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 $\left({\lambda }_{min}\right)$. However, damping reduces the amplitude of the refocused wave which may affect the resolution in the presence of large noise.

波的时间反转 (TR) 被用于许多领域，并奠定了实用设备的基础，例如，用于破坏脑肿瘤和肾结石、检测固体缺陷、查明地下矿藏和协助在远距离通讯中。理论上，波的 TR 特性在阻尼存在时会丢失。然而，发现基于波 TR 的技术对阻尼具有鲁棒性。特别是，水锤波的 TR 特性最近被用于开发计算快速且耐噪声的方法，用于在存在粘性和粘弹性阻尼的情况下进行管道缺陷检测。本文的目的是了解基于 TR 的方法在应用于阻尼水锤波时如何保持适用性。正在研究的问题是 (i) 有和没有阻尼的无界管道，以及 (ii) 有和没有阻尼的有界管道。考虑了稳态摩擦、非稳态摩擦和粘弹性阻尼效应。结果表明，水锤波显着地重新聚焦在源的正确位置和时间。无界和有界管道系统中波的空间分辨率约为最小探测波长的一半$\left({\lambda }_{分钟}\right)$. 然而，阻尼会降低重新聚焦波的幅度，这可能会影响存在大噪声时的分辨率。