# Characteristics of Low-Frequency Acoustic Wave Propagation in Ice-Covered Shallow Water Environment
## 2 Elastic Model
### Elastic Parabolic Equation Method with Ice Covers  
The parabolic equation can deal with the large-angle problem by approximating the far field and ignoring the inward propagating wave. The accuracy and stability of the solution are greatly improved by the split step method, and the improved self-starter can deal with various boundary conditions effectively.  
Figure 1: Schematic diagram of polar sound field model.  
Based on the relationship between stress and displacement, the elastic parabolic equation is derived from the equation of motion for elastic medium, which are written as the (\(u_{r}\),\(w\)) formulation of elasticity, for \(u_{r}=\partial u/\partial r\)[13],  
\[(\lambda+2\mu)\frac{\partial^{2}u_{r}}{\partial r^{2}}+\frac{\partial}{ \partial z}\Big{(}\mu\frac{\partial u_{r}}{\partial z}\Big{)}+\rho\omega^{2}u _{r}+(\lambda+\mu)\frac{\partial^{3}w}{\partial r^{2}\partial z}+\frac{ \partial u}{\partial z}\frac{\partial^{2}w}{\partial r^{2}}=0 \tag{1}\]  
\[\mu\frac{\partial^{2}w}{\partial r^{2}}+\frac{\partial}{\partial z}\Big{(}( \lambda+2\mu)\frac{\partial w}{\partial z}\Big{)}+\rho\omega^{2}w+(\lambda+ \mu)\frac{\partial u_{r}}{\partial z}+\frac{\partial\lambda}{\partial z}u_{r}=0 \tag{2}\]  
where \(u_{r}\) is the derivative of the horizontal displacement with respect to range and \(w\) is the vertical displacement. \(\omega\) is a time-harmonic point source of angular frequency. \(\lambda\) and \(\mu\) are Lame elastic parameters and defined as,  
\[\lambda=\rho\Big{(}c_{p}^{2}-2c_{s}^{2}\Big{)} \tag{3}\]  
\[\mu=\rho c_{s}^{2} \tag{4}\]  
where \(c_{p}\) is a compressional wave, \(c_{s}\) is a shear wave, and \(\rho\) is medium density.  
Through the derivation and transformation of Equations (1) and (2) in the form of a matrix operator, the standard form of a elastic parabolic equation is written. It is transformed into the form of inward and outward propagation, ignoring the outward propagation component. The solution of the equation is given by the differential:  
\[\left(\begin{array}{c}u_{r}\\ w\end{array}\right)\bigg{|}_{r+\Delta r}=e^{ik_{0}\Delta r\sqrt{l+X}}\left( \begin{array}{c}u_{r}\\ w\end{array}\right)\bigg{|}_{r} \tag{5}\]  
with  
\[X={k_{0}}^{-2}\Big{(}L^{-1}M-{k_{0}}^{2}I\Big{)} \tag{6}\]  
where \(\Delta r\) is a step in the range, \(I\) is the identity matrix, \(k_{0}\) is the reference wavenumber, and \(L\) and \(M\) are a matrix containing medium parameters and depth, respectively [14].  
By applying the split-step Pade approximation for Equation (3), it is obtained that  
\[\left(\begin{array}{c}u_{r}\\ w\end{array}\right)\bigg{|}_{r+\Delta r}=e^{ik_{0}\Delta r}\prod_{j=1}^{n} \frac{1+\alpha_{j,n}}{1+\beta_{j,n}}\left(\begin{array}{c}u_{r}\\ w\end{array}\right)\bigg{|}_{r} \tag{7}\]  
where \(\alpha_{j,n}\) and \(\beta_{j,n}\) should meet the requirements of stability, convergence, and accuracy.  
The discretization of depth operator \(X\) is based on the [PERSON] method [15]. After discretization, a large sparse banded matrix can be obtained, which can be solved by the block pursuit method based on Gaussian principal elimination. For the storage of the matrix, according to the characteristics of only six elements in each row, six groups of data can be used to store the coefficient matrix. Applying the initial condition at \(r=0\) by using the self-starter [16] and the sparse matrix, the solution can be obtained by solving step by step.  
After obtaining the whole sound field, the sound propagation characteristics are expressed by the sound pressure transmission loss:  
\[TL=-20*\log\frac{P}{P_{0}} \tag{8}\]  
where \(P\) represents the sound pressure amplitude obtained by simulation, and \(P_{0}\) represents the sound pressure at the sound source.