The wave equations about displacement, velocity, stress and strain in functionally gradient material (FGM) with constituents varied continuously and smoothly were established. Four kinds of waves are of linear second-order partial differential equation of hyperbolic type and have the same characteristic curve at the plane of X,t. In general, the varying mode of stress is different from that of displacement and velocity at the front of wave. But in a special case that the product of density ρ and elastic modulus E of the material remains unchanged, the three wave equations have a similar expression and they have a similar varying mode in the front of wave. The wave equations about displacement, velocity, stress and strain in functionally gradient material (FGM) with constituents varied continuously and smoothly were established. Four kinds of waves are of linear second-order partial differential equation of hyperbolic type and have the same characteristic curve at the plane of X, t. In general, the varying mode of stress is different from that of displacement and velocity at the front of wave. But in a special case that the product of density ρ and elastic modulus E of the material remains unchanged, the three wave equations have a similar expression and they have a similar varying mode in the front of wave.