# A Green's Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method
## 3 Theoretical Model Verification  
A typical sound propagation model in the Pekeris waveguide is established to verify the Green functions described by the ISM, as shown in Figure 3, which consists of a point source that is 30 m in depth, where the upper surface is the pressure released, the depth of homogeneous water is 100 m, and the density and the sound speed are \(\rho_{a}\) = 1024 kg/m\({}^{3}\) and \(c_{a}\) = 1500 m/s, respectively. The seabed in the Pekeris waveguide is regarded as an infinite half-space liquid with the density and the sound speed of \(\rho_{b}\) = 2000 kg/m3 and \(c_{b}\) = 1800 m/s, respectively, and the sound absorption in the seabed is also considered. The depth of each receiver is \(z_{i}\) and the horizontal interval is 1 m.  
From the configuration of the Pekeris waveguide shown in Figure 3, it is clear that the upper boundary is the Dirichlet boundary (i.e., the reflection coefficient is \(V_{1}=-1\)). The seabed is an infinite liquid half space, and the interaction between the sound field and the fluid-fluid interface can be described by using the reflection coefficient. As shown in Figure 4, Equation (19) is utilized to calculate the reflection coefficient of the plane wave at the interface versus the incident angle for different absorption coefficients of the seabed. It can be observed that the reflection coefficient is positively related to the incident angle \(\theta\). When the liquid seabed does not contain sound absorption, there is an incident angle (i.e., the critical angle) so that the sound field on the interface appears to be a total reflection, the reflection coefficient is constant at 1, and the critical angle is \(\theta=\arcsin(\text{c}_{\text{a}}/\text{c}_{\text{b}})\approx 56^{\circ}\). In contrast, when the medium contains acoustic absorption with \(\alpha_{\lambda}=0.5\) dB/ \(\lambda\), \(\alpha_{\lambda}=1.0\) dB/ \(\lambda\), and \(\alpha_{\lambda}=2.0\) dB/ \(\lambda\), the reflection coefficient is a value that varies with the angle of incidence, and total reflection no longer occurs. The reflection coefficients are approximately the same for angles less than the critical angle (i.e., \(56^{\circ}\)). For angles above the critical angle, the reflection coefficients corresponding to different absorptions vary considerably. Therefore, the spherical waves carry plane waves in different directions and incident angles on the fluid-fluid interface with the sound absorption. The reflection coefficients of the plane waves are different in each direction. Although in the range of 0-10\({}^{\circ}\) the reflection coefficient can all be approximated as 0.41 (i.e., it is reasonable that the reflected sound field at the receiver in the range of \(r\leq 0.18(z_{s}+z)\) can be calculated using Equation (1)), the distance of the receiver is too close to be used to accurately describe the reflected sound field in the Pekeris waveguide. Therefore, to make the ISM more accurate for sound field calculations over the whole distance, it is necessary to decompose the spherical wave into plane waves with different incident angles. The  
Figure 4: Curves of plane wave reflection coefficients versus angle of incidence for different absorptions. Figure 3: Schematic diagram of the configuration of the Pekeris waveguide. reflected sound field of the spherical wave from the point source is solved according to Equation (18), and finally, the reflected sound field of a physical source and image sources are summed by Equation (1). Thus, the sound field of a point source (i.e., the GF) in the Pekeris waveguide at any field point is obtained by this rigorous ISM.