  We study the damping of plasma waves in linear Josephson junction chains as
well as in two capacitively coupled chains. In the parameter regime where the
ground capacitance can be neglected, the theory of the antisymmetric mode in
the double chain can be mapped onto the theory of a single chain. We consider
two sources of relaxation: the scattering from quantum phase slips (QPS) and
the interaction among plasmons related to the nonlinearity of the Josephson
potential. The contribution to the relaxation rate $1/\tau$ from the
nonlinearity scales with the fourth power of frequency $\omega$, while the
phase-slip contribution behaves as a power law with a non-universal exponent.
In the parameter regime where the charging energy related to the junction
capacitance is much smaller than the Josephson energy, the amplitude of QPS is
strongly suppressed. This makes the relaxation mechanism related to QPS
efficient only at very low frequencies. As a result, for chains that are in the
infrared limit on the insulating side of the superconductor-insulator
transition, the quality factor $\omega\tau$ shows a strongly non-monotonic
dependence on frequency, as was observed in a recent experiment.
