## 2 Theory
### Theory of Directional Point Source Equivalence  
The elastic structure is placed in the free field, and the finite element multi-physics coupling method is used to calculate the near-field sound scattering field, as shown in Figure 1a. The coupling equation for finite element calculation can be expressed as:  
\[\left(\left[ \begin{array}{cc}K_{s}&K_{c}\\ 0&K_{a}\end{array} \right]+j\omega\left[ \begin{array}{cc}C_{s}&0\\ 0&C_{a}\end{array} \right]-\omega^{2}\left[ \begin{array}{cc}M_{s}&0\\ -\rho_{0}K_{c}^{T}&M_{a}\end{array} \right]\right)\cdot\left[ \begin{array}{c} u_{i}\\ p_{i}\end{array} \right]=\left[ \begin{array}{c}F_{st}\\ F_{at}\end{array} \right], \tag{1}\]  
where \(J\) is an imaginary unit; \(\omega\) is the angular frequency; \(K_{s}\), \(C_{s}\), and \(M_{s}\) represent the stiffness, damping, and mass matrices of the unconstrained part of the structural grid, respectively; \(K_{a}\), \(C_{a}\), and \(M_{a}\) represent the stiffness, damping, and mass matrices of the unconstrained part of the fluid grid, respectively; \(K_{c}\) represents the coupling stiffness matrix; \(F_{at}\) is the acoustic excitation under the acoustic system; \(F_{st}\) is the excitation on the structures; and, there is no incentive on the structure of this article, so \(F_{at}\) is 0. If the sound pressure of discrete points on the spherical surface (S) of the surrounding structure is \(\{p_{a}\}\), then it is the linear superposition of the incident and scattered sound fields. When no elastic structure exists, the same finite element calculation can be used to obtain the incident sound pressure of discrete point S on the spherical surface as \(\{p_{in}\}\). Subsequently, the sound scattering pressure on spherical surface S is \(\{p_{sc}\}=\{p_{a}\}-\{p_{in}\}\). According to the sound scattering pressure \(\{p_{sc}\}\) of spherical surface S, the sound field on the spherical surface can be expanded by spherical waves to calculate the sound pressure S2 of the far field. The sound scattering field in the free field satisfies the spherical coordinate system's Helmholtz equation. The formal expression of the scattered sound pressure can be obtained as (omit time factor \(e^{i\omega t}\))  
\[\psi(r_{0},\theta,\varphi)=\sum_{l=0}^{\infty}\sum_{n=0}^{l}\left[A_{nl}\cdot \cos(n\varphi)+B_{nl}\cdot\sin(n\varphi)\right]\cdot P_{l}^{n}(\cos\theta) \cdot h_{l}^{(2)}(kr_{0}), \tag{2}\]  
Figure 1: Schematic of equivalent theory of directional point source. (**a**) Part of FEM and spherical wave expansion; (**b**) Part of directional point source equivalence; and, (**c**) Part of ray theory. where \(P_{l}^{n}(\cos\theta)\) is the Legendre function, \(h_{l}^{(2)}(kr_{0})\) is the Hankel function of the second kind, and \(A_{nl},B_{nl}\) is the coefficient. The orthogonality of the Legendre function is applied to solve formula (2). \[ \begin{array}{c}\int\limits_{0}^{\pi}\psi(r_{0},\theta,\varphi)\cdot P_{k}^{n }(\cos\theta)\cdot\sin\theta d\theta=\\ \quad\quad\sum\limits_{l=0}^{\infty}\sum\limits_{n=0}^{l}[A_{nl}\cdot\cos(n \varphi)+B_{nl}\cdot\sin(n\varphi)]\cdot\int\limits_{0}^{\pi}P_{k}^{n}(\cos \theta)\cdot P_{l}^{n}(\cos\theta)\cdot\sin\theta d\theta\cdot h_{l}^{(2)}(kr_ {0})\end{array}  \tag{3}\]  
Using the orthogonality of the Legendre function, when \(k\
eq l\), the integral result is 0. When \(k=l\), the left side of Equation (3) is equal to \(a_{nl}\). \[a_{nl}=\sum\limits_{l=0}^{\infty}\sum\limits_{n=0}^{l}\left[A_{nl}\cdot\cos(n \varphi)+B_{nl}\cdot\sin(n\varphi)\right]\cdot\frac{2}{2l+1}\cdot\frac{(l+n)!} {(l-n)!}\cdot h_{l}^{(2)}(kr_{0}) \tag{4}\]  
In order to better transform formula (4), let the function of \(A_{nl}\) and \(B_{nl}\) be equal to \(b_{nl}\). \[b_{nl}=a_{nl}\cdot\frac{2l+1}{2}\cdot\frac{(l-n)!}{(l+n)!}=\sum\limits_{l=0}^{ \infty}\sum\limits_{n=0}^{l}\left[A_{nl}\cdot\cos(n\varphi)+B_{nl}\cdot\sin(n \varphi)\right]\cdot h_{l}^{(2)}(kr_{0}) \tag{5}\]  
where the orthogonality of the Legendre function can be expressed as  
\[\int_{0}^{\pi}P_{k}^{n}(\cos\theta)\cdot P_{l}^{n}(\cos\theta)\cdot\sin\theta d \theta=