A magnon-phonon interaction model is developed on the basis of two-dimensional square Heisenberg ferromagnetic system. The transverse acoustic phonon damping and longitudinal acoustic phonon damping are studied through Matsubara Green function theory, and transverse acoustic phonon damping and longitudinal acoustic phonon damping curves are calculated on the main symmetric point and line in the first Brillouin zone. It’s found that in the first Brillouin zone, on line Δ, there is no damping for the transverse acoustic phonon, on line Z, there is no damping for longitudinal acoustic phonon. In the first Brillouin zone, the transverse acoustic phonon damping is one order larger than the longitudinal acoustic phonon damping. The influences of transverse acoustic phonon damping and longitudinal acoustic phonon damping for different parameters on them are also illustrated in this article. According to the relationship between phonon damping and phonon lifetime, and the relationship between phonon damping and phonon density of state, the lifetime and density of state for transverse acoustic phonon and longitudinal acoustic phonon is discussed.