## 2 Materials and Methods  
### Modal Analysis  
The modal analysis method was studied in detail in [17] for a cylindrical tubular waveguide. Solutions of the wave equation in a cylindrical waveguide can be easily found using potentials and the variable separation method, which gives the following general form for the displacement vector (\(\hat{u}\)) and the stress tensor (\(\hat{\sigma}\)) in a cylindrical coordinate system:  
\[\hat{u}(r,\theta,z)=\widetilde{u}(r,\theta)e^{\hat{y}\kappa z}=u(r)e^{jn\theta}, \tag{1}\]  
\[\hat{\sigma}(r,\theta,z)=\widetilde{\sigma}(r,\theta)e^{\hat{y}\kappa z}= \sigma(r)e^{jn\theta}e^{\hat{y}\kappa z}, \tag{2}\]  
where the cylindrical system is used (with coordinates (\(r\), \(\theta\), \(z\))), harmonic time variation \(e^{-j\omega t}\) is assumed, \(k=k(\omega)\) is the wavenumber, \(\omega\) is the angular frequency, and integer \(n\) is a separation constant called the circumferential order, which determines the symmetry of the solutions in the azimuthal direction. Modal analysis is based on two properties of normal modes: orthogonality (i.e., the existence of a scalar product that is zero for any two different modes) and completeness (i.e., the ability of a set of normal modes to represent arbitrary waveforms in a waveguide). In the case of the existence of external fields, the orthogonality relation of two arbitrary modes \(\mathbb{N}\)\(1\) and \(\mathbb{N}\)\(2\) is formulated as  
\[\hat{\
abla}\cdot(\hat{u}_{1}\hat{\sigma}_{2}^{*}-\hat{u}_{2}^{*}\hat{\sigma} _{1})=-\hat{u}_{1}\hat{f}_{2}^{*}+\hat{u}_{2}^{*}\hat{f}_{1}, \tag{3}\]  
where \(\hat{f}_{1,2}(r,\theta,z)\) are external forces, \(\hat{u}_{1,2}\) is the displacement vector, \(\hat{\sigma}_{1,2}\) is the stress tensor for modes \(\mathbb{N}\)\(1\) and \(\mathbb{N}\)\(2\), and \(\hat{\
abla}\) is tridimensional divergence operator. The second condition is completeness, which is based on the premise that an arbitrary disturbance in a waveguide can be represented as a set of waveguide modes:  
\[\hat{u}_{1}(r,\theta,z)=\sum\
olimits_{p}a_{p}(z)\widetilde{u}_{p}(r,\theta), \ \hat{\sigma}_{1}(r,\theta,z)=\sum\
olimits_{p}a_{p}(z)\widetilde{\sigma}_{p}(r, \theta), \tag{4}\]  
where the modes are numbered using index \(p\), and \(a_{p}(z)\) are the amplitude coefficients of the propagating modes. The next task is to calculate the set of coefficients \(a_{p}(z)\) if the waveguide is under an arbitrary action consisting of the vector force field \(\hat{f}_{e}(r,\theta,z)\) acting on the pipe volume (region D) and the stress \(\hat{\sigma}_{e}(r,\theta,z)\) applied to the pipe surface (region \(\partial\)D). Using the divergence theorem, the authors of [17] obtained an expression for the coefficients \(a_{p}\):  
\[a_{p}(z)=\frac{e^{\hat{y}\kappa z}}{4P_{p}}\int_{R_{g}}e^{-\hat{y}\kappa_{p}z^ {\prime}}\Big{[}f_{p}^{s}(z^{\prime})+f_{p}^{v}(z^{\prime})\Big{]}dz^{\prime}, \tag{5}\]  
where \(f_{p}^{v}(z)\) and \(f_{p}^{s}(z)\) are the contributions to the mode amplitude due to the volumetric forces and surface tractions:  
\[f_{p}^{s}(z)=-j\omega\oint_{\partial D}e^{-\hat{y}\kappa_{p}\theta}\cdot \Big{[}u_{p}^{*}(r)\cdot\hat{\sigma}_{e}(r,\theta,z)\Big{]}e_{n}dl, \tag{6}\]  
\[f_{p}^{v}(z)=-j\omega\iint_{D}e^{-\hat{y}\kappa_{p}\theta}\cdot\Big{[}u_{p}^{ *}(r)\cdot\hat{f}_{e}(r,\theta,z)\Big{]}dS, \tag{7}\]  
where \(n_{p}\) is the circumferential order for mode \(p\), and \(k_{p}\) is the wavenumber for mode \(p\). Integration is carried out in region \(R_{g}\), where elements \(f^{s}\) and \(f^{e}\) corresponding to the emission are not equal to zero, \(z\) is the coordinate of the point at which the ultrasonic signal is observed, \(P_{p}\) is the acoustic power, and the normal unit vector \(e_{n}\) is taken on each surface, thereby indicating the waveguide's interior.