本研究以 Hsu 等人 (2006) 所推導之完整型緩坡方程式(Complementary Mild-Slope Equation，CMSE) 為基礎，利用波譜分割法建立數值模式，用以模擬不規則波之變形。於模式中利用指數函數處理波譜切割的問題，並以能量守恆之觀點於控制方程式中加入非線性淺化、碎波能量消散及非線性三波交互作用效應。控制方程式中同時考慮底床坡度效應與波浪角度之變化，使模式能更完整地描述不規則波浪通過真實底床之變形。本研究先藉由規則波和不規則波之試驗比較來驗證模式之正確性；同時模擬沙漣底床地形，以了解控制方程式中高階地形參數對模式之影響；最後，將模式應用於模擬平面波場並與水工試驗比較，得知模式可以合理地模擬不規則波浪之綜合變形效應，由此說明模式之合理性。

In this study, a numerical model based on Complementary Mild Slope Equation (CMSE) proposed by Hsu et al. (2006) was developed to simulate wave transformation of irregular waves over a general finite seabed. The exponential function is used to cut the wave spectrum into several wave components waves. A combined energy coefficient is added in the governing equation to solve nonlinear wave shoaling, wave breaking, energy dissipation, and wave-wave interaction. Both of them takes into accounts the effect of the bottom slope and wave propagation direction, that the model is validated thought experimental data for regular and irregular wave propagation over sloping bottom. The numerical results have shown good agreement with results of laboratory experiments. The numerical results show that the model is capable of describing the wave energy dissipation in numerical calculation for the case of sinusoidal ripped seabed. The model is also applied to calculate irregular waves travelling over a submerged elliptic shoal on a sloping bottom. The comparison of plenary wave height variations demonstrates that the present model is able to produce irregular wave transformation over submerged structure including shoaling, refraction, diffraction, reflection, wave breaking and energy dissipation.

1.Arcilla, A. S., J.A. Roelvink, B.A. O’Connor, A.J.H.M. Reniers, and J.A. Jimenez, The Delta flume, “93 experiment, Proc. Coastal Dynamics Conf.,’’ Barcelona, Spain, pp. 488-502 (1994).
2.Armstrong, J.A., N. Bloembergen, J. Ducuing and P.S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Physical Review, Vol. 127, pp. 1918-1939 (1962).
3.Battjes, J.A. and J.P.F.M. Janssen, “Energy Loss and Set-up due to Breaking of Random Waves,” Proceedings of 16th International Conference on Coastal Engineering, ASCE, Hamburg, pp. 569-587 (1978).
4.Berkhoff, J.C.W., N. Booy, and A.C. Radder, “Verification of Numerical Wave Propagation Models for Simple Harmonic Linear Water Waves,” Coastal Engineering, Vol. 6, pp. 255-279 (1982).
5.Berkhoff, J.C.W., “Computation of Combined Refraction-Diffraction,” Proceedings of 13th International Conference on Coastal Engineering, ASCE, Canada, pp. 471-490 (1972).
6.Biesel, F., “Study of Wave Propagation in Water of Gradually Varying Depth,” U.S. National.Bureau of Standards, Gravity Waves, NBS Circular 521, pp. 243-253 (1952).
7.Black, K.P. and M.A. Rosenberg, “Semi-Empirical Treatment of Wave Transformation Outside and Inside the Breaking Line,” Coastal Engineering, Vol. 16, pp. 313-345 (1992).
8.Bretherton, F.P., “Resonant Interactions between Waves: The Case of Discrete Oscillations,” Journal of Fluid Mechanics, Vol. 20, pp. 457-480 (1964).
9.Bretschneider, C. L., ‘‘Significant Waves and Wave Spectrum,’’ Ocean Industry, pp. 40-46 (1968).
10.Chen, Y.Y., B.D. Yang,., L.W. Tang, S.H. Ou, and R.C. Hsu, ‘‘Transformation of progressive waves propagating obliquely on gentle slope,’’ Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 130, pp. 162-169 (2004).
11.Davies, A.G., and A.D.Heathershaw, ‘‘Surface-Wave Propagation over Sinusoidally Varying Topography,’’ Journal of Fluid Mechanics, Vol. 144, pp. 419-443 (1984).
12.Eldeberky, Y., “Nonlinear Transformation of Wave Spectra in the Nearshore Zone,” Ph.D. thesis, Department of Civil Engineering, Delft University of Technology, The Netherlands (1996).
