本文以演進型緩坡方程式 (evolution equation for mild-slope equation，EEMSE ) 之數值模式，將底床坡度展開至α2階，結合輻射邊界條件，推導出非線性演進型緩坡方程 (nonlinear evolution equation of mild-slope equation)，用以模擬波浪在斜坡上的波動效應與變形，其效應包括淺化、折射、繞射、反射、碎波與能量消散等。模式計算結果與 Guza和Bowen (1976) 以及楊等人 (2004) 的理論結果進行比較，用以驗證本文模式的正確性與適用性。
底床坡度為1/10、1/5及1/3之情況，本文計算結果與楊等人 (2004)之理論結果有相同的趨勢，唯有在底床坡度為1/5與1/3時，深水和淺水海域兩者之波形相符，而中間性水深漸有差異，此差異隨著底床坡度變陡而變大。與 Guza 和Bowen (1976) 底床坡度為1/40以及1/10之波形比較發現，模式在α1階級數之數值計算結果與理論值相符合。

The evolution equation for mild-slope equation model is expended to second order in bottom slope to derive a nonlinear evolution equation of mild-slope equation. The nonlinear evolution equation of mild-slope equation model is used to simulate wave transformations such as shoaling, refraction, diffraction, reflection, wave breaking and energy dissipation.
In the condition of bottom slope on 1/10, 1/5 and 1/3, the proposed model can modify wave transformations under the sloping bottom by examining Yang’s (2004) theory and calculate the accurate results. The bottom slope on 1/40 and 1/10,
proposed model calculate the accurate results for wave phase with Guza and Bowen’s (1976) theory.

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