# A Green's Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method
## 2 Theoretical Model of the Rigorous ISM
Different from the processing method in ideal shallow water conditions with a rigid seabed (i.e., the \(V_{2}\) is always 1), the reflection coefficient \(V_{2}\) in the Pekeris waveguide depends on the incident angle and incident waveform (i.e., plane wave), and it is more similar to the reflection coefficient of the plane wave in the theoretical calculation. However, the sound field in Equation (1) is the sound superposition of spherical waves reflected by physical source and the image sources--it is essential to make the waveform and reflection coefficient consistent. In this rigorous ISM, the solution to this problem requires the decomposition of spherical waves into plane waves with various angles. In the plane at \(z=0\), the sound field of a spherical wave can be expanded into a double Fourier integral with \(x\) and \(y\)[24], as follows:  
\[\frac{e^{ikr}}{r}=\int\limits_{-}^{+}\int\limits_{\infty}^{+}A(k_{x},k_{y})e^{ i(k_{x}x+k_{y}y)}dk_{x}dk_{y}, \tag{2}\]  
where \(r\) is the distance vector from the point source to the receiver, \(i\) is the imaginary part of the complex number, and the integral variables \(k_{x}\) and \(k_{y}\) are the vector projections of the wavenumber \(k\) in the direction of the \(x\) and \(y\) axes, respectively, and the spectral density of the wave, which is obtained by the inverse Fourier transform, as follows:  
\[A(k_{x},k_{y})=\frac{1}{(2\pi)^{2}}\int\limits_{-}^{+}\int\limits_{\infty}^{ \infty}\frac{e^{ikr}}{r}e^{-i(k_{x}x+k_{y}y)}dk_{x}dk_{y}, \tag{3}\]  
To obtain the value of \(A(k_{x},k_{y})\), we now establish the transformation between polar and Cartesian coordinates such that  
\[x=r\cos\psi,\ y=r\sin\psi,dxdy=rdrd\psi, \tag{4}\]  
\[k_{x}=l\cos\varphi,k_{y}=l\sin\varphi,l=\sqrt{k_{x}^{2}+k_{y}^{2}}, \tag{5}\]  
where \(l\) is the projection of the wave vector \(k\) on the xoy plane. By substituting Equation (4) and Equation (5) into Equation (3), we find  
\[A(k_{x},k_{y})=\frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}d\psi\int_{0}^{\infty}e^{ir[k-l \cos(\varphi-\psi)]}dr \tag{6}\]  
The integral over \(r\) is elementary. Moreover, if we assume that there is some absorption in the medium, no matter how small (i.e., that \(k\) has a positive imaginary part), then the substitution of the upper limit yields zero, and finally, \(A(k_{x},k_{y})\) is solved by integration operations and can be expressed as follows:  
\[ \begin{array}{c}A(k_{x},k_{y})=\frac{i}{(2\pi)^{2}}\int_{0}^{2\pi}\frac{1}{k- l\cos(\varphi-\varphi)}d\psi\\ =\frac{i}{2\pi}\frac{1}{\sqrt{k^{2}-l^{2}}}\\ =\frac{i}{2\pi}\frac{1}{\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}}},\end{array}  \tag{7}\]  
Thus  
\[\frac{e^{ikr}}{r}=\frac{i}{2\pi}\int\limits_{-\infty}^{+\infty}\frac{e^{i(k_{ x}x+k_{y}y)}}{\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}}}dk_{x}dk_{y}, \tag{8}\]  
Here, we extend the sound field describing the xoy plane to the entire three-dimensional (3D) space and with a general focus on the sound field in water (i.e., \(z\geq 0\)). As a result, Equation (8) can also be written in the following form:  
\[\frac{e^{ikR}}{R}=\frac{i}{2\pi}\int\limits_{-\infty}^{+\infty}\frac{e^{i(k_{ x}x+k_{y}y+k_{z}z)}}{k_{z}}dk_{x}dk_{y}, \tag{9}\]  
As shown in Figure 2a, we perform the wavenumber domain conversion in this 3D space and the components \(k_{x}\) and \(k_{y}\) of the wave vector \(k\) can be replaced by integration over the angles \(\vartheta\) and \(\varphi\) as follows:  
\[k_{x}=k\sin\vartheta\cos\varphi,k_{y}=k\sin\vartheta\sin\varphi,k_{z}=k\cos \vartheta=\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}}, \tag{10}\]  
The angle range of \(\varphi\) is (0, 2\(\pi\)). According to Equation (10), when \(k_{x}=k_{y}=0\), \(k_{z}=k\) is obtained. Additionally, \(\mathrm{k_{z}=io}\) when \(\mathrm{k_{y}\to\pm\infty}\), so that the angle range of \(\vartheta\)  
Figure 2: Diagram of the interaction between plane wave and liquid–liquid interface: (**a**) decomposition of the wavenumber vector \(k\) in three-dimensional (3D) space; (**b**) sound reflection of plane waves on the liquid–liquid interface. can be expressed as (0, \(\pi/2-\text{i}\infty\)).