  A theoretical analysis is made of the transformation of the dispersion
relation of waves in artificial crystals under the influence of loss, including
the case of photonic and phononic crystals. Considering a general dispersion
relation in implicit form, an analytic procedure is derived to obtain the
transformed dispersion relation. It is shown that the dispersion relation is
generally shifted in the complex $(k,\omega)$ plane, with $k$ the wavenumber
and $\omega$ the angular frequency. The value of the shift is obtained
explicitly as a function of the perturbation of material constants accounting
for loss. The method is shown to predict correctly the transformation of the
complex band structure $k(\omega)$. Several models of the dispersion relation
near a symmetry point of the Brillouin zone are analyzed. A lower bound for the
group velocity, related to the local shape of the band around symmetry points,
is derived for each case.
