# A Comprehensive Review of Acoustic Methods for Locating Underground Pipelines
## 2 Theoretical Study on Wave Propagation Mechanism in Pipe Systems
[PERSON] [57] studied the vibro-acoustic energy flow through a straight pipe, revealing that four wave types are responsible for most energy transmission including three axisymmetric waves (n = 0) and one flexural wave (n = 1). Of the axisymmetric waves, the fluid-dominated wave (s = 1) and the shell-dominated wave (s = 2) have coupled axial and radial motion. The third one is a torsional wave (s = 0), which is uncoupled from the fluid and does not have significant radial motion [58]. These mode shapes in cylindrical co-ordinates are demonstrated in the Figure 1. More recently, research carried out by University of Southampton has focused mainly on wave propagation in pipe structures regarding these four modes at low frequencies well below the pipe ring frequency. [PERSON] and [PERSON] [60] derived motion equations of two axisymmetric waves in a fluid filled pipe based on a simplified form of [PERSON]'s equation [61], and calculated the relative sizes of these two waves in different boundary conditions. Following the findings, an externally applied piezoelectric sensor, which is sensitive only to axisymmetric waves, was proposed to monitor the radial wall motion. Further study measuring the axisymmetric vibrational power in empty and fluid-filled pipes was carried out in [62], where practical measurements using four methods was investigated through various combinations of axially aligned accelerometers and circumferential sensors. The aforementioned research work has mainly focused on the pipes in vacuo. To understand the wave propagation mechanism in buried pipe in practical applications, the analytical model in [60] was extended to a fluid-filled pipe surrounded by infinite elastic medium [63]. To simplify the calculation, the outer medium was treated as fluid that can sustain both shear and longitudinal waves. The expressions  
Figure 1: Illustration of the low-frequency mode shapes in cylindrical co-ordinates: (a) wave motion; (b) longitudinal mode; (c) torsional mode; (d) flexural mode [59]. for the wavenumbers of the fluid-dominated and shell-dominated waves were then derived in terms of stiffness components of the contained fluid and pipe wall, a pipe wall mass component, and the impedances of the waves in the surrounding medium. In this way, the effects of the surrounding medium can be investigated through depicting the real and imaginary parts of component impedances. Experimental measurements were subsequently conducted to validate the theoretical predictions [64]. A special case with submerged plastic water pipes was further investigated in [65]. Based on the PVDF (Polyvinylidene fluoride) wire sensor proposed in [60], a novel configuration of ring sensors was modified and installed to monitor the acoustic pressure in the main pipe. In the previous model established in [63], the surrounding infinite soil was treated as fluid. The wavenumber prediction model of axisymmetric waves was improved by including the shear coupling at a lubricated pipe/soil interface [66; 67]. However, at low frequencies, the pipe is more likely to be in perfect bond with the surrounding soil. Simplified dispersion relationships for the fluid-dominated wave [58] and the shell-dominated wave [68] have been presented, where both shear coupling and perfect bonding at the pipe/soil interface were considered. The co-ordinate systems as well as stresses acting at the pipe-soil interface of the analytical model used in this dispersion relationships are demonstrated in Figure 2.  
Recent studies have shown that the fluid-dominated wave can radiate effectively into surrounding soil, based on which the location of buried pipes can be inferred through monitoring the ground surface vibration using for example geophones [69]. To clarify the wave propagation in surrounding soil, the ground surface vibration due to the fluid-dominated wave in buried water pipes was studied [70], following an investigation of surface displacements accompanying transmission of plane progressive waves in a buried gas pipe [71].