本研究主要推導緩坡長波方程式，探討波浪在通過一透水淺灘上的島嶼時，其周邊波浪的運動情形。本文同時進一步探討在不同的海底地形與滲透性對於波浪折繞射的影響。本解析解主要是由一系列貝索函數(Bessel function)和複雜的參數所組成。在不透水底床條件下，本文引文所獲結果退化成Yu 和 Zhang (2003)的解析解，其解利用兩個獨立的參數將任意海底地形的描述轉換為徑向函數。在本文所推導的解析解中同樣在涉及拋物線底床(α=2)時，會產生奇異點(singular point)的情形，此時則須利用特殊條件來處理奇異點附近的數值解。本文除了將所推導之具透水性的解析解退化為不具透水性質的解析解，驗證了不透水底床條件下解的正確性。此外，我們的解析解得以處理波浪在通過透水底床時的問題，進而衍生可應用在一些具體的工程問題上。本文經由解析解計算結果顯示透水底床之孔隙率的增加會減弱波浪在島嶼後方匯聚現象的產生；而透水底床之線性摩擦係數的增加會對波浪在島嶼後方增加匯聚現象的產生。且波浪在通過透水孔隙底床時，其繞射所產生之會聚現象，在α=1拋物線型底床時較為顯著。

In this study, an analytic solution of long water wave propagating over permeable media around an island is derived and the effects of bottom topography and permeability on the refraction and diffraction are discussed. The solution is essentially consisting of a series form involving the Bessel function of real orders and complex arguments. The arbitrary bottom topography is described by a radial function with two independent parameters. In our solution, the singularity of parabolic shoal (α=2) can be removed by a special treatment. Furthermore, our solution can be reduced to the solution of wave propagation on impermeable media, which was derived in the literature, in some specific settings. Therefore, our solutions are validated by comparing with the prescribed solution. Then the effect of permeability is studied and its application to ocean engineering is discussed.

1. Berkhoff, J.C.W., 1972. Computation of combined refraction-diffraction. Proceedings of 13th International Conference on Coastal Engineering, ASCE, pp. 471-490.
2. Biot, M. A., 1956a. Theory of propagation elastic waves in a fluid staturated porous solid. I: low-frequency range. Journal of the Acoustical Society of America, Vol. 28, pp. 168-178.
3. Biot, M. A., 1956b. Theory of propagation elastic waves in a fluid saturated porous
solid.Ⅱ: higher frequency range. Journal of the Acoustical Society of America, Vol. 28, pp. 179-191.
4. Chen, H.B., Tsai, C.P. ＆ Jeng, C.C., 2007. Wave transformation between submerged breakwater and seawall. Journal of Coastal Research, pp. 1069-1074.
5. Dalrymple, R.A., Losada, M.A. & Martin, P.A., 1991. Reflection and transmission from porous structures under oblique wave attack. Journal of Fluid Mechanics, Vol.224, pp. 625-644.
6. Forchheimer, P., 1901. Wasserbewegung Durch Boden, Z, Vere-ines deutscher Ingenieure, Vol. 45, pp. 1782-1788.
7. Fujima, K., Goto, C., 1994. Characteristics of long waves trapped by conical island. Proceedings of. Japan Society. of Civil Engineering, Vol. 497 (II), pp. 101–110 in Japanes.
8. Fujima, K., Yuliadi, D., Goto, C., Hayashi, K., Shigemura, T., 1995. Characteristics of long waves trapped by conical island. Coastal Engineering. in Japan, Vol. 38, pp. 111–132.
9. Hsu, J. R. C., Jeng, D. C. & Tsai, C. P., 1993. Short-crested wave-induced soil response in a porous seabed of infinite thickness. International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 78, pp. 553-576.
10. Homma, S., 1950. On the behavior of seismic sea waves around circular island.
Geophysical Magazine, Vol. XXI, pp. 199–209.
11. Jonsson, I.G., Skovgaard, O., Brink-Kjaer, O., 1976. Diffraction and refraction calculations for waves incident on an island. Journal of Marint Research. 34, pp. 469–496.
