There are some formulas about wind engineering (aerodynamic admittance function):

$$ a=\frac{\Gamma\left(\frac13\right)}{\sqrt{\pi}\Gamma\left(\frac56\right)}\approx 1.339 $$

$$ c=\frac{1}{a}\approx 0.7468 $$

von Kármán 风谱：

$$ S_u\left(k_1\right)=\frac{4\sigma_u^2 L_u}{\left[1+\left(2\pi a L_u k_1\right)^2\right]^{\frac56}}\Longrightarrow S_u\left(k_1,k_2\right)=\frac43 \pi \left(a\sigma_u L_u\right)^2 \frac{1+\left(2\pi a L_u k_1\right)^2+\frac{11}{3}\left(2\pi a L_u k_2\right)^2} {\left[1+\left(2\pi a L_u\right)^2\left(k_1^2 +k_2^2\right)\right]^{\frac73}} $$

$$ S_w\left(k_1\right)=\frac{4\sigma_w^2 L_w\left[1+\frac{32}{3}\left(2\pi a L_wk_1\right)^2\right]}{\left[1+\left(4\pi a L_w k_1\right)^2\right]^{\frac{11}{6}}} \Longrightarrow S_w\left(k_1,k_2\right)=\frac{\frac{128}{9}\pi^3\left(2aL_w\right)^4\left(k_1^2+k_2^2\right)}{\left[1+\left(4\pi a L_w\right)^2\left(k_1^2+k_2^2\right)\right]^{\frac73}} $$

Bullen相关函数模型：

$$ B\left(\Delta y\right)=\frac{\left(\frac{\Delta y}{m}\right)^n}{2^n \Gamma\left(n\right)}\left[2K_n\left(\frac{\Delta y}{m}\right)-\frac{\Delta y}{m}K_{n-1}\left(\frac{\Delta y}{m}\right)\right] $$

其中\(\Delta y\)为展向间距，\(m\)和\(n\)为控制形状和尺寸的参数，\(K\)为第二类修正Bessel函数，Matlab：Besselk()。

Jakobsen相干函数模型：

$$ Coh_J\left(k_1,\Delta y\right)=\mathrm{exp}\left(-2\pi A_J \Delta y\right) $$

$$ A_J=\frac{\left[\sqrt{c_2^2+\left(c_3 k_1 L\right)^2}\right]^{c_1}}{2\pi L} $$

其两波数形式为：

$$ \Phi_J\left(k_1,k_2\right)=\frac{1}{\pi}\frac{A_J}{A_J^2+k_2^2} $$

双指数模型：

$$ Coh_{exp}\left(k_1,\Delta y\right)=\frac{\lambda \mathrm{exp}\left(-2\pi A_J \Delta y\right)-A_J \mathrm{exp}\left(-2\pi \lambda \Delta y\right)}{\lambda-A_J} $$

$$ \lambda=\frac{\sqrt{a_1\left(2\pi b k_1\right)^{a_2}+a_3}}{2\pi b} $$

两波数形式：

$$ \Phi_{exp}\left(k_1,k_2\right)=\frac{\lambda A_J}{\pi}\frac{\lambda+A_J}{\left(\lambda^2+k_2^2\right)\left(A_J^2+k_2^2\right)} $$

展向修正项形式：

$$ F\left(k_1\right)=\frac{\lambda}{\lambda+A_J} $$

三参数Kimura模型：

$$ Coh_K\left(k_1,\Delta y\right)=\frac{2^{\frac16}}{\Gamma\left(\frac56\right)}\left[A_K^{\frac56}K_{\frac56}\left(A_K\right)- \frac{A_K^{\frac{11}{6}}K_{\frac16}\left(A_K\right)}{B_K}\right] $$

$$ A_K=\frac{c \Delta y}{\gamma L}\sqrt{1+\left[\left(\frac{2\pi}{c}\right)\left(\alpha L k_1\right)\right]^{\beta}} $$

$$ B_K=1+\frac83 \left[\left(\frac{2\pi}{c}\right)\left(\alpha L k_1\right)\right]^{\beta} $$

两波数形式为：

$$ \Phi_K\left(k_1,k_2\right)=\frac{2\Gamma\left(\frac43\right)}{\Gamma\left(\frac56\right)}\sqrt{\pi}A_K^{\frac53} \left\{\frac{2\pi k_2^2 \left(1+\frac{5}{3B_K}\right)+A_K^2 \left(1-\frac{1}{B_K}\right)}{\left[\left(2\pi k_2\right)^2+A_K^2\right]^{\frac73}}\right\} $$

展向修正项形式：

$$ F\left(k_1\right)=\frac{\Gamma\left(\frac56\right)}{2\pi^{\frac32}\Gamma\left(\frac43\right)}\frac{A_K}{A_J\left(1-\frac{1}{B_K}\right)} $$