# A Comprehensive Review of Acoustic Methods for Locating Underground Pipelines
## 2 Theoretical Study on Wave Propagation Mechanism in Pipe Systems  
Theoretical studies on the behaviours of wave propagation in pipe structures have received much attention to date, and have been carried out by many researchers. Better understanding of the wave propagation mechanism in pipe systems is essential not only for pipeline locating, but also for leakage detection [15; 16; 17] and condition assessment [18; 19; 20; 21]. Here, a comprehensive literature review of fundamental studies on wave propagation mechanism in pipe structures is presented to provide physical insights into the development of acoustic detection techniques. Earlier research on wave propagation in tubes or liquid cylinders [22; 23; 24] provides the theoretical basis for localization of water pipes, with attention to low-order mode waves or relatively low frequency range. [PERSON] investigated radial vibrations of hollow cylinders with thick wall with particular emphasis on axisymmetric motion [25]. Further investigation was carried out by [PERSON] [26] on the plane-stain vibration of hollow cylinders covering the cases from a thin shell to the Pochhammer [27] case of a solid cylinder. Both axisymmetric and non-axisymmetric motions were then studied in the framework of the theory of elasticity. This led to a three-dimensional solution to a characteristic equation of wave propagation in hollow circular cylinder [28], and corresponding numerical results [29]. [PERSON] has solved the exact formulation of the characteristic equation for tubes with arbitrary thickness, focusing on axisymmetric waves [30; 31]. Based on this theoretical analysis, [PERSON] and [PERSON] conducted research on two mode waves (ET0 and ET1) in a liquid-filled elastic tube at low frequencies [32], where the dispersion characteristics and particle displacement profiled in both plastic and aluminum tubes were presented. [PERSON] et al. [33] have investigated axisymmetric waves propagation in fluid-loaded cylindrical shells, where five cases were considered including a cylindrical shell in vacuum, a cylindrical shell surrounded by a liquid, a hypothetical liquid column, a cylindrical shell filled with liquid, and a cylindrical shell immersed in an infinite liquid. The numerical results of dispersion curves, attenuation due to radiation, displacement distribution and stress distribution were presented. To validate the theory foundation, experiments were then carried out [34], where the dispersion curves was inverted from experimental time waveforms through [PERSON]'s method [35]. To address a special case where the thickness is much smaller than the radius, various approximations have been adapted to develop various thin shell theories, which range from [PERSON]'s first approximation [36; 37], [PERSON]'s theory [38; 39; 40; 41] to 'Timoshenko-type'theories [42]. A representative work based on Timoshenko-type theory was carried out by [PERSON] and [PERSON] [43], extending to high modes at higher frequencies in a fluid-filled cylindrical shell. The study was confined to axisymmetric waves with purely sinusoidal variation, in comparison with the wave propagation through a cylindrical bore contained with fluid in [44]. Two modes were found to exist at all frequencies, which challenges [PERSON]'s erroneous conclusion [45] that only one mode exists at lower frequencies. [PERSON] and [PERSON] [46] investigated more comprehensive dispersion and energy distribution of waves in shells containing fluid on the basis of Donnell-Mushtati shell equations, where branches in the real, imaginary and complex planes were observed. The vibration behaviour of shells excited by an internal monopole source was studied further in [47; 48; 49]. More references regarding different thin shell theories can be found in [50], with useful writings from the standpoint of comparison of various thin shell theories such as [51; 52; 53; 54; 55; 56] being recommended in the book.