臺灣博碩士論文加值系統

在物理和工程學裡，水波的繞射與散射是常見且重要的問題，這篇論文利用了格林函數和其相關方法，對這兩種問題做了分析和研究。在水波繞射問題上，海更斯原理常常被當作一個類比，卻沒有數學基礎，我們利用格林函數的方法推導出水波的狹縫繞射公式，解釋了海更斯原理，我們的方法和光學裡的克希荷夫繞射公式的推導是類似的。而對於散射問題，我們假設水底幾乎是平坦的，但具有很小的起伏，我們同樣利用格林函數的方法，並結合微擾理論，推得出水波被地面散射的公式，為了驗證這個公式，我們利用其他方法來計算一個特別的例子，而其計算結果與我們的公式是一致的。

Diffraction and scattering of water wave are common and important problems in physics and engineering. In this thesis, we use Green''s function method to analysis the two problems. For diffraction, we apply Green''s function method to derive the formula of slit diffraction to explain Huygens principle, which is usually just an analogy in water wave diffraction. We use the similar way that Kirchhoff derive his diffraction formula in optics. For scattering, we assume that the bottom of water is roughly flat with small variations. We use Green''s function combined with perturbation method and obtain the formula of the scattering problem. We also use matching method to calculate a special case and then we find that the result is consistent with our formula.

口試委員會審定書 i
誌謝 iii
摘要 v
Abstract vii
1 Introduction 1
2 Governing Eqautions 3
3 Flat Bottom Diffraction 7
3.1 Green’s Function Method . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Slit Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.1 The Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2.2 Exapmle of a Simple Diffraction Problem . . . . . . . . . . . . . 11
3.3 Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.1 Rewrite the Diffraction Formula . . . . . . . . . . . . . . . . . . 14
3.3.2 Approximation of The Integral . . . . . . . . . . . . . . . . . . . 16
3.3.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Uneven Bottom Scattering 21
4.1 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Scattering on Periodic Bottom . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Scattering on Any Topography of Bottom . . . . . . . . . . . . . . . . . 24
5 Conclusion 27
Appendices 29
A Expansion of Green’s function 29
B Pillbox Scattering 33
Bibliography 41

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