A method to compute the second-order free surface elevation, depth integrated force and mud line moment for a slender circular vertical cylinder is presented. The method is valid for unidirectional irregular waves and includes inertia loads and viscous loads.

We first derive the linear transfer functions for free surface elevation, depth-integrated force and moment from the complex Fourier amplitudes of the velocity potential. Next, the second-order contributions are expressed through closed form quadratic transfer functions, which are further diagonalized through eigen decomposition. Hereby the second-order contributions can be computed as products of pseudo time series calculated by FFT, with the eigenvectors acting as transfer functions on the linear Fourier amplitudes.

For a sample 3-hour sea state, we find that eight eigen vectors are sufficient to achieve an accuracy of 1% for the maximum peak value of force and moment and 1.3% for free surface elevation, relative to the standard deviation of each signal. These results are obtained 2500 faster than with the conventional approach and we demonstrate that the computational effort of the new method scales like $O(NlogN)$, similar to linear wave loads, where $N$ is the number of frequencies. For the eight mode approximation, the error bound of 1% for loads and 4% for free surface elevation are found to hold across various values of the normalized peak wave number from shallow to deep water. The accuracy is adjustable through the number of modes and is found to be independent of the time series length. The methods potential in practical design is discussed.

提出了一种计算细长圆柱体的二阶自由面高程、深度积分力和泥线力矩的方法。该方法适用于单向不规则波浪，包括惯性载荷和粘性载荷。

我们首先从速度势的复傅立叶振幅中推导出自由表面高程、深度积分力和力矩的线性传递函数。接下来，二阶贡献通过闭合形式的二次传递函数表示，通过特征分解进一步对角化。因此，二阶贡献可以计算为通过 FFT 计算的伪时间序列的乘积，特征向量充当线性傅立叶幅度上的传递函数。

对于一个 3 小时海况样本，我们发现相对于每个信号的标准偏差，8 个特征向量足以实现力和力矩的最大峰值的 1% 精度和自由表面高程的 1.3% 精度。这些结果的获得比传统方法快 2500 倍，我们证明新方法的计算工作量类似于$哦(N日志N)$，类似于线性波浪载荷，其中 $N$是频率的数量。对于八模态近似，发现载荷误差范围为 1%，自由表面高程误差范围为 4%，适用于从浅水到深水的各种归一化峰值波数值。精度可通过模式数进行调整，并且与时间序列长度无关。讨论了在实际设计中的潜在方法。