13.Eldeberky, Y. and J.A. Battjes, “Parameterization of Triad Interactions in Wave Energy Models,” Proceedings of Coastal Dynamics Conference ’95, Gdansk, Poland, pp. 140-148 (1995).
14.Elgar, S., M.H. Freilich and R.T. Guza, “Model-data Comparisons of Moments of Nonbreaking Shoaling Surface Gravity Waves,” Journal of Geophysical Research, Vol. 95, pp. 16055-16063 (1990).
15.Elgar, S., R.T. Guza and M.H. Freilich, “Observations of Nonlinear Interactions in Directionally Spread Shoaling Surface Gravity Waves,” Journal of Geophysical Research, Vol. 98, pp. 20299-20305 (1993).
16.Elgar, S., T.H.C. Herbers, V. Chandran and R.T. Guza, “Higher-order Spectral Analysis of Nonlinear Ocean Surface Gravity Waves,” Journal of Geophysical Research, Vol. 100, pp. 4997-4983 (1995).
17.Goda, Y. and K. Nagai, ‘‘Report of the Port and Harbour,’’ Research Istitute, No. 61, pp.64 (1968)
18.Goda, Y., “Random Seas and Design of Maritime Structures,” University of Tokyo Press, 323pp. (1985).
19.Hsu, T.W. and C.C., Wen “On Radiation Boundary Conditions and Wave Transformation Across Surf Zone,” China Ocean Engineering, Vol. 15, pp. 405-416 (2001a).
20.Hsu, T.W. and C.C. Wen “A Parabolic Equation Extended to Account for Rapidly Varying Topography,” Ocean Engineering, Vol. 28, pp. 1479-1498 (2001b).
21.Hsu, T.W. , T.Y. Lin, C.C. Wen, and S.H. Ou, “A Complementary mild-slope equation derived using higher-order depth function for wave obliquely propagating on sloping bottom,” Physic of Fluid, Vol. 18 , 0987106 (2006).
22.Hsu, T.W. , S.C. Hsiao,S.H. Ou, S.H. Wang, B.D Yang, and S.E. Chou, “An Application of Boussinesq Equations to Bragg reflection of Irregular Waves,” Ocean Engineering, Vol. 34 , pp. 870-883 (2007).
23.Isobe, M., “A Parabolic Equation Model for Transformation of Irregular Waves due to Refraction, Diffraction and Breaking,” Coastal Engineering in Japan, Vol. 30, pp. 33-47 (1987).
24.Isobe, M., Y. Shibata, T. Izumiya, and A. Watanabe, “Set-up Due to Irregular Waves on a Reef,” 第 35 回海岸工學講演會論文集, pp. 192-196 (1988). (In Janpanese)
25.Izumiya, T. and M. Endo, “Wave Reflection and Transmission Due to a Submerged Breakwater,” 第 36 回海岸工學講演會論文集, pp. 638-642 (1989). (In Janpanese)
26.Kubo, Y., Y. Kotake, M. Isobe, and A. Watanabe, “ Time dependent Mild Slope Equation for Random Waves,” Proceedings of 23th International Conference on Coastal Engineering, ASCE, pp. 419-431 (1992).
27.Lee, C., G. Kim, and K.D. Suh, “Extend mild-slope equation for random waves,” Coastal Engineering, Vol. 48, pp. 277-287 (2003).
28.Li, B., “An Evolution Equation for Water Waves,” Coastal Engineering, Vol. 23, pp. 227-242 (1994a).
29.Li, B., “A Generalized Conjugate Gradient Model for the Mild Slope Equation,” Coastal Engineering, Vol. 23, pp. 215-225 (1994b).
30.Li, B., D.E. Reeve, and C.A. Fleming, “Numerical Solution of the Elliptic Mild-Slope Equation for Irregular Wave Propagation,” Coastal Engineering, Vol. 20, pp. 85-100 (1993).
31.Longuet-Higgins, M. S., “On the Statistical Distributions of the Height of Sea Waves,” Journal of. Marine Research., Vol. IX, No. C5, pp. 245-266 (1952).
32.Luth, H.R., G. Klopman, and N. Kitou, ‘‘Kinematics of Waves Breaking Partially on an Offshore Bar,’’ Rep. H1573, 13 pp., Delft Hydraulics, Delft, Netherlands (1993).
