# A Green's Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method
## 1 Introduction  
The Green's function (GF) is the fundamental element in the acoustic calculation, which describes the sound field generated by a point source under certain boundary conditions or initial conditions, that is, the acoustic transfer relationship between the vibration source and the receiver is established. Therefore, the GF is widely used in the acoustic computational method, such as the Helmholtz integral method, which is generally a concise formulation of the partial differential equation for acoustic problems by using the Gaussian theory and automatically meeting the boundary conditions, the radiated sound field in different fluid environments (infinity field, half space, closed space, etc.) can be investigated efficiently by adjusting the GF of the Helmholtz integral method [1, 2, 3]. Additionally, the GF provides us with a lot of convenience of an acoustic problem where we can not only inverse the vibration characteristics of the source through the identified acoustic characteristics but can also easily calculate the pressure level at any point according to the configured radiator in the corresponding fluid [4]. Due to the GF having these advantages in acoustic computation, it is very significant for acoustic holography [5, 6, 7], beamforming [8], acoustic radiation and scattering [9, 10, 11], acoustic measurement [12], etc. By introducing the GF into the Helmholtz integral equation, acoustic calculations become simpler and more flexible. In the equation, boundary conditions on the surface of the source, such as the sound pressure (Dirichlet condition), the normal velocity (Neumann condition), and the boundary normal conductance (Robin condition), can be selectively given in theintegral equation to match the elastic radiator for the acoustic radiation calculation [10], and a suitable GF can be chosen to establish the transfer relationship between the source on the structure surface and the sound field in the fluid environment. Thus, we can solve the radiated sound field or invert the vibration characteristics of the radiator via the scheme of combining the Helmholtz integral equation and the GF. Subsequently, computational methods such as boundary elements [13], equivalent sources [14], and hydroelasticity theory [15] have been extended to promote further developments in the fields of acoustic cloaking design, near-field acoustic field measurements, and target identification for underwater vehicles. However, when the integral method is utilized for the structural radiation in shallow water, the universal acoustic problems of the near and far field are inevitable. Consequently, the GF must be adjusted and adapt to such problems [16]. Different from the GF in infinity field or half space, the solution of the GF in shallow water involves different mathematical treatment and the corresponding applicable conditions. Physically, the GF in shallow water is the solution to the inhomogeneous Helmholtz equation for a point source, which satisfies the sound conditions of the upper and lower boundaries, which leads to the fact that the difficulty in deriving the GF in shallow water lies in establishing the interaction between the sound field and the boundaries, especially for seabeds with different types. Currently, the most prominent models for solving the GF in shallow water are the ray method [17], the normal mode (NM) method [18], the wavenumber integration (WI) method [19], and the parabolic equation (PE) method [20]. The theoretical foundation of all sound propagation models is the acoustic wave equation; it is only that the mathematical processing strategies used for nonhomogeneous wave equations are different, in addition to the form of the solution [21]. Therefore, each model has the applicable conditions that make it suitable for a practical problem.