Three-dimensional analytical solution for transient guided wave propagation in liquid-filled pipe systems

The objective of this study is to investigate the three-dimensional (3-D) analytical solution for transient guided wave propagation in liquid-filled pipe systems using the eigenfunction expansion method (EEM). The eigenfunctions corresponding to finite liquid-filled pipe systems with a traction-free lateral boundary and rigid smooth end boundaries are obtained. Additionally, the orthogonality of the eigenfunctions is proved in detail. Subsequently, the exact 3-D analytical transient response of finite liquid-filled pipe systems to external body forces is constructed using the EEM, based on which, the approximate 3-D analytical transient response of the systems to external surface forces is derived. Furthermore, the analytical solution for transient guided wave propagation in finite liquid-filled pipe systems is extended explicitly and concisely to infinite liquid-filled pipe systems. Several numerical examples are given to illustrate the analysis of the spatial and frequency distributions of the radial and axial displacement amplitudes of various guided wave modes; the numerical examples also simulate the transient displacement of the pipe wall and the transient pressure of the internal liquid from the present solution. The present solution can provide some theoretical guidelines for the guided wave nondestructive evaluation of liquid-filled pipes and the guided wave technique for downhole data transfer.