## 2 Anisotropic Maritime Atmospheric Spectrum
### Radial Component of the WSF  
Now, consider the radial component of the WSF of the Gaussian-beam waves. Here, one cannot use the same method used for the longitudinal component, because it can cause big errors. Therefore, we just substitute Equation (5) into Equation (10) and expand \(I_{0}(\cdot)\) as Maclaurin series representation and the radial component of the WSF can be expressed, as follows:\[ \begin{split}\sigma_{r}^{2}(\rho,L)=& 8\pi^{2}k^{2}LA(\alpha)\tilde{C}_{n}^{2}\xi^{2- \alpha}\sum_{n=1}^{\infty}\frac{(\frac{\Lambda\rho}{2})^{2n}}{n!(1)_{n}}\int_{ 0}^{1}\xi^{2n}d\xi\int_{0}^{\infty}\kappa^{2n+1}\exp(-\frac{\Lambda L\kappa^{2 }\xi^{2}}{k})\\ &\times(\kappa^{2}+\tilde{\kappa}^{2})^{-\frac{\alpha}{2}}[1+a_{1 }\frac{\kappa}{\kappa_{H}}+a_{2}(\frac{\kappa}{\kappa_{H}})^{3-\frac{\alpha}{2 }}]\exp(-\frac{\kappa^{2}}{\tilde{\kappa}_{H}^{2}})d\kappa,\end{split}  \tag{16}\]  
By using Equation (12), we can obtain the radial component of the WSF for Gaussian-beam wave, and it is written, as follows:  
\[ \begin{split}\sigma_{r}^{2}(\rho,L)=& 4\pi^{2}k^{2}LA(\alpha)\tilde{C}_{n}^{2}\xi^{2- \alpha}\sum_{n=1}^{\infty}\frac{(\frac{\Lambda\rho}{2})^{2n}}{n!(1)_{n}}\kappa _{0}^{2n+\tilde{\tau}2-\alpha}\int_{0}^{1}\xi^{2n}d\xi\\ &\times[\Gamma(n+1)U(n+1,n+2-\frac{\alpha}{2},\frac{\tilde{\kappa }_{0}^{2}}{\tilde{\kappa}_{H}^{2}}(1+\Lambda T_{H}\xi^{2}))\\ &+b_{1}[\Gamma(n+\frac{3}{2})U(n+\frac{3}{2},n+\frac{5}{2}-\frac{ \alpha}{2},\frac{\tilde{\kappa}_{0}^{2}}{\tilde{\kappa}_{H}^{2}}(1+\Lambda T _{H}\xi^{2}))\\ &+b_{2}[\Gamma(n+\frac{5}{2}-\frac{\alpha}{4})U(n+\frac{5}{2}- \frac{\alpha}{4},n+\frac{7}{2}-\frac{3\alpha}{4},\frac{\tilde{\kappa}_{0}^{2} }{\tilde{\kappa}_{H}^{2}}(1+\Lambda T_{H}\xi^{2}))],\end{split}  \tag{17}\]  
where \(T_{H}=L\tilde{\kappa}_{H}^{2}/k\). By using Equation (13), the final and new expression of the radial component of the WSF can be divided into two parts and the first part \(\sigma_{r1}^{2}\) is rewritten, as follows:  
\[\begin{split}\sigma_{r1}^{2}(\rho,L)=& 4\pi^{2}k^{2}LA( \alpha)\tilde{C}_{n}^{2}\xi^{2-\alpha}\sum_{n=1}^{\infty}\frac{(\frac{\Lambda \rho}{2})^{2n}}{n!(2n+1)}\kappa_{0}^{2n+\tilde{\tau}2-\alpha}\\ &\times[\frac{\Gamma(\frac{\alpha}{2}-n-1)}{\Gamma(\frac{\alpha }{2})}+b_{1}\frac{\Gamma(n+\frac{3}{2})}{\Gamma(n+1)}\frac{\Gamma(\frac{ \alpha}{2}-n-\frac{3}{2})}{\Gamma(\frac{\alpha}{\[{{}_{2}F_{1}}(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{1}t^{b-1}(1 -t)^{c-b-1}(1-zt)^{-a}dt,\quad c>b>0, \tag{21}\]  
Additionally, the second part of the radial component can be rewritten, as follows:  
\[ \begin{split}\sigma_{r2}^{2}(\rho,L)=& 4\pi^{2}k^{2}LA(\alpha)\tilde{C}_{n}^{2}\zeta^{2- \alpha}\frac{\Lambda^{2}\rho^{2}\kappa_{H}^{\tilde{\zeta}-\alpha}}{12}\\ &\times[\Gamma(2-\frac{\alpha}{2})M_{1}(\alpha)+a_{1}\Gamma( \frac{5-\alpha}{2})M_{2}(\alpha)+a_{2}\Gamma(\frac{14-3\alpha}{4})M_{3}(\alpha )],\end{split}  \tag{22}\]  
\[ \begin{split} M_{1}(\alpha)=&{{}_{2}F_{1}}(2-\frac{ \alpha}{2},\frac{3}{2};\frac{5}{2};-\Lambda T_{H}),\\ M_{2}(\alpha)=&{{}_{2}F_{1}}(\frac{5}{2}-\frac{ \alpha}{2},\frac{3}{2};\frac{5}{2};-\Lambda T_{H}),\\ M_{3}(\alpha)=&{{}_{2}F_{1}}(\frac{7}{2}-\frac{3 \alpha}{4},\frac{3}{2};\frac{5}{2};-\Lambda T_{H}).\end{split}