  We determine the hydrodynamic modes of the superfluid analog of a smectic-A
phase in liquid crystals, i.e., a state in which both gauge invariance and
translational invariance along a single direction are spontaneously broken.
Such a superfluid smectic provides an idealized description of the
incommensurate supersolid state realized in Bose-Einstein condensates with
strong dipolar interactions as well as of the stripe phase in Bose gases with
spin-orbit coupling. We show that the presence of a finite normal fluid density
in the ground state of these systems gives rise to a well-defined second-sound
type mode even at zero temperature. It replaces the diffusive permeation mode
of a normal smectic phase and is directly connected with the classic
description of supersolids by Andreev and Lifshitz in terms of a propagating
defect mode. An analytic expression is derived for the two sound velocities
that appear in the longitudinal excitation spectrum. It only depends on the
low-energy parameters associated with the two independent broken symmetries,
which are the effective layer compression modulus and the superfluid fraction.
