Abstract: The axisymmetric lateral vibration equations of the soil described by fractional derivative viscoelastic model are established by the theory of fractional derivative as well as viscoelastic and soil dynamics, and the lateral vibration of the soil described by a fractional derivative viscoelastic model is analyzed and obtained the lateral resistance of the soil by potential functions and separation of variables. The horizontal vibration equations of a single pile in soil are established, and the lateral dynamic impedance of the pile is obtained by considering the continuous conditions at the soil-pile and the orthogonality of trigonometric functions. The lateral resistance factor of the soil described by a fractional derivative viscoelastic model and the horizontal dynamic impedance of a single pile in soil are investigated with the numerical examples. The result indicated that the lateral resistance factor of the soil and the horizontal dynamic impedance of a single pile in soil described by a fractional derivative viscoelastic model can reduced to the cases of classic viscoelasticity and elasticity, the influence of the modulus ratio on the lateral dynamic impedance of the pile has great difference between fractional derivative viscoelastic and elastic soil.