## 1 Introduction  
In pipeline systems, a number of different materials can be used for the pipe walls, with the most common being steel and thermoplastics such as polyethylene (PE). PE pipelines have a wide range of applications, including transportation of liquids on ships and aircraft, long-distance transportation of natural gas, storage and transportation of liquids in the chemical industry, and urban water supply. They are of particular importance in the last of these applications owing to their advantages of high strength, high resistance to corrosion and wear, good stability over a wide range of temperatures, and lack of toxicity [1]. However, fluid-filled PE pipelines often suffer from problems caused by excessive vibration and noise [2; 3]. Vibration can cause long-term fatigue damage to the pipeline system, while noise not only reduces the stability and safety of the entire pipeline system, but can also have a deleterious effect on the environment for people in the vicinity of the pipeline. Burst water supply pipelines not only result in losses of large amounts of water and consequent serious disruption to daily life, but can also lead to secondary consequences such as traffic jams and even disasters such as landslides if they are not repaired rapidly. Therefore, it is of great importance to investigate the acoustic transmission characteristics of fluid-filled PE pipelines and to develop methods to control the noise and vibration that are generated in these systems. Before investigating acoustic propagation in fluid-filled pipelines, it is useful to consider the problem of elastic vibrations of a cylindrical shell. In the nineteenth century, [PERSON] (1945) investigated the vibration of cylindrical shells and obtained the free-vibration frequency of an infinite cylindrical shell in vacuum. More recently, the dispersion characteristics of pipelines have been extensively investigated ([PERSON] et al., 2016; [PERSON] et al., 2017; [PERSON] et al., 2018; [PERSON] et al., 2019). [PERSON] (2019); [PERSON] (2019); [PERSON] (2019) investigated vibration problems for cylindrical shells in liquids. The first to investigate acoustic propagation in a fluid-filled pipeline was [PERSON] (1945), who came to the conclusion that this propagation is influenced by the strength of longitudinal waves in the wall compared with that of bending waves. Later, [PERSON] and [PERSON] (2019) investigated the dispersion properties of a sound field in a fluid-filled pipeline, and analyzed the first four normal modes of waves in an axisymmetric rigid pipeline. Using a short-pulse signal, [PERSON] et al. (2016) experimentally investigated the dispersion of longitudinal waves in a liquid-filled cylindrical shell and found that the liquid in the pipeline slightly reduced the group velocity and cutoff frequency of the longitudinal mode in the tube wall. [PERSON] et al. (2017) conducted an experimental investigation of acoustic propagation in a liquid-filled pipeline, examining the effects of different pipe-wall materials on the sound field in the pipe. However, their experiment suffered from the limitations that the end of the pipeline was not muffled and the sound pressure was measured at only one point at a given time. [PERSON] et al. (2018) investigated acoustic propagation in a fluid-filled pipeline both experimentally and numerically. In their experiment, sound field measurements could not be obtained throughout the fluid, because they used only two PZT (Lead zirconate titanate piezoelectric ceramic) circular transducers mounted on the pipe wall, one at the transmitting end and the other at the receiving end. [PERSON] (2019), [PERSON] et al. (2020) and [PERSON] et al. (2018) conducted systematic theoretical and experimental investigations of acoustic propagation in liquid-filled elastic pipelines, but their experimental methods and equipment were not very different from those used in previous studies. To date, there have been few theoretical and experimental studies focusing on sound below the cutoff frequency.