於二度空間，解析斜坡坡度α之底床上前進的週期性規則表面重力波問題，本文乃基於陳(2003Ⅰ、Ⅱ)之攝動展開至εmαn階的非線性解析理論，ε表示波浪非線性階次量的參數（an ordering parameter）如Pierson(1962)所示，α表示底床坡度，擴展求解至εmαn(m+n=4)之波動流場的解析解，以闡述波動流場特性分別受到波浪尖銳度及底床坡度與其互制之影響，且引入試驗結果印證之，同時並說明此解析模式較前人研究更具完整適足性。在此非線性解析模式中，所考慮的波動流場除需滿足基本控制式，及其在固定不透水的斜坡底床（α≠0）與自由表面處的邊界條件外，同時忽略反射波而維持波浪週期不變下，對斜坡底床為界的波動水域內之各斷面處，於波浪週期平均下之質量通量的平衡，亦即波浪週期平均下各斷面處之定常性一併考慮處理。藉此，則本文之非線性解析模式，將逐次地列出至εmαn階的所有控制式，以展現波浪非線性量與底床坡度對平緩斜坡上之波浪變化特性之描述。

　　對於所欲解析的斜坡底床上前進的表面波動，其波動流場的物理量均以波浪非線性量參數ε與底床坡度α的雙冪級數形式表示之，並逐階求解至εmαεmαn(m+n=4)階解，其中至εiαj(i+j=3)的非線性解析解完全與陳(2003Ⅱ)所得者相同；而對本文所進一步求得至εmαn(m+n=4)階之解，在ε1α3階解中，得出本模式中尚未被定出的振幅影響係數e3；於ε2α2階解中，除包含時空變動項外，同時又含有往昔未被呈現出的受坡度所影響之波降量，且在此量的加入下與實驗值比較更為吻合，此顯示了往昔尚未被論及的波降與底床坡度之量化的密切關連；於ε3α1階解中，有坡度衍生出的時空變動項；在ε4α0階解中，則包含受波浪非線性量所影響之波降量與時空變動項及回流速度項。本文所得的非線性解於深海處不受底床的影響，與在斜坡坡度α=0時退化成等水深前進波至ε4α0之解，都能獲得具體的印証。

　　為更明確地描述流體質點的運動特性，本文再進一步將Eulerian 系統所得的非線性解轉換至 Lagrangian系統至ε2α1階的非線性解，並求得流體質點運動的參數化方程式。藉此參數化方程式，沿著波浪前進的方向，從深海至淺水受深海波浪尖銳度與底床坡度之影響，波浪形狀的變形及其整個流場的時空演變均被解析；尤其在接近碎波點，波形產生捲入(plunging)或溢出(spilling)之整個時空連續變形亦能清楚的顯示其詳細演變過程。本文由所得之Lagrangian 系統所分析出的相關波動特性，藉由繪製一系列的關係圖，可明確地瞭解這些物理量與深海波浪尖銳度、水深及底床坡度的關係。再者，由波形的斜率非對稱性參數的探討，可知其與底床坡度及深海波浪尖銳度有關。當相對水深越淺時，波形的斜率非對稱性變化越大。另外，底床坡度與深海波浪尖銳度對質量傳輸及波降的影響，在攝動展開下會出現在O(ε2α2)階次量，此乃前人之解析解中尚未展現出的特性。而本文所得之解在退化至有限水深時則與Longuet-Higgins(1953)解析非旋轉性前進波的質量傳輸速度結果一致。更值得一提的是，經由逐階求解非線性解所得之波降解η (即於碎波前之波浪週期平均水位下降)，可知 η 為深海波浪尖銳度λ0a0與底床坡度α的函數。而本文考慮底床坡度修正量之計算結果較Longuet -Higgins & Stewart(1962)之結果為佳，且與Saville(1961)之試驗值較為吻合。因此，對於碎波前之週期平均水位下降除與深海波浪尖銳度有關外，底床坡度之因素亦需加以考量，尤其在較陡之斜坡底床上，底床坡度影響項之效應也愈強。最後由本解析模式適用範圍之討論，結果顯示依流速勢函數之各階修正量的比值，取其不同的限制條件，則解析模式的適用範圍亦有所不同。

　　For wave propagating on sloping bottoms, this study extends the model of Chen(2003Ⅰ、Ⅱ) to analyze the influences of the wave steepness and the bottom slope. Based on the principle of wave motion with nonlinear free-surface boundary conditions, rigid bottom boundary condition and the conservation of time-averaged mass flux, the nonlinear solution for the velocity potential is derived as a two-parameter function of the nonlinear ordering parameter ε and the bottom slope α perturbed to the fourth order (εmαn,m+n=4) in the Eulerian coordinate system.

　　The solutions up to εiαj(i+j=3) order are exactly the same as those of Chen (2003Ⅱ). For the higher order solutions obtained by this study, it is found that bottom slope α is related to the changes of wave elevations and wave profile in the ε1α3 order solution . The effect of bottom appears in the quantity of α3∫x x0e3dx´, which has not yet been obtained by Chen (2003Ⅱ). The velocity potential to ε2α2 order includes a second harmonic and a time–independent potential, all affected by the second-order bottom slope effect. The corresponding free surface elevation contains a new mean sea-level set-down α2η2,2,0 term due to the slope effect and a second harmonic η2,2,2. The velocity potential to the ε3α1 order includes a third harmonic term that depends on the bottom slope. Finally the velocity potential to ε4α0 order includes an oscillatory term, which is the form of potential of the Stokes fourth harmonic, and the time-independent potential. To check the validity of the nonlinear analytical solution, it is shown analytically that, in the limit of deep water or constant depth, the nonlinear solution approaches the well known classical Stokes fourth-order solution of progressive waves.

　　The transformation between Eulerian and Lagrangian coordinates is used to calculate the water particle motion up to second order. Up to now, it is known that the Eulerian solution of Stokes wave up to the third order cannot be transformed into the corresponding Lagrangian solution. It still need to make further research. The Lagrangian velocity in this paper is calculated only to the order of ε2α1. The displacement of the water particle X and Y to the ε2α1-order approximation can thus be written by integration. These enable the description of the features of wave shoaling in the direction of wave propagation from deep to shallow water and the deformation of a wave profile, particularly, the successive process of the wave plunging or spilling on a beach near the breaking point. From theoretical analysis, it is apparent that the asymmetry parameter is the function of the bottom slope and the square of incident wave steepnessλ0a0. The absolute value of the asymmetry parameter |s| increases with decreasing water depth for a given slope. In addition, for a given wave steepness, |s| also increases as the bottom slope α increases. From theoretical analysis, the mass transport velocity on the sloping bottom is proportional to the bottom slope and the incident wave steepness. In the limiting case where the water depth is constant and finite, the Stokes’ drift is reduced to that derived by Longuet-Higgins (1953). This study also obtained the wave-induced set-down in the shoaling zone, correct to the ε2α2 order, is a function of both the square of the bottom slope and the square of the incident wave steepness λ0a0. Comparing the theoretical set-down with the experimental data of Saville (1961), the present theory shows good agreement with the experimental results while the L-H/S theory tends to underestimate the set-down. Thus, it can be concluded that this averaged surface depression depends on the slope of the bed, and this dependence increases significantly for steep slopes. The present theoretical formulae developed will be applicable for cases of arbitrary bottom slopes. By examining the ratio of the high-order term to the low-order term, which must be less than 1 in order for the series for ψ, the region of validity of the present study can be obtained.

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