A theoretical approach to non-destructive evaluation of deep foundations using guided waves has been formulated. Guided wave propagation in an infinitely long cylindrical pile embedded in soil is developed from dynamic equations of elasticity. Considering axisymmetric motion in the pile, the frequency equation for longitudinal modes is derived. The frequency equation represents a transcendental relationship between the non-dimensional frequency, W , and non-dimensional wave number, xa . The solution to the frequency equation is satisfied by an infinite number of modes that form branches in W-xa space. The first five branches of the longitudinal family of modes in a concrete pile embedded in soft/loose and hard/dense soils were numerically evaluated. Sensitivity analyses show that the real branches in the W-xra plane are essentially independent of the shear modulus and density of the surrounding soil, and correspond closely to the branches of longitudinal modes in a free-standing pile. However, the imaginary components of the branches in the W-xia plane are higher for soils with increased shear modulus and density.; The attenuation of the longitudinal guided wave modes is represented directly by the imaginary part of the wave number, xi , in nepers per length. The phase and group velocities vary with the frequency or wave number, contrary to the assumptions made in the theory of the one-dimensional approach. The power and displacement profiles of a given guided wave mode generally become more oscillatory as the order of the branch increases and as the frequency increases. For non-dimensional frequencies less than 2, the L(0,1) modes display the lowest attenuation and are easily induced in a pile, as evidenced by the popularity of the impulse response test in practice. Modes on the L(0,2) branch have the least attenuation compared to all other modes for non-dimensional frequencies above 12, and their relative simple mode shapes suggest that they are likely to be induced in practice. It is shown that locations where the group velocity is a maximum, the modes are isolated, which make it feasible for these particular modes to be propagated in practice.
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