Time-reversal of water-hammer waves

Time reversal (TR) of waves is being used in many fields and underpins the foundation of practical equipment that, for example, is used to destroy brain tumour and kidney stones, to detect defects in solids, to pinpoint underground mineral deposits, and to assist in long-distance communication. Theoretically, the TR property of waves is lost in the presence of damping. However, techniques which are based on TR of waves are found to be robust to damping. In particular, the TR property of water-hammer waves was recently used to develop computationally fast and noise-tolerant methods for pipeline defect detection in the presence of viscous and visco-elastic damping. The objective of this paper is to understand how TR-based methods remain applicable when applied to damped water-hammer waves. The problems being investigated are (i) an unbounded pipe with and without damping, and (ii) a bounded pipe with and without damping. Steady friction, unsteady friction, and visco-elastic damping effects are considered. The results show that water-hammer waves remarkably refocus at the correct position and timing of the source. The spatial resolution of the waves in both unbounded and bounded pipe systems is about half the minimum probing wavelength (Formula presented.). However, damping reduces the amplitude of the refocused wave which may affect the resolution in the presence of large noise.