  We study the long-wavelength excitations of the inner crust of neutron stars,
considering three phases: cubic crystal at low densities, rods and plates near
the core-crust transition. To describe the phonons, we write an effective
Lagrangian density in terms of the coarse-grained phase of the neutron
superfluid gap and of the average displacement field of the clusters. The
kinetic energy, including the entrainment of the neutron gas by the clusters,
is obtained within a superfluid hydrodynamics approach. The potential energy is
determined from a model where clusters and neutron gas are considered in phase
coexistence, augmented by the elasticity of the lattice due to Coulomb and
surface effects. All three phases show strong anisotropy, i.e., angle
dependence of the phonon velocities. Consequences for the specific heat at low
temperature are discussed.
