This paper presents a theoretical study of the excitations in a two-dimensional (2D) supersolid state of a Bose-Einstein condensate (BEC) with either dipole-dipole or soft-core interactions. The 2D supersolid state has three gapless excitation branches, two of which are related to longitudinal sound waves, similar to those in one-dimensional (1D) supersolids, and the third is a transverse wave arising from the non-zero shear modulus of the 2D crystal. The authors present numerical calculations for the excitations and dynamic structure factors, characterizing density fluctuations, and analyze their behavior across the discontinuous superfluid to supersolid transition. They show that the speeds of sound are described by a hydrodynamic theory that incorporates generalized elastic parameters, including the shear modulus. The study reveals that dipolar and soft-core supersolids exhibit distinct characteristics, falling into the bulk incompressible and rigid lattice limits, respectively. The paper also discusses the limiting behavior of the systems, highlighting the contrasting importance of compressibility and lattice elasticity in the properties of the two supersolids.This paper presents a theoretical study of the excitations in a two-dimensional (2D) supersolid state of a Bose-Einstein condensate (BEC) with either dipole-dipole or soft-core interactions. The 2D supersolid state has three gapless excitation branches, two of which are related to longitudinal sound waves, similar to those in one-dimensional (1D) supersolids, and the third is a transverse wave arising from the non-zero shear modulus of the 2D crystal. The authors present numerical calculations for the excitations and dynamic structure factors, characterizing density fluctuations, and analyze their behavior across the discontinuous superfluid to supersolid transition. They show that the speeds of sound are described by a hydrodynamic theory that incorporates generalized elastic parameters, including the shear modulus. The study reveals that dipolar and soft-core supersolids exhibit distinct characteristics, falling into the bulk incompressible and rigid lattice limits, respectively. The paper also discusses the limiting behavior of the systems, highlighting the contrasting importance of compressibility and lattice elasticity in the properties of the two supersolids.