Longitudinal waves in piles with exponentially varying cross sections

Longitudinal waves in piles with exponentially varying cross sections were analytically studied.The pile,subjected to longitudinal impact force at the top,was assumed to be a homogeneous elastic circular bar with finite length and exponentially varying radii.The velocity of shear wave in the surrounding soil was also assumed to vary exponentially,and the soil-pile interaction was taken into account.The transfer function and pulse response of the soil-pile system under longitudinal vibration were derived by direct and inverse Laplace transforms,respectively.The frequency response function,frequency-domain and time-domain expressions for the longitudinal vibration velocity at the pile top were obtained further.The characteristics of longitudinal waves for piles with gradually varying and equal cross sections were investigated.It was found that reflected waves could be produced by the gradual variation of the cross section of pile,which varied with the variation of the cross section of pile,mainly affecting the low frequency resonant peaks.It was shown by parametric studies and the proposed theoretical formula that the attenuation of longitudinal waves in the pile depended on three ratios,that is,the ratio of shear wave velocity in soil to longitudinal wave velocity in pile,the ratio of pile length to diameter of pile top,and the ratio of the soil density to the pile density.