過去幾十年內，Boussinesq模式被廣泛地應用在模擬波浪由深水區域傳遞至淺水區域的非線性變化現象。本文採用Lynett和Liu (2004) 所發展之多層Boussinesq 模式 (multi-layer Boussinesq model) 研究狹長形狀港池之港池振盪現象。如以多層Boussinesq方程式中之雙層Boussinesq方程式 (two-layer Boussinesq equation) 為例，其線性波及非線性波之模擬能力可達 kh~8 及kh~6。

　　為了瞭解此高階Boussinesq模式之適用範圍，本文利用一系列的數值模擬來進行驗證。其中包含長距離傳遞之孤立波、Berkhoff (1982) 典型波浪變形試驗及Beji和Battjes (1994) 非線性波浪通過梯形潛堤之試驗。結果指出模式在模擬具有開放邊界之問題時有極佳之模擬效果。

　　港池振盪方面，本文最初選定Ippen 和 Goda (1963) 及Rogers和Mei (1978)之線性振盪及非線性振盪試驗進行模擬，上述兩種試驗之港池均為狹長之形狀。在模擬過程中，不論是線性或非線性振盪，均因港池入口轉角處 (corner) 所產生之數值短波出現而使得模式在模擬過程發散，因此本文採用了與試驗不同之港池長寬比進行模擬。在模擬線性振盪之過程中發現，不論港內振盪量的大小，港池長度 L 及入射波波長 l 之關係為整個振盪機制之首要。特別值的一提的是，只要 kl 呈現一定比例，振盪現象即會發生，而港池寬度 b 的改變在振盪現象中似乎不會造成太大之影響。而在非線性振盪之模擬中，本文可以得到兩點結論。首先，當造波處距港池開口為不同距離及港池具有不同長度時均會對於港內之非線性振盪現象造成影響，這也與Rogers和Mei (1978) 所提出之結論相同；其次，如入射波浪包含高頻之成份波時，該量也會於港內自成一振盪現象。

　　本文最後提出高階Boussinesq方程式在模擬港池振盪時之幾點結論與建議，而本文之數值結果仍有待透過試驗去進一步地加以證實。

　The Boussinesq model has been widely used in simulating nonlinear waves transformation from deep water to shallow water in the past decade. In this thesis, the multi-layer Boussinesq model developed by Lynett and Liu (2004) is applied to investigate the phenomenon of the harbor resonance with the shape of the harbor being long and narrow. In particular, this multi-layer model exhibits accurate linear characteristics up to a kh~8 and nonlinear accuracy to kh~6 for two-layer model.

　To validate this higher-order Boussinesq model, a series of numerical tests are carried out including the solitary wave propagating a long distance to ensure the stability of the model, the classical experiments conducted by Berkhoff (1982) and the experiment about the nonlinear wave transformation over submerged trapezoidal bar conducted by Beji & Battjes (1994). Our results indicated that this model could simulate the problems with open boundary very well.

　For the harbor resonance, we have initially chosen the linear resonance experiment and nonlinear resonance experiment conducted by Ippen & Goda (1963) and Rogers & Mei (1978), respectively. Both of their experiments were conducted in a long and narrow harbor. However, numerical short waves are found due to the corner points at the harbor entrance, which leads to the code overflow. Similar situations are also encountered in the simulation of nonlinear harbor oscillation. Different aspect ratios of harbor geometry are then used in our simulation. In the case of linear harbor resonance, it is found that the relation between the harbor length L and the wavelength of the incident wave l is the main mechanism for whether the resonance is agitated or not. Specifically, the harbor resonance occurs as kl reaches to certain number and the change of the harbor width b seems not to play an important role. In the case of the nonlinear resonance, two findings are concluded as follows. Firstly, the distance between the incident waves to be initiated and the length of the harbor will affect the nonlinear resonance inside the harbor. This result agrees with the results presented by Rogers & Mei (1978). Secondly, the higher harmonics of incident wave will also resonate inside the harbor.

　In summary, this thesis provides some results and suggestions in the simulation of the harbor resonance using higher-order Boussinesq model. However, further experiments are needed to verify our numerical findings.

