# A Green's Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method
## 2 Theoretical Model of the Rigorous ISM
With the use of the variable transformation formula \(\frac{dk_{x}dk_{y}}{k_{z}}=ksin\partial d\theta d\varphi\), Equation (9) can also be written in the following form:  
\[\frac{e^{ikR}}{R}=\frac{ik}{2\pi}\int_{0}^{(\pi/2)-i\infty}\int_{0}^{2\pi}e^{i( k_{x}x+k_{y}y+k_{z}z)}\sin\theta d\theta d\varphi, \tag{11}\]  
Up to this point, we know that different orders of plane waves are decomposed from the spherical wave and reflecting at the fluid-fluid interface with different incident angles. Each order of plane wave on the interface satisfies the Snell law for reflection and should be multiplied by the corresponding reflection coefficient \(\text{V}(\vartheta)\). Assuming \(m=0\), as shown in Figure 2b, for the first time the spherical wave of the sound source interacts with the interface, the physical source \(O_{01}\) is mirrored by the interface to the first image source \(O_{02}\). In this case, the path of the sound field of the source \(O_{02}\) is \(R_{02}=\sqrt{r^{2}+z_{02}^{2}}\). Based on Equations (10) and (11), the sound field of the spherical wave radiated by the image source \(O_{02}\) on the fluid-fluid interface can be found from the following equation  
\[\text{V}_{2}(\vartheta)\frac{e^{ikR_{02}}}{R_{02}}=\frac{ik}{2\pi}\int_{0}^{( \pi/2)-i\infty}\int_{0}^{2\pi}e^{ik(x\sin\vartheta\cos\varphi+y\sin\vartheta \sin\varphi+z_{02}\cos\vartheta)}\text{V}_{2}(\vartheta)\sin\theta d\vartheta d\varphi, \tag{12}\]  
On the xoy plane, let \(x=rcos\varphi_{1},y=rsin\varphi_{1}\), which results in the following:  
\[\int_{0}^{2\pi}e^{ik\sin(rcos\varphi+y\sin\vartheta)}d\varphi = \int_{0}^{2\pi}e^{ikrsin\varphi(\cos\varphi+\sin\varphi)}d\varphi \tag{13}\] \[= \int_{0}^{2\pi}e^{ikrsin\varphi(\cos\varphi-\vartheta)}d\varphi,\]  
According to the definition of the \(n\)th order Bessel function  
\[J_{n}(x)=\frac{1}{2\pi i^{n}}\int_{0}^{2\pi}e^{ix\cos\theta}\cos(n\theta)d\theta, \tag{14}\]  
Here, we take \(n=0\), and the above equation can be rewritten as follows:  
\[\int_{0}^{2\pi}e^{ik\sin\varphi(x\cos\varphi+y\sin\vartheta)}d\varphi=2\pi J_ {0}(kr\sin\varphi), \tag{15}\]  
According to Equation (15), Equation (12) can be expressed as follows:  
\[\frac{e^{ikR}}{R}=\frac{i}{2\pi}\int\limits_{-\infty}^{+\infty}\frac{e^{i(k_{ x}x+k_{y}y+k_{z}z)}}{k_{z}}dk_{x}dk_{y}, \tag{9}\]0  
Similarly, we can obtain the reflected sound field of other image sources. The expression is also conveniently transformed by changing the limits of integration and replacing the Bessel function by Hankel functions. For this, we note that  
\[\frac{e^{ikR}}{R}=\frac{i}{2\pi}\int\limits_{-\infty}^{+\infty}\frac{e^{i(k_{ x}x+k_{y}y+k_{z}z)}}{k_{z}}dk_{x}dk_{y}, \tag{9}\]1 \[\frac{e^{ikR}}{R}=\frac{i}{2\pi}\int\limits_{-\infty}^{+\infty}\frac{e^{i(k_{ x}x+k_{y}y+k_{z}z)}}{k_{z}}dk_{x}dk_{y}, \tag{9}\]2  
We apply the relation to each image source. The spherical waves are decomposed into plane waves with various orders by using the double Fourier integration as follows:  
\[\frac{e^{ikR}}{R}=\frac{i}{2\pi}\int\limits_{-\infty}^{+\infty}\frac{e^{i(k_{ x}x+k_{y}y+k_{z}z)}}{k_{z}}dk_{x}dk_{y}, \tag{9}\]3The reflected sound field of the spherical wave is decomposed into the plane wave by Equation (18). We consider the reflection at an interface separating two homogeneous fluid media with the reflection coefficients, which can be written in the form:  
\[V_{{}_{2}}(\vec{\varphi})= \begin{cases}\dfrac{\ell\cos\vartheta-\sqrt{\beta^{2}- \sin^{2}\vartheta}}{\ell\cos\vartheta+\sqrt{\beta^{2}-\sin^{2}\vartheta}},&| \sin\vartheta|\leq\beta\\ 1&|\sin\vartheta|>\beta\end{cases} , \tag{19}\]  
where \(\ddagger=\rho_{b}/\rho_{a}\), \(\beta=c_{a}/c_{b}\), \(\vartheta\) represent the incident angle of the plane wave. The seabed surface of shallow water is covered with a layer of non-condensable solid material, which is deposited by several rivers, so it is reasonable to simulate the seabed sediment as a liquid. Further, the attenuation also becomes a significant loss factor for waterborne energy.