# Study of Low-Frequency Hydroacoustic Waves' Behavior at the Shelf of Decreasing Depth
## 4 Model Calculations  
With the purpose of studying the spatial distribution of the hydroacoustic energy, model calculations were performed. The simulation tool is the spectral element method (SEM), which is a high-order finite element method developed for the local and global scales of seismic wave propagation [7]. We simulated the propagation of an acoustic signal at the shelf of decreasing depth using the SEM software package with the SPECFEM2D open-source code [8; 9].  
SPECFEM2D integrates a simplified form of the wave Equation (3) using a high degree Lagrange interpolation polynomial. In a spatially inhomogeneous area of liquid, the wave equation for the pressure \(P(x,t)\) has the form  
\[\frac{1}{k}\ddot{P}=\
abla\cdot\left(\frac{\
abla P}{\rho}\right), \tag{3}\]  
where \(k(x)\) is the adiabatic volumetric modulus of liquid elasticity.  
In linearly elastic bodies, the strain tensor \(\varepsilon(x,t)\) is calculated by the displacement vector u as  
\[\varepsilon=\frac{1}{2}\Big{(}\
abla\overset{\rightarrow}{u}+\left(\
abla \overset{\rightarrow}{u}\right)^{T}\Big{)}. \tag{4}\]  
The stress tensor can be expressed through the strain tensor \(\overset{\rightarrow}{o}(\overset{\rightarrow}{x},t)\) according to Hooke's law.  
\[\overset{\rightarrow}{o}=\overset{\rightarrow}{c}:\varepsilon, \tag{5}\]  
where the colon denotes the double tensor reduction operation. The elastic properties of the medium are described by a fourth-order elastic tensor.  
SPECFEM2D allows the users to perform 2D and 2.5D (axisymmetric) simulations of the propagation of acoustic, elastic, visco-elastic, and poro-elastic acoustic waves. In addition, SEM combines the flexibility of the finite element method with the accuracy of the spectral method. Therefore, this software package is appropriate for performing numerical simulations in ocean acoustics with complex media and topography.  
The model is shown in Figure 6. The ocean surface is set at z = 0 m. All the physical parameters of the medium are listed in Table 3. The calculation area has a width of 3000 m and a depth of 100 m below the sea surface. All sides, except the top and axis, are covered with perfectly matched layers (PML) [10]. The source of Ricker pressure pulses with a dominant frequency of 22 Hz is located at (rs, zs) = (0, -18 m). The grid of the model consists of 118,385 spectral elements constructed using the open source Gmsh software [11]. Each numerical simulation takes an hour on a PC with the Linux operating system.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Distance from** & **Sea** & **Energy Density** & **Hydroacoustic** & **Energy Density of** & **Density of** & **Density of Seismic** \\
**Transmitter,** & **Depth,** & **Transmitter,** & **Energy** & **the Transmitter/** & **Seismic** & **Acoustic Energy/** \\
**m** & **m** & **Transmitter,** & **Doal/m3** & **Doal/m3** & **Hydroacoustic** & **Acoustic/m3** & **Hydroacoustic** \\ \hline
335 & 37 & 0.124 & \(3.1\times 10^{-4}\) & 83.8 & 0.91\(\times 10^{-8}\) & 4.2 \\ \hline
502 & 37.5 & 0.124 & \(1.7\times 10^{-4}\) & 68.8 & \(1.15\times 10^{-8}\) & 8.8 \\ \hline
674 & 36.2 & 0.176 & \(8.3\times 10^{-5}\) & 31.8 & \(3.51\times 10^{-8}\) & 46.8 \\ \hline
970 & 30 & 0.176 & \(4.0\times 10^{-5}\) & 22.1 & \(3.04\times 10^{-8}\) & 61.5 \\ \hline
1148 & 20 & 0.176 & \(3.3\times 10^{-6}\) & 2.2 & \(3.37\times 10^{-8}\) & 100 \\ \hline
1252 & 10 & 0.176 & \(1.9\times 10^{-8}\) & 1.4 & \(2.54\times 10^{-8}\) & 100 \\ \hline
1300 & 5 & 0.176 & \(4.3\times 10^{-9}\) & 0.3 & \(2.32\times 10^{-8}\) & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The calculated data.