This paper presents a theoretical study of the excitation spectrum of a two-dimensional (2D) supersolid state of a Bose-Einstein condensate (BEC) with either dipole-dipole or soft-core interactions. The 2D supersolid state exhibits three gapless excitation branches, arising from the spontaneous breaking of continuous symmetries. Two of these branches correspond to longitudinal sound waves, similar to those in 1D supersolids, while the third branch is a transverse wave due to the non-zero shear modulus of the 2D crystal. The study uses numerical calculations to analyze the excitation spectrum and dynamic structure factor, characterizing density fluctuations across the discontinuous superfluid-to-supersolid transition. The speeds of sound are described by a hydrodynamic theory incorporating generalized elastic parameters, including the shear modulus. The paper shows that dipolar and soft-core supersolids exhibit distinct characteristics, corresponding to the bulk incompressible and rigid lattice limits, respectively. The study considers two systems: a dipolar BEC in a planar trap and a 2D BEC with soft-core interactions. The dipolar system has a long-range dipole-dipole interaction, while the soft-core system has a finite-range interaction. Both systems are analyzed using the Bogoliubov-de Gennes (BdG) equations. The results show that the hydrodynamic model accurately describes the speeds of sound, providing insight into the behavior of the two systems across the transition. The dipolar system exhibits a transverse excitation branch with no weight in the dynamic structure factor, while the soft-core system has a transverse branch sandwiched between the second and first sound modes. The hydrodynamic theory is validated by comparing the speeds of sound with BdG results, showing excellent agreement. The paper also discusses the limiting behaviors of the systems, identifying the bulk incompressible and rigid lattice limits. The results highlight the importance of compressibility and lattice elasticity in the properties of the two supersolids. The study provides a foundation for understanding the equilibrium and dynamical properties of higher-dimensional supersolids, which are being explored in experiments with dipolar gases.This paper presents a theoretical study of the excitation spectrum of a two-dimensional (2D) supersolid state of a Bose-Einstein condensate (BEC) with either dipole-dipole or soft-core interactions. The 2D supersolid state exhibits three gapless excitation branches, arising from the spontaneous breaking of continuous symmetries. Two of these branches correspond to longitudinal sound waves, similar to those in 1D supersolids, while the third branch is a transverse wave due to the non-zero shear modulus of the 2D crystal. The study uses numerical calculations to analyze the excitation spectrum and dynamic structure factor, characterizing density fluctuations across the discontinuous superfluid-to-supersolid transition. The speeds of sound are described by a hydrodynamic theory incorporating generalized elastic parameters, including the shear modulus. The paper shows that dipolar and soft-core supersolids exhibit distinct characteristics, corresponding to the bulk incompressible and rigid lattice limits, respectively. The study considers two systems: a dipolar BEC in a planar trap and a 2D BEC with soft-core interactions. The dipolar system has a long-range dipole-dipole interaction, while the soft-core system has a finite-range interaction. Both systems are analyzed using the Bogoliubov-de Gennes (BdG) equations. The results show that the hydrodynamic model accurately describes the speeds of sound, providing insight into the behavior of the two systems across the transition. The dipolar system exhibits a transverse excitation branch with no weight in the dynamic structure factor, while the soft-core system has a transverse branch sandwiched between the second and first sound modes. The hydrodynamic theory is validated by comparing the speeds of sound with BdG results, showing excellent agreement. The paper also discusses the limiting behaviors of the systems, identifying the bulk incompressible and rigid lattice limits. The results highlight the importance of compressibility and lattice elasticity in the properties of the two supersolids. The study provides a foundation for understanding the equilibrium and dynamical properties of higher-dimensional supersolids, which are being explored in experiments with dipolar gases.