  A one-dimensional Bose-Einstein condensate may experience nonlinear periodic
modulations known as "cnoidal waves". We argue that such structures represent
promising candidates for the study of supersolidity-related phenomena in a
non-equilibrium state. A mean-field treatment makes it possible to rederive
Leggett's formula for the superfluid fraction of the system and to estimate it
analytically. We determine the excitation spectrum, for which we obtain
analytical results in the two opposite limiting cases of (i) a linearly
modulated background and (ii) a train of dark solitons. The presence of two
Goldstone (gapless) modes -- associated with the spontaneous breaking of
$\mathrm{U}(1)$ symmetry and of continuous translational invariance -- at large
wavelength is verified. We also calculate the static structure factor and the
compressibility of cnoidal waves, which show a divergent behavior at the edges
of each Brillouin zone.
