Abstract: The relationship between the slowness-curved-surface and the softening of the acoustic mode is discussed.The acoustic mode softening of cubic systems corresponds to the maximum slowness.The eigenvector in the soft mode corresponds to the polarization of acoustic mode.By solving the Christoffel equation,the maximum slowness and the polarization of the acoustic mode are obtained.The symmetry change of proper ferroelastic phase transition of cubic systems is determined by the solution of the Christoffel equation and the Curie principle.

Key words: slowness-curved-surface, eigenvector of the soft mode, proper ferroelastic phase transition, Curie principle, cubic systems