Solution

$12.1.30$ The amplitude of the wave when reflected from a metal wall moving towards
it increases by a factor of $k$. Determine the wall speed.

Using the relation between the magnetic and electric fields of an electromagnetic wave $E = B c$, and the
Lorentz transformation for electric and magnetic fields:

\begin{equation} E = \frac{E_0 - (-v) E_0/c}{\sqrt{1-v^2/c^2}} = E_0 \sqrt{\frac{1+v/c}{1-v/c}} \end{equation}

And finally the amplitude of the reflected wave in the Earth frame is:

\begin{equation} E_1 = E_0 \frac{1+v/c}{1-v/c} = E_0 k \end{equation}

Then, the velocity of the wall is:

\begin{equation} v = c \frac{k - 1}{k + 1} \end{equation}