## 5 Principles Related to Traveling Wave Detection Methods  
### Principles and Algorithms of Variational Modal Decomposition  
VMD is a signal decomposition technique aimed at decomposing a signal into multiple intrinsic mode functions so that a local frequency component of the original signal can be represented by these intrinsic mode functions. Based on the variational principle, VMD obtains intrinsic mode functions by minimizing the total variation energy of the signal [32; 33; 34]. The variational calculation model of VMD is shown in Equation (1).  
\[\left\{\underset{\{u_{k}\},\{\omega_{k}\}}{\underset{\{\lambda_{k}\}}{\underset {\{\sum_{k}\}}{\underset{\{\sum_{k}\}}{\underset{\{\sum_{k}\}}{\underset{ \partial_{t}\}}{\underset{\{\left[\left(\delta(t)+\frac{j}{\pi t}\right)u_{k}(t )\right]}e^{-j\omega_{k}t}}}\right\|_{2}^{2}}}}}\right\} \tag{1}\]  
In the equation, \(\{u_{k}\}\) represents the collection of various IMF components decomposed from the original signal, \(\{\omega_{k}\}\) represents the collection of center frequencies of each IMF component, \(\delta(t)\) represents the unit impulse function, and \(\partial_{t}\) represents the partial derivative operator.  
The expanded Lagrangian expression is shown in Equation (2):  
\[\begin{split} L(\{u_{k}\},\{\omega_{k}\},\lambda):=\\ \underset{k}{\underset{\{\sum_{k}\}}{\underset{\{\partial_{t} \}}{\underset{\{\left[\left(\delta(t)+\frac{j}{\pi t}\right)u_{k}(t)\right]}e ^{-j\omega_{k}t}}}}}\Big{|}_{2}^{2}\\ +\left\|f(t)-\underset{k}{\sum_{k}u_{k}(t)}\right\|_{2}^{2}+ \left\langle\lambda(t),f(t)-\underset{k}{\sum_{k}u_{k}(t)}\right\rangle\end{split} \tag{2}\]  
By introducing the Lagrange multiplier \(\lambda(t)\) and the quadratic penalty term \(\alpha\), the problem is transformed from a constrained form into an unconstrained variational model. This extended Lagrangian expression not only simplifies the computation but also addresses the issue of estimating the fundamental frequency bandwidth.  
For the \"saddle point\" of the extended Lagrangian expression, the alternating direction method of multipliers (ADMM) is used for computation. It iteratively optimizes the variational model by updating \(\hat{u}_{k}^{n+1}\) and \(\omega_{k}^{n+1}\) until the optimal solution is obtained. The expressions for the updated mode components and central frequencies are given by Equations (3) and (4):  
\[\hat{u}_{k}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\underset{i\
eq k}{\sum_{i} }\hat{u}_{i}(\omega)+\frac{\hat{\lambda}(\omega)}{2}}{1+2\alpha(\omega-\omega _{k})^{2}} \tag{3}\]  
\[\omega_{k}^{n+1}(\omega)=\frac{\int_{0}^{\infty}\omega|\hat{u}_{k}(\omega)|^{ 2}\text{d}\omega}{\int_{0}^{\infty}\left|\hat{u}_{k}(\omega)\right|^{2}\text{ d}\omega} \tag{4}\]In the equations, \(\hat{f}(\omega)-\sum_{i\
eq k}\hat{u}_{i}(\omega)\) represents the current residual, \(\hat{u}_{k}^{n+1}(\omega)\) denotes the Wiener filter of the current residual, \(\omega_{k}^{n+1}(\omega)\) stands for the central frequency of the current mode function, and the Fourier transforms of \(f(t)\), \(u(t)\), \(\lambda(t)\), and \(\hat{u}_{k}^{n+1}(t)\) are, respectively, denoted by \(\hat{f}(\omega)\), \(\hat{u}_{i}(\omega)\), \(\hat{\lambda}(\omega)\), and \(\hat{u}_{k}^{n+1}(\omega)\).