## 1 Introduction  
Reliable maintenance of high-pressure gas vessels, especially those containing fuel gases, has increased exponentially over the past decade, because the gases are extremely hazardous under conditions that can cause them to ignite. The most important way to prevent disasters caused by gas leak fires is to detect leaks and their leak location and make subsequent repairs. There are several pipeline leak detection methods being used to detect and locate leaks in a limited area of pipeline [1]. These include flow rate monitoring [2], pressure fluctuation analysis [3], fiber optic sensing [4], acoustic sensing [5; 6; 7; 8], and infra-red imaging [9]. Among the various methods, acoustic methods have proven to be very effective in detecting leaks by eliminating disturbing signals through time and frequency domain analysis and characterization of acoustic emission parameters [10; 11; 12]. Acoustic emissions (AEs) are elastic or stress waves generated by a material as the result of a sudden release of energy from a localized source within the solid. As well as the material degradation related to deformation and crack and fracture development, and fluid flow through the hole or crack created in the container system also generate AEs. When a fluid leak occurs due to the pressure difference between the inside and outside of the container, the fluid ejected from the leak hole can be characterized by Lighthill's stress [13]. In the turbulence flow, a fluctuating Reynolds stress (FRS), corresponding to the interaction between the fluid and the solid wall, is dominant in Lighthill's stress. The Reynolds stress is associated with a direction perpendicular to the wall and, consequently, leads to the excitation of AEs that propagate through the container. Many experimental investigations have been conducted to characterize the AE caused by gas leaks. [PERSON] et al. investigated the AE signal as a function of internal pressure, hole cross-sectional area, and leak rate [14]. They found that a low frequency sensor (30 kHz) is suitable for detecting small leaks and that the signal amplitude is nearly linear with internal pressure. Yoshida et al. investigated AEs during gas leaks from pipes with straight and stepwise pinholes in the 100-1200 kHz frequency range [PERSON] et al. (2018). For the straight pinhole, they found that the mean amplitude of the AE signal increased gradually as the pressure of the outgoing air increased, with no variation until 0.3 MPa. For stepwise pinholes, the results were dependent on the shallow depth: compared to the straight pinhole case, shorter shallow depths showed low mean amplitudes with similar pressure dependence, while longer shallow depths showed variation in the mean amplitude in the range of 0.11-0.15 MPa. [PERSON] et al. also identified the effect of the hole geometry on the amplitude and peak frequency of AE due to air leakage [PERSON] et al. (2018)[PERSON] Three types of pinholes used in the leakage test, namely straight-type, stepwise-type and cone-type, gave different results[PERSON] [PERSON] et al. measured AE signals generated by pipe vibrations due to gas leaks and used the Fast Fourier Transform (FFT) to determine the resonant frequency of the observed signals [PERSON] et al. (2018). The observed frequencies were confirmed by applying [PERSON]'s non-linear theory. Previously, we presented a mathematical model for AEs generated by point source (PS) in cylindrical structures [PERSON] and [PERSON] (1983). The PS as an internal defect was represented by a concentrated force (CF) with both spatial and temporal characteristics[PERSON] In cylindrical coordinates, the CF generates three scalar potentials responsible for one compressional (P) wave and two shear waves (horizontally and vertically polarized; SH and SV, respectively), which we designated as the concentrated force-incorporated potentials (CFIPs).