33.Madsen, P.A. and O.R. Sørensen, “Bound Waves and Triad Interactions in Shallow Water,” Ocean Engineering, Vol. 20, No. 4, pp. 359-388 (1993).
34.McCowan, J., “On the Highest Wave of Permanent Type,” Philosophical Magazine and Journal of Science, Vol. 38, pp. 351-358 (1894).
35.Nagai, K., “Computation of Refraction and Diffraction of Irregular Sea,” Report of the Port and Harbor Research Institute., Vol. 11, No. 2, pp. 47-119, June (1972).
36.Nagayama, S., “Study on the change of wave height and energy in the surf zone.” B. Eng. thesis, Yokohama National University, Japan. (In Japanese) (1983).
37.Phillips, O.M., “On the Dynamics of Unsteady Gravity Waves of Finite Amplitude, Part 1,” Journal of Fluid Mechanics, Vol. 9, pp. 193-217 (1960).
38.Radder, A.C., “On the Parabolic Equation Method for Water Wave Propagation,” Journal of Fluid Mechanics, Vol. 95, No. 1, pp. 159-176 (1979).
39.Rojanakamthorn, S., M. Isobe, and A. Watanabe, “A Mathematical Model of Wave Transformation Over a Submerged Breakwater,” Coastal Engineering in Janpan, Vol. 32, No. 2, pp. 209-234 (1989).
40.Rojanakamthorn, S., M. Isobe, and A. Watanabe, “Modeling of Wave Transformation on Submerged Breakwater,” Proceedings of 22th International Conference on Coastal Engineering, ASCE, pp. 1060-1073 (1990).
41.Shuto, N., “Nonlinear Long Waves in a Channel of Variable Section,” Coastal Engineering in Japan, Vol. 17, pp. 1-12 (1974).
42.Sommerfeld, A., “Mechanics of Deformable Bodies,” Vol. 2 of Lectures on Theoretical Physics, Academic Press, New York (1964).
43.Suh, K. D., C. Lee, and W.S. Part, “Time-dependent equations for wave propagation on rapidly varying topography,” Coastal Engineering, Vol. 32, pp. 91-117 (1997).
44.Tang, F.L.W. and C.F. Lin, “Practical Method for Evaluation Directional Spectra After Shoaling and Refraction,” Proceedings of the 20th Twentieth Coastal Engineering Conference, pp. 780-793 (1986). (In Taipei)
45.Tsai, C.P., H.B. Chen, and R.C. Hsu, “Calculations of Wave Transformation Across the Surf Zone,” Ocean Engineering, Vol. 28, No. 8, pp 941-955 (2001).
46.Watanabe, A. and M. Dibajnia, “A Numerical Model of Wave Deformation in Surf Zone,” Proceedings of 21th International Conference on Coastal Engineering, ASCE, Malaga, Spain, pp. 578-587 (1988).
47.林朝福，簡榮生，「波譜分割在淺海波譜變形之應用」，第十三屆海洋工程研討會論文集，台北，pp. 85-103 (1991)。
48.陳陽益，湯麟武，「平緩坡度底床上前進的表面波」，第十四屆海洋工程研討會論文集，新竹，pp.1-22 (1992)。
49.廖哲民，「應用能譜觀念由緩坡方程式求解斜坡上波場變形之計算方法」，國立成功大學水利及海洋工程研究所，碩士論文 (1996)。
50.陳陽益，「平緩坡度底床上前進的表面波」，第十九屆海洋工程研討會論文集，台中，pp.112-121 (1997)。
51.楊炳達，陳陽益，湯麟武，歐善惠，「前進波列斜向傳遞於等緩坡度底床之研究」，第二十二屆海洋工程研討會論文集，高雄，pp.1-8 (2000)。
52.許泰文，廖建明，鄧秋霞，謝文斌，「以緩坡方程式模擬不規則波之變形」，第二十四屆海洋工程研討會論文集，台中，pp. 70-77 (2002)。
53.廖建明，許泰文，謝文斌，「由緩坡方程式求解不規則波浪變形之研究」，第二十五屆海洋工程研討會論文集，基隆，pp. 235-242 (2003)。
54.翁文凱，黃奕全，江銘祥，「近岸波場變形研究-平面波場波浪變形試驗研究(1/2)」，第二十九屆海洋工程研討會論文集，台南，pp. 59-64 (2007)。