega_p^2/\omega^2}}}.$$$$\boxed{v_g=\frac{c^2k}{\omega}=c\sqrt{1-\omega_p^2/\omega^2},\quad v_\phi v_g=c^2}.$$

(b) Interpretation and further calculation

$$\omega=2\omega_p\Rightarrow\boxed{v_\phi=2c/\sqrt3,\quad v_g=\sqrt3c/2}.$$$$v_\phi>c\text{ describes phase crests};\quad\boxed{v_g<c\text{ for this packet model}}.$$

Physical check

A narrow-packet group speed is defined here; determining a signal front requires the full causal response of the medium.