本文利用流函數法推導緩坡方程式，發展高階流函數緩坡方程式，用以改善傳統勢能理論 (potential theory) 在底床邊界微分發生奇異點之限制問題。本文以連續方程式及能量守恆原理，利用變分法推導流函數之高階緩坡方程式，將水深積分從底部積分至自由液面，得到含有水位和流函數之高階緩坡方程式。當水位為零時，本文之方程式與前人所推導之流函數緩坡方程式一致。文中並以有限元素法建立二維數值模式，其中形狀函數使用二次方程式，以避免邊界條件二次微分為零之不合理現象。所發展之數值模式用來模擬波浪通過各種地形底床之變形。模式計算結果與前人勢能理論解析之緩坡方程式進行比較及驗證，得到合理的結果。本文分別針對波浪通過斜坡底床、單一潛堤及沙漣底床之計算條件，探討其適用範圍。本文之流函數緩坡方程式同時應用於波浪通過系列潛堤之地形底床，模擬波浪之布拉格共振 (Bragg resonance)，反射率和結果顯示的預測值與實驗數據，表現良好的一致性。

The mild-slope equation was derived using stream function method. A numerical model was developed by means of finite element method to simulate wave deformation over different types of sea bottom. Computational cases including Booij’s ramp, parabolic mound and sinusoidal bed were used to test the accuracy and valid range of the present model. The model is also applied to a series of submerged breakwaters for describing Bragg resonance of water waves. Comparison of the numerical results with experiments shows that the present model is in a good agreement with experimental data.

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