12. Lan Y.J. & Lee J.F., 2010. On waves propagating over submerged poro-elastic structure. Ocean Engineering, Vol. publish online.
13. Lee, C.P., T.L., Tsai, C.P. & Jeng, D.S., 2002. Ocean waves propagating over a porous seabed of finite thickness. Ocean Eng, Vol. 29, pp. 1577-1601.
14. Lee, 1987. Wave interaction with permeable structure. Dissertation, Ocean Engineering Program, Department of Civil Engineering, Oregon State University, Corvallis, Oregon, U.S.A..
15. Liu, L.F., 1973. Damping of water waves over porous bed, Journal of Hydraulics Division, Vol. 99, pp. 2263-2271.
16. Madsen, O,S., 1974. Wave transmission through porous structures. Journal of Waterway, Harbors and Coastal Engineering Division, Vol. 100, pp. 169-188.
17. Rojanakamthorn, S., Isobe, S. M. & Watanabe, A., 1989. A mathematical model of wave transformation over a submerged breakwater. Coastal Engineering in Japan, Vol. 32, pp. 209-234.
18. Rojanakamthorn, S., Isobe, S. M. & Watanabe, A., 1990. Modeling of wave transformation on submerged breakwater. Proceedings of 22th International Conference on Coastal Engineering, pp. 1060-1073.
19. Silva, R., Salles, P. & Palacio, A., 2002. Linear waves propagating over a rapidly varying finite porous bed. Coastal Engineering, Vol. 44, pp. 239-260.
20. Sollitt, C.K. & Cross., R.H., 1972. Wave transmission though permeable breakwaters, Proceedings of 13th International Conference on Coastal Engineering, ASCE, pp. 1827-1846.
21. Tae-Hwa Jung., Changhoon Lee., Yong-Sik Cho., 2010. Analytical solutions for long waves over a circular island. Coastal Engineering, Vol. 57, pp. 440–446.
22. Ting C.L., Lin, M.C. & Cheng, C.Y., 2004. Porosity effects on surface waves over permeable submerged breakwaters: an experimental investigation. Coastal Engineering, pp. Vol. 50, pp. 213-224.
23. Tsai, C.P., 1995. Wave-induced liquefaction potential in a porous seabed in front of a breakwater. Ocean Engineering, Vol. 22, pp.1-18.
24. Tsai, C.P. & Lee, T.L., 1995. Standing wave induced pore pressures in a porous seabed. Ocean Engineering, Vol. 22, 505-517.
25. Vastano, A.C., Reid, R.O., 1967. Tsunami response for islands: verification of a numerical procedure. J.Marine Res. 25, pp. 129–139.
26. Ward, J.C., 1964. Turbulent flow in porous media. Journal of the Hydraulics Division. ASCE, Vol. 90, pp. 1-12.
27. Yu, Xiping., Zhang, B., 2003. An extended analytic solution for combined refraction and diffraction of long waves over circular shoals. Ocean Engineering, Vol. 30, pp. 1253-1267.
28. Yamamoto, T., Koning, H.L., Sellmeiher, H. & Van Hijum, E.V., 1978. On the response of a poro-elastic bed to water waves. Journal of Fluid Mechanics, Vol. 87, pp. 193-206.
29. Zhang, S., Jin, J., 1996. Computation of special functions. John Wiley & Sons.
30. Zhu, S., Zhang, Y., 1994. New solution for the propagation of long water waves over variable depth. Journal of Fluid Mechanics, Vol. 278, pp. 391–406.
31. Zhu, S., Zhang, Y., 1996. Scattering of long waves around a circular island mounted on a conical shoal. Wave Motion, Vol. 23, pp. 353–362.
32. 丁肇隆、林銘崇、林其蒼、李芳承，2003，「波浪通過透水底床之變形」，第二十四屆海洋工程研討會論文集，第562-569頁。
33. 張人懿，2010，「利用含虛擬時間積分法之延伸型緩波方程式模擬波浪通過透水結構物之變形」，國立成功大學水利及海洋工程學所博士論文。