緩坡方程式已普遍應用於近岸波場數值計算，然而由於係數為超越函數之故，其解析解較少被提出。本文提出Berkhoff(1972)緩坡方程式的解析解，考慮求解問題有二：一為波浪斜向入射平面變水深地形，一為波浪通過變水深淺灘上方圓柱型島嶼。針對第一個問題以複變函數表示斜向波浪運動效應而將二維平面問題簡化一個維度，隨後對控制方程式中超越函數係數進行Taylor展開使成為羃級數形式，並應用Frobenius級數解法配合邊界條件得到解析解。針對第二個問題本文將求解領域分為外圍等水深開放性海域及內圍淺灘上方海域。外圍等水深海域控制方程式為Helmholtz方程式，解的形式早已被求出；內圍變水深淺灘海域則以Berkhoff(1972)的緩坡方程式為求解對象，待控制方程式之超越函數係數被展開為Taylor級數後利用Frobenius級數解法配合邊界條件求得解析解。經由和數值解以及其它解析解的比較顯示本文解析解可得到正確結果。目前為止緩坡方程式解析解大多受限於淺水波條件或特定水深地形，本文方法則可求解大範圍週期波浪入射各種簡單變水深地形之散射波場，應用上受到較少限制。

Mild-slope equations have been widely applied to calculate linear wave fields in engineering, mainly solved by using numerical methods. Analytic solutions of the mild-slope equations are rarely proposed, however, partially due to the existence of transcendental coefficients in the equations. In this study, analytic solutions of Berkhoff’s(1972) mild-slope equation are presented concerning two different sets of boundary-value problem which are often taken as verifying subjects for numerical models of the mild-slope equations. Dimensions for the problem of wave obliquely incident to plane topography of varying water depth is reduced to one by assuming the appearance of spatially periodic wave motion in lateral direction. The transcendental coefficients in the governing equation are then expanded into series forms, and the Frobenius method is applied to yield an analytic solution. In dealing with the problem of wave scattering around a circular island mounted on a submerged shoal of varying water depth, wave field in the open sea of constant water depth is governed by the Helmholtz equation, while the mild-slope equation is solved in the region of varying water depth between the island and open sea. The transcendental coefficients in governing equation are expanded into Taylor series to allow the usage of the Frobenius method, as proceded in the first problem, in order to obtain an analytic solution. The present method is proved as valid while comparing to numerical and other analytic solutions. While previous analytic solutions for the mild-slope equations are mostly restricted to shallow water wave condition and specific topographies of varying water depth, the present method may be regarded as a more general tool in seeking analytic solutions of the mild-slope equation, as no prior assumption is set on wave condition and topography to the boundary-value problems in consideration.

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