# Modeling Acoustic Emission Due to an Internal Point Source in Circular Cylindrical Structures
## 4 Simulations  
In ref. [28], simulations of the displacement fields in TIC were confined to the case for the azimuthal independence (\(m=0\)) of wave propagation. In this paper, we extended the simulation to the case of azimuthally dependent tangential displacements, i.e., \(2\pi\)-periodic solutions. Stainless steel cylindrical structures (\(a=0.50\) m, \(l=2.0\) m, \(\rho=7.80\times 10^{3}\) kg/m\({}^{3}\), \(c_{P}=5.98\) km/s, \(c_{s}=3.30\) km/s, and \(\
u=155.4\) kHz) were used as the test specimens. First, we determined \(k_{  
with a minor SH wave and very weak SV wave. In Figure 4, the displacement amplitudes generated by \(P_{z}\) excitation differ significantly from those generated by the \(P_{j}\) excitation, in which the \(P_{z}\) excitation produces only the P wave. For the \(P_{j}\) excitation, the maximum values of \(u_{rj}\) and \(u_{zj}\) at the (0.5 m, 45\({}^{\circ}\), 1 m) position were 6.7 and 17.4 nm, respectively, while for the \(P_{z}\) excitation they were 22.1 and 127.0 nm, respectively. The amplitudes of the displacements due to \(P_{z}\) excitation are much stronger than those due to the \(P_{j}\) excitation. The angular dependence of the displacement was also calculated as a function of \(m\) (\(m\) = 0, 1, and 2), as shown in Figure 5. For \(m\) = 0, when the PS is located at the center of the circular plane, the angular dependences of \(u_{rj}\), \(u_{zj}\) and \(u_{ij}\) arise only from \(\Xi\) and \(\Sigma\), defined in Equations (32) and (35), respectively. For \(P_{z}\) excitation, the displacements of \(u_{rz}\) and \(u_{zz}\) are free from these factors. When the distances from the PS to the circumference are not equivalent, the angular dependences of the radial and axial displacements are highly significant. These effects are due not only to \(\Xi\) and \(\Sigma\), but also to the superposition of the Bessel functions involved in the displacement Equations. When \(m\) is nonzero, additional azimuthal factors, \(m\), \(\cos(m\theta)\), and \(\sin(m\theta)\), result in complex angular dependency. It should be noted that at a certain angle, some \(a_{ij}\) values of \(u_{rj}\), \(u_{zj}\), and \(u_{rz}\) in Equation (64) become too small to cause the sudden increase in displacement, hereafter referred to as \"computational divergency\". Figure 6 shows the angular dependence of a relatively  
Figure 4: The 2\(\pi\)-periodic (\(m\) = 0) displacements at the (0.5 m, 45\({}^{\circ}\), 1 m) position generated by the PS of 1.0 N located at the center of the cylinder (\(P_{j}\): upper, \(P_{z}\): lower): (**a**,**c**) radial (black line), axial (red line), and tangential (green line) components; and (**b**,**d**) P (black line), SH (red line), and SV (green line) waves. strong \(u_{zz}\) generated by two PSs located at \(x_{0i}=0,\ x_{0j}=0.45\) m and \(z_{0}=1\) m, and \(x_{0i}=0.35\) m, \(x_{0j}=0.35\) m and \(z_{0}=1\) m. For the first PS, the distance from the PS to the point at \(\xi_{i}=0,\ \xi_{j}=0.5\) m, and \(\eta=0\) on the outer surface is the shortest with \(\varsigma=0.05\) m.  
For this case, \(\tan^{-1}\left(\frac{\xi_{i}}{\xi_{j}}\right)=90^{\circ}\). For the second PS, the distance along \(\tan^{-1}\left(\frac{\xi_{i}}{\xi_{j}}\right)=45^{\circ}\) is the shortest from PS \(\left(\varsigma=0.005\text{ m}\right)\). Spectra of the angular dependences of the displacement fields are symmetric with respect to \(\theta=\tan^{-1}\left(\frac{\xi_{i}}{\xi_{j}}\right)\). As shown in Figure 6, the \(u_{zz}\) displacements at a point very close to the PS \(\left(\varsigma=0.005\text{ m}\right)\) are extremely large. Figure 5: Angular dependences of the maximum displacements on the circumference, with \(r=0.5\) m and \(z=1\) m for the cylinder: radial (black line), axial (red line), and tangential (green line) displacements generated by (**a**) \(P_{i}\) and (**b**) \(P_{x}\) of \(1.0\) N, located at \(x_{0i}=0,\ x_{0j}=0\), and \(z_{0}=1\) m.  
Remarkably, \(P_{z}\) excitation of 1 N produces a P wave with a maximum amplitude of tens of centimeters.