The lattice wave solution and the dispersion relation of the lattice vibration equation of the one-dimensional three-atom chain are obtained by solving simple harmonic approximation and then the lattice vibration characteristics of the chain are studied. The intrinsic link of the dispersion relation of the one-dimensional three-atom chain and one-dimensional single atom chain are discussed. The results show that the dispersion relation of the one-dimensional triatomic chain contains an acoustic branch and two optical branches. The spectral range and frequency band gap are related to the mass of the three atoms in the original cell. When the atomic masses of the original cell are equal，the dispersion relationship of one-dimensional triatomic chain is the same as the lattice vibration mode described by the one-dimensional single atom chain dispersion relation. Furthermore，two related problems that are easily misunderstood in solid physics teaching are explained.