Abstract: For the one-dimensional analysis of sites on a bedrock subjected to the excitation of steady-state and vertically propagating shear seismic waves,closed-form solutions for the displacements are presented for a single layer with exponentially increasing shear modulus with depth.A general form of analytical solution of the seismic amplification is developed.The study of an example shows that the characteristics of soil sites,the vibration frequency and the impedance ratio of the half-space and the bottom surface of the soil layer affect the seismic amplification.The predominant period of a nonlinear site,i.e.four times the total shear wave travel time through the layer,is proved to be a good estimate.