自由液面格林函數為船舶耐海性能分析與邊界積分法中重要之理論基礎，其中三維平移－脈動型源點函數可用以描述船舶具前進速度及週期性脈動條件下所產生之流場特性。Bessho-Form三維平移－脈動源格林函數屬單重積分形式，在理論實作上具有便利性。然而，在實際積分過程中，波浪項被積函數於特定條件下（如：積分變數θ→±π⁄2 ）容易產生高頻振盪，將造成數值積分收斂困難，進而影響速度勢及其導數之計算精度。
本研究以 Bessho-Form 三維平移－脈動型源點函數為研究對象，針對其波浪項及空間導數項進行數學推導與數值分析。首先，將波浪項拆分為T_1與T_2兩部分，並針對一般情況、平移非脈動及脈動非平移進行討論，以建立不同物理條件下之積分表示式。其次，針對被積函數於複數平面中所產生之高頻振盪與奇異性問題，進行積分路徑規劃，並結合最陡下降法(Steepest Descendent Method，SDM)建立一積分路徑以取代發生高頻振盪的部分路徑，最陡下降法積分路徑概念在於相位函數虛部維持定值，同時令實部沿路徑快速下降，使被積函數振盪受抑制，並快速收斂、提升積分效率。對於端點奇異或發散之積分區域，則配合適當之變數轉換與數值處理方式，以改善積分穩定性。
本研究與文獻算例的結果比較顯示，在不同速度與脈動頻率條件下，本文所計算之速度勢及其方向導數計算結果與文獻做比較，呈現良好一致性，顯示本文所建立之積分路徑規劃與最陡下降法具有初步之穩定性與可行性。最後，藉由波浪圖之觀察，可發現在部分特定條件下（ y=0 ）仍可能存在局部不穩定現象，後續可進一步探討相關參數對積分路徑與數值結果之影響，並將點對點速度勢計算延伸至點對線或點對面積分，以提升船體與海洋結構物受力分析中之應用性。

The boundary integral equation based on the free-surface Green function is widely used in solving seakeeping problems. The used Green function is employed to present the flow field generated by a ship advancing at a constant forward speed while undergoing harmonic oscillation. Among several formulations of Three-Dimensional Translating-Pulsating Source Green Function, Bessho form is expressed in a single-integral, which greatly simplifies the complexity of numerical integration and raises the implementation of the forward-speed hydrodynamic problems. However, it is specially noted that involved wave-term integrand may exhibit severe oscillatory behavior, and results in the poor numerical convergence and reduced accuracy in evaluating integration.
This study investigates the numerical evaluation of the Bessho-Form Three-Dimensional Translating-Pulsating Source Green Function through mathematical derivation and numerical analysis. The wave term and its spatial derivatives are formulated under different physical conditions. To suppress the oscillatory behavior of the integrand, the Steepest Descent Method (SDM) is employed to construct an appropriate integration path. In addition, appropriate variable transformations and numerical treatments are introduced to improve the stability and efficiency of the numerical integration.
The proposed numerical method is validated through numerical computations by comparison with the benchmark results reported by Inglis and Price. The computed results show good agreement over a range of forward speeds and oscillation frequencies, confirming the validity of the proposed numerical method. The present study establishes a numerical foundation for future extensions to line- and panel-source integrations, facilitating further applications in seakeeping analysis and forward-speed hydrodynamic problems.

[1] Haskind, M. D. (1946). The oscillation of a ship in still water. Prikladnaya Matematika i Mekhanika, 10(1), 33–66. (Translated by the Society of Naval Architects and Marine Engineers [SNAME], 1953, SNAME Technical and Research Bulletin No. 1-12)
[2] Haskind, M. D. (1946). The hydrodynamic theory of ship oscillation in rolling and pitching. Bulletin de l'Académie des Sciences de l'URSS, Classe des Sciences Techniques, (1), 23–34. (Translated by the Society of Naval Architects and Marine Engineers [SNAME], 1953, SNAME Technical and Research Bulletin No. 1-12)
[3] Chang, M. S. (1977, September). Computations of three-dimensional ship motions with forward speed. In Proceedings of the 2nd International conference on numerical ship hydrodynamics (pp. 124-135).
[4] Inglis, R.B. (1981). A Three Dimensional Ship Motion Theory―Comparison between Theoretical Prediction and Experimental Data of the Hydrodynamic Coefficients with Forward Speed. Trans. of the Royal Institution of Naval Architects, 124.
[5] Kobayashi, M. (1981). On the hydrodynamic forces and moments acting on a three dimensional body with a constant forward speed. Journal of the Society of Naval Architects of Japan, 1981(150), 175-189.
[6] Hess, J. L., & Smith, A. M. O. (1964). Calculation of nonlifting potential flow about arbitrary three-dimensional bodies. Journal of ship research, 8(04), 22-44.
[7] Webster, W. C. (1975). The flow about arbitrary, three-dimensional smooth bodies. Journal of Ship Research, 19(04), 206-218.
[8] Newman, J. N. (1986). Distributions of sources and normal dipoles over a quadrilateral panel. Journal of Engineering Mathematics, 20(2), 113-126.
[9] Bessho, M. (1977). ON THE FUNDAMENTAL SINGULARITY IN THE THEORY OF SHIP MOTIONS IN A SEAWAY.
[10] Inglis, R.B. (1980). Calculation of velocity potential of a translating, pulsating source. Trans RINA, 123, 163-175.
[11] Havelock, T. H. (1958). The effect of speed of advance upon the damping of heave and pitch. Trans. Inst. Naval Architect I, 131-135.
[12] Xu, Y., & Dong, W.-c. (2011). Study on characteristics of 3-D translating-pulsating source green function of deep-water havelock form and its fast integration method. China Ocean Engineering, 25(3), 365-380. https://doi.org/10.1007/s13344-011-0031-8
[13] Miao, G. P., Liu, Y. Z., Yang, Q. Z., & Liu, Z. Y. (1995, June). On the 3-D pulsating source of Michell's type with forward speed. In ISOPE International Ocean and Polar Engineering Conference (pp. ISOPE-I). ISOPE
[14] Ohkusu, M., & Iwashita, H. (1987). Radiation and diffraction wave pattern of ships with forward speed.
[15] Iwashita, H. (1992). Evaluation of the added-wave-resistance Green function distributing on a panel. Mem. Fac. Engrg. Hiroshima Univ, 11(2), 21-39.
[16] Iwashita, H., & Ohkusu, M. (1989). Hydrodynamic forces on a ship moving with forward speed in waves. Journal of the Society of Naval Architects of Japan, 1989(166), 187-205.
[17] Iwashita, H. (1997). An Improved Computation Method of the Seakeeping Green Function. Institut für Schiffbau der Universität Hamburg.
[18] Yao, C.-b., & Dong, W.-c. (2014). A fast integration method for translating-pulsating source Green’s function in Bessho form. Journal of Zhejiang University SCIENCE A, 15(2), 108-119.
[19] Yao, C., & Dong, W. (2014). Study on fast integration method for Bessho form translating–pulsating source Green’s function distributing on a panel. Ocean engineering, 89, 10-20.