  We studied response of the Josephson vortex lattice in layered
superconductors to the high-frequency c-axis electric field. We found a simple
relation connecting the dynamic dielectric constant with the perturbation of
the superconducting phase, induced by oscillating electric field. Numerically
solving equations for the oscillating phases, we computed the frequency
dependences of the loss function at different magnetic fields, including
regions of both dilute and dense Josephson vortex lattices. The overall
behavior is mainly determined by the c-axis and in-plane dissipation
parameters, which is inversely proportional to the anisotropy. The cases of
weak and strong dissipation are realized in
$\mathrm{Bi_{2}Sr_{2}CaCu_{2}O_{x}}$ and underdoped $\mathrm{YBa_{2}Cu_{3}
O_{x}}$ correspondingly. The main feature of the response is the
Josephson-plasma-resonance peak. In the weak-dissipation case additional
satellites appear in the dilute regime mostly in the higher-frequency region
due to excitation of the plasma modes with the wave vectors set by the lattice
structure. In the dense-lattice limit the plasma peak moves to higher frequency
and its intensity rapidly decreases, in agreement with experiment and
analytical theory. Behavior of the loss function at low frequencies is well
described by the phenomenological theory of vortex oscillations. In the case of
very strong in-plane dissipation an additional peak in the loss function
appears below the plasma frequency. Such peak has been observed experimentally
in underdoped $\mathrm{YBa_{2}Cu_{3} O_{x}}$. It is caused by frequency
dependence of in-plane contribution to losses rather then a definite mode of
phase oscillations.
