於三度空間中，本文以極座標系統來描述原為規則前進波列傳遞在同心圓底床上的折射現象。主控折射現象的波向線微分通式，由其在卡氏直角座標下的表示式經座標轉換，及直接依極座標下的Fermat’s原則與波浪守衡性本質等，三種不同的方式分別導述出，且結果完全一致而得到確證。將此通式應用到同心圓底床地形情況時，其波向線與沿波向線的時空位相函數之顯函數表示式被決定出下，涵括波峰線的等位相線以及沿任一波向線上的折射係數，亦被明確地定出。據此，對原為均勻的規則入射波列，傳遞在一同心圓狀隆起淺灘上的折射特性，包括等位相線因為同心圓底床的影響而發生的轉折斷裂現象，以及由這些轉折斷裂後的等位相線所構成的包絡線等，其所有物理特性皆可用動態與靜態兩種不同觀點，配合函數式的組合使用來得到完整的描述；而在淺灘後方由此包絡線界定出來的紛亂遮蔽區域內，所會出現的波向線相交的所謂聚焦(caustic)現象的通盤組合情形，及在波向線相交處之折射係數的量化問題，亦被合理明確地解決。由此，對往昔迄今所沿用的古典波向線原理，之所無法解決波向線相交時的聚焦(caustic)問題之癥結點，因而被顯然地揭露且得到解決。

This study discusses the three-dimensional refraction of progressive wave trains propagating over a bottom of circular concentric contours and the results are expressed in a polar coordinate. First, a general differential formulation of refraction is derived via three different methods: by transferring from its original Cartesian form to the polar coordinate, by applying the Fermat’s principle in polar coordinate, and by applying the conservation of waves in polar coordinate. All three approaches give the same governing equation; hence, its correctness is verified. Based on this governing equation, the wave ray, the phase function, the constant phase line, and the refraction coefficient are all determined.
In the present refraction problem for an originally uniform wave train propagating over a bottom of circular concentric contours, a few special features, including the cusps of constant phase lines due to the effect of bottom, and the envelope composed of these cusps, are present. All these refraction properties can be expressed in terms of both a snapshot and a time evolution of constant phase lines.
In the lee side of the shoal, there exists a sheltered zone that is enclosed by the envelope of the cusps. In this zone, wave rays intersect and the corresponding caustic problem arises, and all possible combinations of intersecting rays are also specifically described in this study. The difficulty of classical ray theory for the caustic problem is overcome and the caustic phenomenon and its refraction coefficients are determined explicitly in this study.

第一章 緒論 1
1-1 研究動機與目的 1
1-2 前人研究 2
1-3 研究方法 4
1-4 本文的組織架構 7
第二章 極座標表示下之波浪前進折射通式 8
2-1 由卡氏直角座標形式經座標轉換 8
2-2 極座標下依Fermat’s原則之推導 11
2-3 極座標下依波浪守衡性本質之推導 12
第三章 波浪前進在同心圓底床上之折射現象 13
3-1 同心圓底床地形上之波向線通式 13
3-2 同心圓底床地形上之位相函數通式 15
3-3 同心圓底床地形上之等位相線 16
3-4 同心圓底床地形上波浪的折射係數 19
第四章 同心圓狀隆起的淺灘情況 21
4-1 進入淺灘前波向線為直線之區段 23
4-2 在淺灘上由 至最小 的區段 24
4-3 在淺灘上由最小 至 的區段 26
4-4 通過淺灘後波向線為直線的區段 28
4-5 同心圓淺灘底床上的等位相線(含波峰線) 36
4-6 同心圓淺灘底床上沿波向線傳遞之折射係數 45
4-7 包絡線 53
4-8 包絡線後方之聚焦(caustic)現象 69
4-9 檢核 95
第五章 討論與建議 96
5-1 討論 96
5-2 建議 98
參考文獻 99
附錄 102
附錄1 沿波向線上對 分段處理的原因 102
附錄2 折射係數的相關討論 103
附錄3 詳細列出尖點條件的全部內容 110

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