  Phenomena of wave propagation in dynamically varying structures have
reemerged as the temporal variations of the medium's properties can extend the
possibilities for electromagnetic wave manipulation. While the dynamical change
of the electromagnetic medium's properties is a difficult task, the movement of
scatterers is not. In this paper, we analyze the electromagnetic fields trapped
inside two smoothly moving mirrors. We employ the method of characteristics and
take into account the relativistic phenomena to show that the temporally and
spatially local Doppler effects can filter and amplify the electromagnetic
signal, tailoring the $k-$ and $\omega-$content of the transients. It is shown
using the Doppler factor and the change of the distance between neighbor
characteristics that the dynamical movement of the boundaries can lead to
condensated characteristics resulting in field amplification or dilution of the
characteristics resulting in the attenuation of the signal. In the case of
periodically moving mirrors the field distribution is shown that asymptotically
leads to exponentially growing delta-like wave packets at discrete points of
space with a limiting number of peaks due to the fact that the velocity of the
mechanical vibrations can not exceed that of light. The theoretical analysis is
also verified by FDTD simulations and is connected with the theory of mode
locking.
