## 1 Introduction  
Longitudinal ultrasonic guided waves in pipes are widely employed in pipeline detection, as well as pipe condition assessment and monitoring, owing to their long-range propagation and low energy dissipation. The theoretical background of guided waves was first described by a characteristic frequency equation in 1959 [1]. Nondestructive testing that employs guided waves has widespread applications. An example is pipeline corrosion, which is a major problem in the chemical and oil industries. Moreover, there is significant interest in reducing leakage due to corrosion in liquid-filled pipes [2, 3, 4, 5, 6, 7]. That is, if pipes are filled with a liquid, then ruptures of the wall can generate acoustic waves in the low audio frequency range, which can then be detected by the remote monitoring equipment placed in the fluid or on the walls [8, 9]. These detection techniques rely on the propagation of acoustic energy in both the structure and the fluid, as well as the detection and interpretation of energy transfer. However, it is common for these pipes to also be buried underground, or immersed in a fluid, and this can significantly affect the propagation of energy. This often leads to a reduction in the effectiveness of detection techniques, and so it is important to develop a good understanding of the parameters and acoustic propagation characteristics of the media outside the pipe if one is to develop reliable inspection methodologies, particularly when there is an existing leakage in the pipeline. To clearly describe the physical behavior of a cylindrical wave, many efforts have been devoted to investigating the ultrasonic guided waves. [PERSON] et al. [10] studied the propagation characteristics of axisymmetric waves in cylindrical shells immersed in liquid and calculated the dispersion curves of the medium wave, which were consistent with experimental results. [PERSON] et al. [11] employed both numerical and experimental methods to study the propagation characteristics of axisymmetric guided waves in liquid-filled elastic tubes. They found at least two modes of propagation in an elastic shell at any frequency. Based on the fluid model, [PERSON] et al. [12] studied the propagation characteristics of guided waves in a tube, and discussed the differences between complex frequency and complex wave number solutions. [PERSON] et al. [13] focused on the attenuation characteristics of the basic mode of propagation in buried pipes. [PERSON] et al. [14] combined the ultrasonic guided wave with the wavelet analysis to study the influence of a crack on the propagation of an ultrasonic waveguide in the pipeline. [PERSON] et al. [15] experimentally investigated the propagation characteristics of guided waves in buried pipelines and analyzed the influence of the variability of soil conditions on the attenuation characteristics of guided waves. [PERSON] et al. [16] conducted a theoretical and numerical investigation of the influence of the contained liquid on backward waves and associated zero-group-velocity modes. In the above-mentioned references, the predictions presented so far assume that the medium surrounding the pipe is perfect and elastic. It is, however, well known that materials such as soil have internal damping present within their structure, and that this may affect the propagation of an elastic wave. Damping will affect the attenuation of a particular eigenmode, and the effect of the material damping may dominate the attenuation caused by the energy radiating away from a structure [17; 18]. Hydrogeological studies also indicate that the medium outside the pipe has a certain complexity, and the shallow surface of the earth is highly porous; thus, it is impractical to apply the elastic medium directly in pipeline detection. Considering that the porous medium takes into account both solid matrix and pore fluid, the porous medium model is more practical and appropriate.