1.Boussinesq, J., “Theorie des ondes et ramous qui se propagent le long dun canal rectangularire horizontal, en communiquant au liquide contenu dansce canal des vitesses sensiblement pareilles de la surface au,” Journal of Mathematical Pure et Application, 2nd Series, Vol. 17, pp. 55-108 (1872).
2.Berkhoff, J. C. W., “Computation of combined refraction-diffraction,” Proceedings of the Thirteen International Conference on Coastal Engineering, ASCE, Vancouver, Vol. 1, pp. 705-720 (1972).
3.Berkhoff, J. C. W., Booy, N., and Radder, A. C., “Verification of numerical wave propagation models for simple harmonic linear water waves,” Coastal Engineering, Vol. 6, pp. 255-279 (1982).
4.Beji, S. and Battjes, J. A., ”Numerical Simulation of Nonlinear Wave Propagation Over A Bar,” Coastal Engineering, Vol. 23 , pp. 1-16 (1994).
5.Chen, H. S. and Mei, C. C., Oscillations and wave forces in an offshore harbor, Parsons Laboratory, Report No. 190, MIT (1974).
6.Chou, C. R. and Han, W. Y., “Oscillations induced by irregular waves in harbors,” Proceedings of the nineteen Conference on Coastal Engineering. ASCE, New York, pp. 1040-1056 (1994).
7.Gobbi, M. F., Kirby, J. T. and Wei G., “A fully nonlinear Boussinesq model for surface waves. Part2. Extension to O ,” Journal of Fluid Mechanics, Vol. 405, pp. 181-210 (2000).
8.Ippen, A. T. and Raichlen, F., Wave induced oscillations in harbors： The problem of coupling of highly reflective basins, Report No. 49, Hydrodynamics Laboratory, MIT (1962).
9.Ippen, A., Raichlen, T. F., and Sullivan, R. K. Wave induced oscillations in harbors： Effect of energy dissipators in coupled basin system, Report No. 52, Hydrodynamics Laboratory, MIT (1962).
10.Ippen, A. T. and Goda, Y., Wave induced oscillations in harbors： The solution for a rectangular harbor connected to the open-sea, Report No. 59, Hydrodynamic Laboratory, MIT (1963).
11.Ippen, A. T., Estuary and coastline hydrodynamics, McGraw-Hill Book Company, New York (1966).
12.Israeli, M. and Orszag, S. A., “Approximation of radiation boundary conditions,” Journal of Computational Physics, Vol. 41(1), pp. 115-135 (1981).
13.LeMehaute, B., “Discussion of the paper ”Harbor paradox” by J. Miles and W. Muck,” Journal of the Waterway and Harbor Division, ASCE, Vol. 88, No. WW2, pp. 173-185 (1962).
14.Lee, J. J., “Wave-induced oscillations in harbors of arbitrary geometry,” Journal of Fluid Mechanics, Vol. 45, pp. 375-394 (1969).
15.Lee, J. J. and Raichlen, F., “Oscillations in harbors with connected basins,” Journal Waterway, Ports, Coastal and Ocean Engineering Division, ASCE, Vol. 98, No. WW3, pp. 311-332 (1972).
16.Lepelletier, T. G., “Tsunami-Harbor oscillations induced by nonlinear transient long waves.” W. M. Keck Laboratory Rep. No. KH-R-41, California Institute of Technology, Pasadena, Calif (1980).
17.Lepelletier, T. G. and Raichlen, F., ”Harbor oscillations induced by nonlinear transient long waves,” Journal Waterway, Ports, Coastal and Ocean Engineering, Vol. 113(4), pp. 381-400 (1987).
18.Liu, P. L.-F., “Model equations for wave propagation from deep to shallow water,” Advances in Coastal Engineering, World Scientific, Singapore, Vol. 1, pp. 125-157 (1994).
19.Li, B., “A Evolution Equation for Water Waves,” Coastal Engineering, Vol. 23, pp. 227-242 (1994).
20.Li, Y. S., Liu, S.-X., Yu, Y.-X., and Lai, G.-Z., “Numerical modeling of Boussinesq equations by finite element method,” Coastal Engineering, Vol. 37, pp. 97-122 (1999).
21.Lynett, P., “A multiplayer approach to modeling nonlinear, dispersive waves from deep water to the shore,” Ph.D thesis, Cornell University, Ithaca, New York (2002).
