## 4 Typhoon-Induced Wind and Wave Loading
### Typhoon-Induced Wave Loading  
The wave-induced hydrodynamic load on the monopile structure requires explicit modeling of extreme sea states under typhoon conditions. Following the stochastic wave modeling framework, the JONSWAP spectrum (\(S_{I}(f)\)) is employed to characterize the irregular wave field. Considering the directional distribution of wave energy, the directional spectral function can be expressed as Equation (19). \[S(f,\theta)=S_{I}(f)\cdot D_{I}(\theta) \tag{19}\]  
where \(D_{I}(\theta)\) is the directional distribution function adopting a cosine exponential model. \(S(f,\theta)\) is a wave energy distribution model that simultaneously considers frequency and direction. \(D_{I}(\theta)\) and \(S_{I}(f)\) can be expressed by Equations (20) and (21), respectively. \[D_{I}(\theta)=\frac{\Gamma(s+1)}{2\sqrt{\pi}\Gamma(s+1/2)}cos^{2s}\left(\frac {\theta_{w}-\theta_{0}}{2}\right) \tag{20}\]  
where \(\theta_{0}\) is the principal wave direction (taken as 150\({}^{\circ}\)) based on on-site observation data during typhoons); \(s\) is the directional concentration factor (taken as \(s=8\)) based on measured data from the Taizhou Sea Area); \(\theta_{w}\) is the wave propagation direction; \(\Gamma(\cdot)\) is the gamma function. Therefore, the wave angle is expressed as \(\alpha=|\theta_{wind}-\theta_{w}|\), where \(\theta_{wind}\) denotes the wind direction. \[S_{I}(f)=\frac{\alpha g^{2}}{\left(2\pi\right)^{4}}f^{-5}\text{exp}\left(- \frac{5}{4}\left(\frac{f}{f_{p}}\right)^{-4}\right)\gamma^{\text{exp}\left(- 0.5\left(\frac{f-f_{p}}{\sigma f_{p}}\right)^{2}\right)} \tag{21}\]  
where f is the wave frequency, \(f=1/T\), \(T\) is the wave period; \(g\) is the acceleration of gravity; The generalized Phillips' constant \(\alpha\) in the JONSWAP spectrum is formulated as: \(\alpha=5\cdot\left(H_{s}^{2}f_{p}^{4}/g^{2}\right)\cdot\left(1-0.287 ln\gamma \right)\cdot\pi 4\), with key parameters defined as follows: the spectral peak frequency (\(f_{p}\)); the wave peak period (\(T_{p}\)) is 6.5 s; The peak enhancement factor (\(\gamma\)) is determined as follows:  
\[\gamma= \begin{cases}5,&\frac{T_{p}}{\sqrt{H_{s}}}\leq 3.6\\ exp\left(5.75-1.15\frac{T_{p}}{\sqrt{H_{s}}}\right),&3.6<\frac{T_{p}}{\sqrt{H _{s}}}\leq 5\\ 1,&\frac{T_{p}}{\sqrt{H_{s}}}>5\end{cases}  \tag{22}\]  
Significant wave height \(H_{s}=4\) m (100-year return period). The spectral energy density \(S_{I}(f_{i})\) enables calculation of representative wave component amplitudes through: \(H_{i}=2\sqrt{2S_{I}(f_{i})\Delta f}\) Where \(\Delta f\) is the frequency discretization interval. Subsequently, linear wave theory [35] provides the horizontal water particle kinematics:  
\[u_{x}=\sum_{i=1}^{n}\frac{H_{i}w_{i}}{2}\frac{coshk_{i}z_{w}}{sinhk_{i}z_{d}} \text{cos}(k_{i}x-w_{i}t+\varphi_{i}) \tag{23}\]\[\frac{\partial u_{x}}{\partial t}=\sum_{i=1}^{n}\frac{H_{i}w_{i}^{2}}{2}\frac{oshk_ {i}z_{w}}{sinhk_{i}z_{d}}\sin(k_{i}x-w_{i}t+\varphi_{i}) \tag{24}\]  
Wave kinematics adhere to the dispersion relation \(\omega^{2}=gktanh\ (kd)\), where \(k_{i}\) and \(w_{i}\), respectively, characterize the wave number and angular frequency of the i-th component. Key parameters are defined as: water depth \(d\) (seabed--MSL): vertical coordinate \(z_{w}\) (origin at the seabed), and stochastic phase angle \(\varphi_{i}\) (uniformly distributed in \([0,2\pi]\), Monte Carlo-sampled). The spectral peak frequency produces a dominant wavelength (63 m) surpassing five times the monopile diameter, fulfilling the [PERSON] regime criterion [36]. Thus, [PERSON]'s equation is utilized for wave load computation.