The phase diagram of a Bose-Einstein condensate with Raman-induced
spin-orbit coupling includes a stripe phase with supersolid features. In
this work we develop a perturbation approach to study the ground state
and the Bogoliubov modes of this phase, holding for small values of the
Raman coupling. We obtain analytical predictions for the most relevant
observables (including the periodicity of stripes, sound velocities,
compressibility, and magnetic susceptibility) which are in excellent
agreement with the exact (non perturbative) numerical results, obtained
for significantly large values of the coupling. We further unveil the
nature of the two gapless Bogoliubov modes in the long-wavelength limit.
We find that the spin branch of the spectrum, corresponding in this
limit to the dynamics of the relative phase between the two spin
components, describes a translation of the fringes of the equilibrium
density profile, thereby providing the crystal Goldstone mode typical of
a supersolid configuration. Finally, using sum-rule arguments, we show
that the superfluid density can be experimentally accessed by measuring
the ratio of the sound velocities parallel and perpendicular to the
direction of the spin-orbit coupling.