# A Green's Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method
## 2 Theoretical Model of the Rigorous ISM  
The theoretical foundation of the classical ISM for deriving the GF in the Pekeris waveguide is ray acoustics, where the sound field at any field point can be expressed as a superposition summation of the direct wave radiated by the sound source and the sound field reflected by the fluid-fluid interface [22]. As shown in Figure 1a, the point source emits sound waves represented by the radiation rays, and then each ray follows the [PERSON] theorem of reflection and transmission on the upper and lower interfaces (i.e., each ray is multiplied by the reflection coefficient of the corresponding incident angle, and the reflected sound field can be obtained at the certain field point, where the effects of the upper and lower boundaries on the physical source are considered by the image sources). However, the traditional ISM is generally applied when the distance between the source and the receiver to the field is much larger than the wavelength (or far field), and the reflection problem of spherical waves on the interface can be approximated by the ISM. However, when the distance is comparable to the wavelength (i.e., the near field), the results of the traditional ISM have some deviations from the practical sound field, and then it is necessary to strictly observe the wave acoustics method to establish the interaction mechanism of the near-field spherical wave and the interface. As shown in Figure 1b, when the spherical wave radiated from the point source interacts with the liquid-liquid interface, the spherical wave needs to be decomposed into plane waves in different directions, and then the reflection theory of plane waves on the interface is used to obtain the reflected sound field of plane waves. Finally, these reflected sound fields are superimposed to derive the reflected sound field of the spherical wave on the plane liquid-liquid interface; these describe the rigorous ISM. Figure 1: Schematic diagram of the Green’s function (GF) in shallow water: (**a**) the principle of the rigorous image source method (ISM); (**b**) sound reflection of spherical waves on the liquid–liquid interface. The traditional ISM of the propagation model is based on the ray theory, and the upper and lower boundaries in the homogeneous waveguide are considered by the image sources. Figure 1a shows a schematic diagram of the contributions from the physical source and the first three image sources; the remaining terms are obtained by successive imaging of these sources to yield the ray expansion for the total field [23]  
\[ \begin{array}{c}G(r,z)=(V_{1}V_{2}(\theta_{i}))^{m}\sum\limits_{m=0}^{\infty} \left[\frac{e^{ik_{x}R_{m1}}}{R_{m1}}+V_{2}(\theta_{i})\frac{e^{ik_{x}R_{m2}}}{ R_{m2}}+\right.\\ \left.V_{1}\frac{e^{ik_{x}R_{m3}}}{R_{m3}}+V_{1}V_{2}(\theta_{i})\frac{e^{ik_{x} R_{m4}}}{R_{m4}}\right],\end{array}  \tag{1}\]  
where \(V_{1}\) and \(V_{2}\) are the reflection coefficients of the plane wave for upper and lower boundaries, respectively. It should be noted that \(V_{1}=-1\) for the pressure-release surface, however, the bottom of the Pekeris waveguide is not an ideal rigid surface; the reflection coefficient \(V_{2}\) does not remain constant but relates to the angle \(\theta_{i}\), which is the incident angle of each term plane wave decomposed by the spherical wave on the seabed surface. \(k_{a}\) is the sound wavenumber in the water, \(R_{mn}=\sqrt{r^{2}+z_{mn}^{2}}\) and \(m\) is the number of mirrored sources, and the value is \(m=0,1,2,3\dots,\infty\), \(n=1,2,3,4\), \(z_{m1}=2 Hm+z-z_{s}\), \(z_{m2}=2 Hm+z+z_{s}\), \(z_{m3}=2H(m+1)-z-z_{s}\), \(z_{m4}=2H(m+1)-z+z_{s}\). Generally, when \(m\) reaches tens of terms, the total sound field will quickly stabilize at a certain value. Equation (1) is fulfilled by the Helmholtz wave equation because it is composed of multiple spherical waves, each of which satisfies the equation.