## 4 Results and Discussion
### Verification of Angular and Axial Dependences  
The \(\kappa_{\eta}\) parameter is crucial in determining the angular and axial dependencies by associating the coupling constant with the Green's function, as implied in Equation (50). In addition, the verification of the dependencies is necessary to apply the proposed mathematical model to the localization of the leak source in cylindrical storage. In this study, the verification has been extended to the whole region of cylindrical storage. First, the angular dependence of the AE amplitude was studied as a function of the tangential angle at a fixed \(\eta\) distance from the LS. Similarly to the case of the external excitation experiment, twelve sensors were mounted on the cylinder surface at 30\({}^{\circ}\) intervals from 0 to 270\({}^{\circ}\) at \(\eta\) = 10 cm. Among the obtained AE parameters, the maximum amplitude \(A_{\rm obs}\) of each AE hit was selected to analyze the angular dependence. For a given sensor, the mean amplitude was calculated from over 6 k hits. Figure 3 shows the \(A_{\rm obs}\) values observed at \(\eta\) = 10 from five different LSs with D0.2-D1.2 and three different internal pressures. Within the standard deviation, all results show that the minimum of \(A_{\rm obs}\) occurs close to \(\vartheta=90^{\circ}\) and \(\vartheta=270^{\circ}\), and the maximum of \(A_{\rm obs}\) occurs close to \(\vartheta=30^{\circ}\) and \(\vartheta=330^{\circ}\). Note that the calculated maximum and minimum values occur at \(\vartheta=\)16\({}^{\circ}\) and \(\vartheta=\)93\({}^{\circ}\) in the first half of the circle, respectively. To simulate the \(A_{\rm obs}\) values, it is necessary to convert the calculated \(u_{\rm leak}\) value in meters to \(A_{\rm obs}\) in millivolts. The analysis of the angular dependence of \(A_{\rm obs}\) can be performed by normalizing the calculated displacement values and then matching the relative amplitude values observed in the 2\(\pi\) region. The simulated angular dependence of the amplitude is formulated as follows:  
\[A_{\rm sim}(\vartheta)=u_{\rm N}(\vartheta)\big{(}A_{\rm obs,max}-A_{\rm o,ang }\big{)}+A_{\rm o,ang}. \tag{53}\]  
In Equation (53), \(u_{\rm N}(\vartheta)\) is the normalized displacement, defined as \(u_{\rm N}(\vartheta)=u_{\rm leak}(\vartheta)/u_{\rm leak,max}\), which is independent of the input parameters (such as FRS and frequency) in Equation (51), and \(A_{\rm obs,max}\) and \(A_{\rm o,ang}\) are the maximum and minimum values estimated from \(A_{\rm obs}\) at \(\vartheta=16^{\circ}\) and 93\({}^{\circ}\), respectively. As shown in Figure 3a-e, experimental values for D0.2-D0.8 agree well with the simulated values within the standard deviation, but for D1.2P3 and D1.2P4 there is a rather large deviation between \(A_{\rm sim}\) and \(A_{\rm obs}\). The angular dependence illustrates the symmetrical property of \(\pi\) and the minimum amplitudes close to 90\({}^{\circ}\) and 270\({}^{\circ}\). These results prove that the \(\kappa_{\eta}\) of 7.5, which was determined by external excitation, works very well to characterize the AE signal observed due to leakage in the cylindrical geometry. The most striking feature of the observed AE amplitudes is that the value of \(A_{\rm obs}\) is strongly influenced by the hole size. As expressed in Equation (10), the CF due to the leakage is only proportional to the mean velocity U. As expressed in Equations (A23) and (A24) in Appendix B [28; 29; 30; 31], substituting \(Q\) into  
Figure 2: (**a**) Angular dependence of the mean amplitudes (circles) and \(\sigma\)s (bars) of signal obtained by an external excitation (1.000 V\({}_{\rm PP}\) and \(\
u=150\) kHz) and the simulated \(u_{\rm leak}\) displacements (lines) at different values of \(\kappa_{\eta}\) (1: 6.0, 2: 7.0, 3: 7.5, 4: 8.0, 5: 9.0); (**b**) axial dependence of the mean amplitudes of signal and the simulated \(u_{\rm leak}\) displacement (line) with \(\kappa_{\eta}=7.5\). cancels the hole diameter factor \(D\) in Equation (A25). Accordingly, CF is only proportional to \(P_{0}^{2}\).