  We present a theoretical study of the excitations of the two-dimensional
supersolid state of a Bose-Einstein condensate with either dipole-dipole
interactions or soft-core interactions. This supersolid state has three gapless
excitation branches arising from the spontaneously broken continuous
symmetries. Two of these branches are related to longitudinal sound waves,
similar to those in one-dimensional supersolids. The third branch is a
transverse wave arising from the non-zero shear modulus of the two-dimensional
crystal. We present the results of numerical calculations for the excitations
and dynamic structure factor characterising the density fluctuations, and study
their behavior across the discontinuous superfluid to supersolid transition. We
show that the speeds of sound are described by a hydrodynamic theory that
incorporates generalized elastic parameters, including the shear modulus.
Furthermore, we establish that dipolar and soft-core supersolids manifest
distinct characteristics, falling into the bulk incompressible and rigid
lattice limits, respectively.
