# A Green's Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method
## 3 Theoretical Model Verification
The cutoff effect will occur in the sound propagation due to the limitation of the upper and lower boundaries of the shallow water (i.e., the sound field below the cutoff frequency cannot propagate in long distances), and the sound propagation frequencies of the order normal modes in the Pekeris waveguide can be above the cutoff frequency \(f_{n}\) given by [22]  
\[f_{n}=\frac{(2n-1)c_{a}c_{b}}{4H\sqrt{c_{b}^{2}-c_{a}^{2}}}, \tag{23}\]  
The propagation frequencies corresponding to the first 6-order normal modes are presented in Table 1. When the propagation frequency is lower than the first-order propagation frequency (e.g., \(f_{1}=6.78\) Hz) the sound field will decay rapidly with the propagation distance and cannot travel far away. To fully compare the calculation accuracy of the ISM for the sound field in the near and far fields, the selected frequencies are 25 Hz, 50 Hz, 100 Hz, and 200 Hz, and the \(\alpha_{\lambda}=0.5\) dB/\(\lambda\). As shown in Figure 5, the SPLs of the point source in this waveguide are calculated by adopting the rigorous ISM, and these results are compared with the theoretical solutions calculated accurately by the mature Scooter calculation program based on WI techniques; the depth of each receiver is half of the water depth (i.e., \(z_{i}=50\) m). It can be seen that the results between the ISM and theoretical solutions are in good agreement in the near and far fields. The ISM considers the upper and lower boundaries through the image sources generated by mirror reflection, and the total sound field is expressed by the physical source and the various image sources. Further, on the fluid-fluid interaction surface, the spherical wave is decomposed into plane waves with various orders through Equation (18), and then the identified reflection coefficient of the plane wave is used to describe the reflection effect on the sound field. Thus, the ISM is an exact solution and simpler than WI, which can handle the near-to-far field problem well. As shown in Figure 6, where the SPL in the xor plane is provided as a comparison value for the results calculated by the presented method and WI, the frequencies are chosen to be 25 Hz and 100 Hz. The modal interference patterns of the sound field calculated by the two methods are identical over the entire calculated distance. For example, when the frequency is 25 Hz, which is more than the cutoff frequency, there are two modes, as shown in Table 1. Thus, two normal modes appear in the depth direction of the radiated sound field in Figure 6a,b. Additionally, there are more orders of modal interference patterns when the frequency increases to 200 Hz, as shown in Figure 6c,d. \begin{table}
\begin{tabular}{c c c c c c c} \hline \(n\) & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(f_{n}/\mathrm{Hz}\) & 6.78 & 20.35 & 33.92 & 47.49 & 61.06 & 74.62 \\ \hline \end{tabular}
\end{table} 
Table 1: The propagation frequencies of first 6-order normal modes. Sound propagation in the Pekeris waveguide is generally concerned with the far field, where the sound field will interact with the seabed tens to thousands of times, and the sound losses of multiple sound paths will sum to significant levels, so the effect of seabed absorption on sound field propagation must be taken into account. As shown in Figure 7, the SPL is calculated using this method for different cases with three absorption coefficients, where all receivers are 50 m and the selected frequency is 20 Hz. It can be seen that at a short distance, for example, in the 0-700 m range, the variation in SPL for different absorption coefficients is minor. Because there are a small number of interactions between the sound field and the lossy seabed, there is not a significant effect on the whole sound propagation, which is dominated by geometrical spreading loss. However, the sound  
Figure 5: Sound field comparison between near- to far-field model and theoretical solution: (**a**) \(f\) = 25 Hz; (**b**) \(f\) = 50 Hz; (**c**) \(f\) = 100 Hz; (**d**) \(f\) = 200 Hz.