  We examine the low-energy excitations of a dilute supersolid state of matter
with a one-dimensional crystal structure. A hydrodynamic description is
developed based on a Lagrangian, incorporating generalized elastic parameters
derived from ground state calculations. The predictions of the hydrodynamic
theory are validated against solutions of the Bogoliubov-de Gennes equations,
by comparing the speeds of sound, density fluctuations, and phase fluctuations
of the two gapless bands. Our results are presented for two distinct supersolid
models: a dipolar Bose-Einstein condensate in an infinite tube and a dilute
Bose gas of atoms with soft-core interactions. Characteristic energy scales are
identified, highlighting that these two models approximately realize the bulk
incompressible and rigid lattice supersolid limits.
