In this paper, we study the propagation of elastic longitudinal wave in one-dimensional (1D) finite periodic phononic crystal with exponential section, and obtain the expression of frequency response function. Comparing the 1D phononic crystal with constant section, we find that the value of attenuation inside the band gap decreases quickly as the cross-sectional area of output and increases, at the same time, the initial frequency decreases and the cut-off frequency increases, i.e. the width of the band gap will increase. The effects of the lattice constant and the filling fraction on band gap are also analyzed. The change trends of the initial frequency and cut-off frequency are consistent with those of constant section. We hope that the investigation of this paper will be helpful for the phononic crystal in practical application such as vibration shield.