## 1 Introduction  
Understanding the wave propagation in a pipe is of fundamental importance to some engineering systems such as oil and gas pipeline systems and urban water distribution networks. Pipelines are extensively used to transport energy products like crude oil and gas, and for water transport and distribution. Corrosion flaws can begin to develop as pipelines age, and it is therefore vital to develop ways to inspect them efficiently. Many industrial pipelines are buried in the ground and are mostly isolated and inaccessible. Therefore, there is a need to employ remote sensing techniques to monitor the development of flaws in such pipelines. The acoustic wave propagation technique has become a very attractive, non-destructive testing technique for the remote monitoring of pipelines. The wave propagation technique has the advantage of full volumetric inspection, and can be used to find the characteristics of flow in a fluid-filled pipeline. The acoustic wave propagation technique operates in elongated geometrical bodies with constant or variable cross-sections, e.g., pipes. Sound generated in these bodies is constrained to propagate in the pipe wall, as well as in the fluid inside the pipe, and is reflected back to the generation point when a change in impedance (discontinuity) occurs. Therefore, acoustic wave propagation can only occur when there is a medium in which to propagate. Acoustic wave propagation phenomena along pipelines have been investigated theoretically and experimentally. Several modifications of the acoustic wave equation to more complex equations have been established. For instance, for a fluid flowing with varying density and acoustic velocity, [PERSON] [1] previously proposed an equation that is widely used in underwater acoustics problems, whereas [PERSON] [2] derived a generalized wave equation for an inhomogeneous fluid in motion whose flow profile differs in time and position. [PERSON] and [PERSON] [3] proposed a method to analyze the surface waves propagating on a grounded anisotropic slab. They considered the field variation in the direction parallel to the wave propagation. The wave propagation characteristics were derived on the basis of [PERSON]'s equation and boundary conditions. Recent developments showed that acoustic wave propagation using the Finite Element Method (FEM) has been utilized in pipeline detection. The sound propagation within a fluid flow in a pipe can be evaluated, taking the flow profile as a boundary condition for the simulation task. The FEA solvers for linear and non-linear acoustic wave equations vary. This leads to a symmetric matrix formulation for fluid-structure interaction [4], and this is being used in several computation solvers for acoustics. [PERSON] and [PERSON] [5] developed a finite element model of a surface acoustic wave sensor using the ANSYS simulation tool; the model was tested for hydrogen detection. Another numerical FEA approach was proposed for the calculation of non-linear propagation of acoustic waves by [PERSON] et al. [6]. [PERSON] et al. [7] presented an FEA method based on a modified wave equation, and compared their results with previously developed techniques like the Holtz-Kirchhoff integral and ray tracing method. They obtained sets of primary and secondary wave equations, and for numerical implementation, they employed the Finite Element Method to discretize and solve the equation integral scheme with respect to time. The numerical result shows that the equations are valid, but with more pronounced discretization error for the secondary waves. An FE numerical technique for simulating a 3D acoustic-wave correlator was presented by [PERSON] et al. [8]. They presented and discussed the results of the analysis of device responses for variable acoustic modes and differences in electrical response in the system. They concluded that FEA provides an effective means of analysing the correlator's response to various propagating modes. [PERSON] et al. [9] used FEM in modelling and simulation of surface acoustic wave sensor.