Analytic solutions of the mild-slope equation

• Main Authors: Yo-Ming Cheng, 鄭又銘
• Other Authors: Jaw-Fang Lee
• Format: Others
• Language: zh-TW
• Published: 2007
• Online Access: http://ndltd.ncl.edu.tw/handle/35591296983929181989

博士 === 國立成功大學 === 水利及海洋工程學系碩博士班 === 95 === 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.