## 2 Mathematical Modelling of Propagating Signal
### Modelling the Interaction between the Medium and the Acoustic Signal  
The next step is to solve \(\mathbf{p}\) using the Lagrangian of (7) for continuous media and fields with finite degrees of freedom:  
\[\mathcal{L}=\frac{1}{2}\dot{\mathbf{p}}^{2}+\frac{\beta(t)ck}{2}\mathbf{p}^{2 }+\frac{\omega(t)^{2}ctk}{2}\mathbf{p}^{2} \tag{8}\]  
where \(\frac{1}{2}\dot{\mathbf{p}}^{2}\) is the kinetic energy density and \((\frac{\beta(t)ck}{2}\mathbf{p}^{2}+\frac{\omega(t)^{2}ctk}{2}\mathbf{p}^{2})\) is the potential energy density. The next step is to formulate the Hamiltonian of (8) using the Euler-Lagrange concept to obtain  
\[\mathcal{H}(\dot{\mathbf{p}},p,t)=\dot{\mathbf{p}}\frac{\partial\mathcal{L}}{ \partial\dot{\mathbf{p}}}-\mathcal{L}=\frac{1}{2}\dot{\mathbf{p}}^{2}-\frac{ \beta(t)ck}{2}\mathbf{p}^{2}-\frac{\omega(t)^{2}ctk}{2}\mathbf{p}^{2}. \tag{9}\]  
The difference between kinetic and potential energies, as reflected in Equation (9) remains constant before and after the duration over which the signal exists. Thus, the Hamiltonian, denoted as \(\mathcal{H}\), can be expressed as the asymptotic Milne energy, \(E_{M}\), rather than the effective signal strength at any point between the transmit and receive points on a UWAC link. Here, we introduce the concept of path bundles, \(\mathbf{q}\), which consists of the solution to the Hamiltonian in (9) bounded by \(E_{M}\geq 1\). Physically, the vector \(\mathbf{q}\) represents the transmission rays that travel along a specific direction. The corresponding Milne energy can be expressed as,  
\[E_{M}=\frac{1}{2}\dot{\mathbf{q}}^{2}-\frac{\beta(t)ck}{2}\mathbf{q}^{2}-\frac {\omega(t)^{2}ctk}{2}\mathbf{q}^{2} \tag{10}\]  
By solving for \(\mathbf{q}\) in (10), we can express \(\mathbf{q}\) as follows:  
\[\mathbf{q}=\sqrt{\mathbf{q}_{\pm}^{2}\cos^{2}(t-\tau)+\mathbf{q}_{\mp}^{2}\sin ^{2}(t-\tau)} \tag{11}\]  
where \(\mathbf{q}\) physically represents the amplitude of the acoustic ray bundles travelling through water (the path bundles include both the direct rays and reflected rays that constitute the total acoustic signal), and \(\tau\) is the effective time delay experienced by the travelling signal. The derivative of \(\mathbf{q}\) in (11) is set to \(\dot{\mathbf{q}}=0\). Putting \(\dot{\mathbf{q}}=0\) back in (10),  
\[E_{M}=-\frac{\beta(t)ck}{2}\mathbf{q}^{2}-\frac{\omega(t)^{2}ctk}{2}\mathbf{q }^{2}. \tag{12}\]  
The above equation (12) can be solved to obtain  
\[\mathbf{q} =\pm t\sqrt{2E_{M}/(\beta(t)ck+\omega(t)^{2}ctk)}\] \[\mathbf{q}(t) =\pm t\sqrt{2E_{M}\cos(2t-\tau)/(\beta(t)ck+\omega(t)^{2}ctk)} \tag{13}\]  
where \(t=\sqrt{-1}\). Figure 1 demonstrates the evolution of the amplitude of the acoustic path bundles (\(\mathbf{q}\)) over time for different values of the damping constant \(\beta\). It is important to note that we did not vary \(\beta\) as a function of time \(t\). For our simulations, we kept \(\beta\) fixed over the time period in which the amplitude evolution of the signal path bundles was observed. The amplitude evolution of the signal paths was formulated using (12). Finally, the interaction between \(\mathbf{q}\) and water over time can be represented in the form of the transition matrix, \(\mathbf{M_{q}}\), where  
\[\mathbf{M_{q}}=\mathbf{D} \begin{pmatrix}\cos 2\delta&\mathbf{q}_{-}^{2}\sin 2 \delta\\ -\mathbf{q}_{+}^{2}\sin 2\delta&\cos 2\delta\end{pmatrix} \mathbf{D};\mathbf{D}= \begin{pmatrix}\cos\tau&-\sin\tau\\ \sin\tau&\cos\tau\end{pmatrix}  \tag{14}\]  
with \(\delta\) being the effective phase shift experienced by the acoustic signal. With the value of \(\mathbf{D}\), \(\mathbf{M_{q}}\) can be expressed as,  
\[\mathbf{M_{q}}= \begin{pmatrix}\cos\tau\cos\delta+\mathbf{q}_{-}^{2}\sin\tau \sin\delta&\mathbf{q}_{-}^{2}\sin\delta\cos\tau-\sin\tau\cos\delta\\ \sin\tau\cos\delta-\mathbf{q}_{+}^{2}\sin\delta\cos\tau&\cos\delta\cos\tau+ \mathbf{q}_{+}^{2}\sin\tau\sin\delta\end{pmatrix}