本文應用Chen等人 (2005；2006) 使用雙參數攝動法 (two-parameter perturbation method) 所求解之三階勢能函數，從中得到其所對應之水深函數，代入 Laplace 方程式中進行底部至水面之積分，得到新型式之高階緩坡方程式 (higher-order mild-slope equation)，改良現有緩坡方程式線性波的限制。本文之非線性效應包含波浪尖銳度 (wave steepness) 和底床坡度，文中推導之階量為三階，即 ， ， ，式中 為波浪尖銳度， 為週波數， 為波浪振幅， 為底床坡度。本文同時遵循前人求解緩坡方程式之方法，在考慮波浪變形之方程式型式 (包含前進波和反射波)，將此高階緩坡方程式以演進型態緩坡方程式 (Evolution Equation of Mild-Slope Equation, EEMSE) 求解偏微分方程式，若僅考慮入射前進波變形時，則將此高階緩坡方程式轉換成拋物線型態緩坡方程式 (Parabolic Mild-Slope Equation, PMSE)，忽略反射波之變形。
本文所發展之數值模式考慮底床坡度效應，經由本文模式計算波浪通過等斜坡底床地形之結果與前人研究 (Guza 和Bowen, 1976；Chen等人, 2005、2006) 成果比較發現本文模式可以合理有效地反應波浪受到斜坡底床之影響所衍生之波形變化。文中並應用本文模式針對波浪通過不同坡度之波形變化進行探討，分析各階次對線性波波高之影響量。在 階次部分，隨著相對水深變淺，在中間性水深與淺水波情況下，發現有明顯的線性波波高影響量存在，在相對水深 =1.3 左右時，其波高影響量達到最大值，在 階次部分，當相對水深 為深水波與中間性水深情況下，其波高影響量為負修正量，在相對水深 =2.5 左右時，其波高影響量最大，但隨著相對水深繼續變淺，其波高影響量會逐漸變小，並在相對水深 =1.0 左右時，其波高影響量會由負修正量轉變成正修正量，在 階次部分，相對水深 對波高影響量並不大。本文同時模擬波浪通過橢圓形淺灘底床、半圓形斜坡底床地形與潛堤底床地形，模式計算結果與實驗資料相比較呈現合理之一致性。

In this thesis a higher-order mild-slope equation (HOMSE) was derived in order to investigate wave transformations of waves propagating over the sloping bottom more realistically. A new depth function derived by Chen et al. (2005; 2006), which includes the nonlinearity and the bottom slope , was introduced in the depth averaged integral equation, where is the wave steepness, is the wavenumber, and is the wave amplitude. A new depth-integrated mild-slope equation was thus derived using the above mentioned depth function perturbed to the third-order. To model a time-harmonic motion of water waves in varying water depth, both evolution equation of mild-slope equation (EEMSE) and parabolic mild-slope equation (PMSE) were used. The former consists of progressive and reflected waves, while the latter only comprises progressive waves.
The numerical models were developed by the finite difference scheme in corporate with alternative direct implicit (ADI) method, it has the advantage to save computer storage and time consuming when the model is applied in a large coastal area. The validity of the model was examined by typical examples of wave travelling over a sloping bottom. The results reveal that the present model are in good agreement with the theory of Guza and Bowen (1976) and Chen et al. (2005; 2006). The impact of each nonlinear term in the accuracy of the wave height was addressed. The numerical results show that the influence of the nonlinearity depends on the relative water depth . The effect of nonlinearity on wave height variation increases first to and decreases with decreasing. For , it increases until and then decreases with the decrease of . The influence of is not significant. It is concluded that the effect of bottom slope and nonlinearity on wave height may be noticeable.
The present model is applied to predict wave transformations due to an elliptical shoal and s loping bottom with circular iso-depths. Numerical results are in good agreement with laboratory observations. The implementation of HOMSE shows that the present model is capable of describing waves travelling over an arbitrary topography.

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