## 2 Theory and Analysis
### Range-Frequency Interference Structure of Shadow Zone
 \tag{12}\]  
Considering the small difference of the grazing angles of the four multi-way arrivals, it could be approximately regarded as \(\phi_{\text{SB}}-\Phi_{\text{B}}=\phi_{\text{SBS}}-\phi_{\text{BS}}\approx\Delta\phi\) and \(\varphi_{\text{rSB}}=\varphi_{\text{rBS}}\), Equation (12) can be approximately written as:  
\[p_{2}(x,z;f)\approx\frac{S(f)\sqrt{WF}}{4\pi R}W_{\text{ws}}V_{\text{sb}}W_{ \text{sw}}e^{i2\pi ft_{\text{SB}}}e^{i2\Phi_{\text{SB}}}(e^{-i2\pi f\Delta t_ {1}}+e^{i2\Delta\phi}e^{i2\pi f\Delta t_{2}}-2) \tag{13}\]  
Due to the interference caused by the layered structure of the ocean bottom, the term \(e^{-i2\Delta\phi}\) appears in the coherent superposition part of Equation (7). This term results from the interference of vertical phase shifts in the sedimentary layer at different angles; however, due to the small difference of grazing angles between B, SB, BS, and SBS, namely, \(2\Delta\phi\ll 2\pi f\Delta t_{1,2}\), it could be approximately assumed that \(\Delta\phi\approx 0\). Combining Equations (11) and (13), Equation (6) can be approximately expressed as:  
\[ \begin{array}{cc}p(x,z;f)&\approx\frac{S(f)\sqrt{WF}}{4\pi R}\Delta V_{\text{ ws}}+W_{\text{ws}}V_{\text{sb}}W_{\text{sw}}e^{i2\Phi_{\text{SB}}}\Delta\cdot e^{i2\pi ft _{\text{SB}}}\cdot(e^{-i2\pi f\Delta t_{1}}+e^{i2\pi f\Delta t_{2}}-2)\\ &\hskip 14.226378 pt\approx\frac{S(f)\sqrt{WF}}{4\pi R}V_{\text{SB}}\cdot e^{i2 \pi ft_{\text{SB}}}\cdot(e^{-i2\pi f\Delta t_{1}}+e^{i2\pi f\Delta t_{2}}-2) \end{array}  \tag{14}\]  
where, \(V_{\text{SB}}\) is the ocean bottom reflection coefficient dominated by the grazing angle of SB, the interference structure of the shadow zone is determined by \(\phi_{\text{SB}}=k_{\text{s}}dsin\phi_{\text{sSB}}\), and \(\phi_{\text{rSB}}\) is the grazing angle of the sediment layer corresponding to the grazing angle of SB. In summary, when the grazing angles of SB and BS are consistent, the sound intensity received in the shadow zone can be approximately expressed as:  
\[|p(x,z;f)|^{2}\approx\frac{|S(f)|_{2}WF}{(4\pi)^{2}R^{2}}|V_{\text{SB}}|^{2}(e ^{-i2\pi f\Delta t_{1}}+e^{i2\pi f\Delta t_{2}}-2)^{2} \tag{15}\]  
For the approximation performed during the derivation, the numerical results are shown in Figure 4a-c, and the simulation parameters are consistent with Figure 2b. Figure 4a shows that the receiving grazing angles of SB and BS are approximately the same at different receiving ranges, indicating that the condition \(\varphi_{\text{rSB}}=\varphi_{\text{rBS}}\) can be met at this receiving and sending depth. Figure 4b shows that \(\Delta t_{1}\) and \(\Delta t_{2}\) decrease with the receiving range, and \(\Delta t_{2}\) is only slightly larger than \(\Delta t_{1}\), which can be further approximated to \(\Delta t_{1}\approx\Delta t_{2}\). In this case, Equation (15) is equivalent to Equation (1). In Figure 4c, \(2(\phi_{\text{SB}}-\phi_{\text{B}})\) and \(2(\phi_{\text{SBS}}-\phi_{\text{BS}})\) is much smaller than \(2\pi f\Delta t_{1}\) and \(2\pi f\Delta t_{2}\), indicating that \(\Delta\phi\) has little effect on the interference period, and the approximation of \(\Delta\phi\approx 0\) is reasonable. Figure 4d shows that the interference fringe calculated by SB (BS) grazing angle can describe the interference structure of the shadow zone in the range-frequency domain more accurately, while the interference fringe calculated by the B or SBS grazing angle cannot describe it. In terms of the energy, since the grazing angles of SB and BS are the same, the interference period of the energy after being reflected by the ocean bottom is the same, and the energy of B, SB, BS, and SBS are approximately equal, so that the total energy interference of multiple paths is determined by SB and BS. Since this paper focused on analyzing the effect of the interference structure brought by layered ocean bottom reflection and clarifying its dominant angle component, the deduced sound intensity Equation (15)  
and Equation (1) in the shadow zone are different in form.