## 2 Theoretical Modeling
### Boundary Condition  
Continuity relations of normal displacements and stresses exist at the interface between water and porous sea ice. Meanwhile, shear stress in the ice vanishes. Therefore, the boundary conditions [22] are as follows:  
\[(1-\beta)u_{z1}+\beta W_{z1} = u_{z0} \tag{13}\] \[\tau_{zz1} = -(1-\beta)p_{0}\] (14) \[p_{f1} = -\beta p_{0}\] (15) \[\tau_{zx1} = 0 \tag{16}\]  
where \(\beta\) is the porosity of the porous sea ice. When substituting the normal displacement and pressure of water in Equations (3) and (4) and Equations (8)-(13) into Equations (13)-(17), a series of equations about the unknown amplitudes of displacement potentials can be derived, which can be described in the form of matrices:  
\[B_{0}(k_{x})A_{0;1} = 0 \tag{17}\]  
where \(B_{0}(k_{x})\) is the 4 \(\times\) 8 coefficient matrix, \(A_{0;1}\ =\ [1,V,\phi_{f1}^{+},\phi_{f1}^{-},\phi_{s1}^{+},\phi_{s1}^{-},\psi_{1}^{+}, \psi_{1}^{-}]\), representing the vector of amplitudes of displacement potentials of water and the first layer of sea ice.  
For simplicity of calculation, it is assumed that adjacent ice layers have the same porosity. Nevertheless, it is also applicable to adjacent layers with different porosities as long as the boundary conditions are changed. Additionally, the continuity of displacements, stresses and pressure of the ice frame and the fluid in pores at the \(i\)th ice-ice interface can be given as follows:  
\[u_{x(i-1)}\ =\ u_{xi},u_{z(i-1)}\ =\ u_{zi},W_{z(i-1)}\ =\ W_{zi} \tag{19}\]  
\[\tau_{zz(i-1)}\ =\ \tau_{zzi},\tau_{zx(i-1)}\ =\ \tau_{zxi},p_{f(i-1)}\ =\ p_{fi} \tag{20}\]  
where the subscript \(i\) means the \(i\)th ice-ice interface, as Figure 3 shows.  
Similarly, when substituting Equations (8)-(13) into Equations (19) and (20), the following can be derived:  
\[B_{i}(k_{x})A_{i-1;i}\ =\ 0(i\ =\ 2,3,4\ldots n) \tag{21}\]  
where \(B_{i}(k_{x})\) is the \(6\times 12\) coefficient matrix, \(A_{i-1;i}\ =\ [\phi^{+}_{f(i-1)},\phi^{-}_{f(i-1)},\phi^{+}_{s(i-1)},\phi^{-}_{s(i-1)}, \psi^{+}_{i-1},\)\(\psi^{-}_{i-1}\phi^{+}_{f^{\prime}},\phi^{-}_{fi},\phi^{+}_{si},\phi^{+}_{si}, \psi^{-}_{i},\)\(\psi^{-}_{i}\), representing the vector of amplitudes of displacement potentials of adjacent layers of sea ice.  
Finally, the boundary condition of the ice-air interface is similar to that of the water-ice interface:  
\[(1-\beta)u_{zn}+\beta W_{zn}\ =\ u_{za} \tag{22}\]  
\[\tau_{zzn}\ =\ -(1-\beta)p_{a} \tag{23}\]  
\[p_{fn}\ =\ -\beta p_{a} \tag{24}\]  
\[\tau_{zzn}\ =\ 0 \tag{25}\]  
When substituting the normal displacement and pressure of air in Equations (3) and (4) and Equations (8)-(13) into Equations (22)-(25), the following can be obtained:  
\[B_{a}(k_{x})A_{n;a}\ =\ 0 \tag{26}\]  
where \(B_{a}(k_{x})\) is the \(4\times 7\) coefficient matrix,\(A_{n;a}\ =\ [\phi^{+}_{f^{\prime\prime}},\phi^{-}_{f^{\prime\prime}},\phi^{+}_{ sn},\phi^{-}_{sn},\psi^{+}_{n},\psi^{-}_{n},\phi^{+}_{a}]\), signifying the vector of amplitudes of displacement potentials of the \(n\)th layer of sea ice and air.  
Based on the boundary conditions at all interfaces, a set of matrix equations about Equations (18), (21) and (26) is derived. There are a total of \(4+6(n-1)+4=6n+2\) linear equations about amplitudes of displacement potentials. Solving these equations, \(6n+2\) solutions are derived:  
\[V,\phi^{+}_{f^{\prime}},\phi^{-}_{f^{\prime}},\phi^{+}_{si},\phi^{-}_{si}, \psi^{+}_{i},\psi^{-}_{i},\phi^{+}_{a}(i\ =\ 2,3,4\ldots n) \tag{27}\]  
Thus, the reflection coefficient of the plane wave is obtained.