22.Lynett, P. and Liu, P. L.-F., “A numerical study of submarine landslide generated waves and runup,” Royal Society of London A in press. (2002).
23.Lynett, P. and Liu, P. L.-F., “A two layer approach to wave modeling,” Philosophical Transactions of the Royal Society of London, Series A, Physical Science and Engineering, A460. pp. 2637-2669 (2004).
24.Miles, J. and Munk, W., “Harbor paradox,” Journal Waterway and Harbors Division, ASCE. WW3, pp. 111-130 (1961).
25.Madsen, P. A. and Sørensen, O. R., “A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2: A slowly-varying bathymetry,” Coastal Engineering, Vol. 18, pp. 183-204 (1992).
26.Nwogu, O., “An alternative form of the Boussinesq equations for modeling the propagation of waves from deep to shallow water,” Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, Vol. 119(6), pp. 618-638 (1993).
27.Peregrine, D. H., “Long waves on a beach,” Journal of Fluid Mechanics, Vol. 27(4), pp. 815-827 (1967).
28.Raichlen, F. and Ippen, A. T., “Wave induced oscillations in harbors,” Journal of the Hydraulics Division, ASCE, Vol. 91, No. Hy. 2, pp. 1-26 (1965).
29.Rogers, S. R. and Mei, C. C., “Nonlinear Resonant excitation of a long narrow bay,” Journal of Fluid Mechanics, Vol. 88, pp. 161-180 (1978).
30.Stokes, G. G., “On the theory of oscillatory waves,” Trans. Camb. Philos. Soc., Vol. 8, pp. 441-455 (1847).
31.Schäffer, H. A. and Madsen, P. A., “Further enhancements of Boussinesq-type equations,” Coastal Engineering, Vol. 26 pp. 1-14 (1995).
32.Tsay, T. K. and Liu, P. L.-F., “A finite element model for wave refraction and diffraction,” App1. Ocean Res., 5(1), pp. 30-37 (1983).
33.Tsay, T. K., Zhu, W. and Liu, P. L.-F., “A finite element model for wave refraction, diffraction, reflection and dissipation,” Applied Ocean Research, Vol. 11, No. 1, pp. 33-38 (1989).
34.Witting, F. M., “A unified model for the evolution of nonlinear water waves,” Journal of Computational Physics, Vol. 56, pp. 203-236 (1984).
35.Wei, G. and Kirby, J. T., “Time –dependent numerical code for extended Boussinesq equations,” Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 121, pp. 251-263 (1995).
36.Wei, G., Kirby, J. T., Grilli, S. T. and Subramanya, R., “A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves,” Journal of Fluid Mechanics, Vol. 294, pp. 71-92 (1995).
37.Wei, G., Kirby, J. T. and Sinha, A., “Generation of waves in Boussinesq models using a source function method,” Coastal Engineering, Vol. 36, pp. 271-299 (1999).
38.Walkley, M. and Berzins, M., “A finite element method for the two-dimensional extended Boussinesq equations,” International Journal Numerical Methods Fluids, Vol. 39, pp. 865-885 (2002)
39.Woo, S. B. and Liu, P. L.-F., “Finite-Element Model for Modified Boussinesq Equations, Ⅰ：Model Development,” Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 130, pp. 1-16 (2004).
40.Woo, S. B. and Liu, P. L.-F., “Finite-Element Model for Modified Boussinesq Equations, Ⅱ：Applications to Nonlinear Harbor Oscillations,” Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 130, pp. 17-28 (2004).
41.Zhao, L., Panchang, V., Chen, W., Demirbilek, Z. and Chhabbra, N., “Simulation of wave breaking effects in two-dimensional elliptic harbor wave models,” Coastal Engineering, Vol. 42, pp. 359-373 (2001).
42.陳柏旭、蔡丁貴，「局部輻射邊界條件在水波數值上的應用」，中華民國第十二屆海洋工程研討會論文集，第1-18頁 (1990).
43.周宗仁、韓文育、朱忠一，「消波式碼頭對港內水面振動之影響」，中華民國第六屆水利工程研討會論文集，第668-679頁 (1992).
44.周宗仁、林炤圭，「應用邊界元素法解析任意地形及水深之港池水面波動問題」，中華民國第八屆海洋工程研討會論文集，第111-129頁 (1986).
45.歐善惠，林西川，林火旺，蘇青和，「不等水深多孔岸壁港池之共振模式」，中華民國第十二屆海洋工程研討會論文集，第74-93頁 (1990).
46.蘇青和、蔡丁貴、歐善惠，「數值方法及輻射邊界條件在港池共振應用之探討」，中華民國第十三屆海洋工程研討會論文集，第23-37頁 (1991).
47.蘇青和、蔡丁貴、張金機，「花蓮港港灣設施改善計畫之研究-第三子計畫 數值模擬」，港灣技術研究所研究報告，專刊128號 (1996).