重力波之Lagrangian解析及其與Eulerian近似解之相互轉換

Abstract

本文應用Eulerian與Lagrangian座標系統間的轉換關係，推導出在Lagrangian座標下均勻水深上規則前進重力波之控制方程式，並利用攝動展開技巧解析獲得五階解。本文所得Lagrangian近似解的質點運動週波率包含二部份，一部份為含深度 的 函數，可表現質點運動的週波率隨深度增加而增加的特性；另一部份為不隨空間中任何一個位置點而改變的常數。本文進一步比較Eulerian及Lagrangian座標的波浪運動週波率的關係，發現本文的Lagrangian近似解的質點運動週波率可轉換至Eulerian座標下固定點觀測波動的週波率。由解析結果可知，Lagrangian系統中奇數階近似解主要進行週波率修正，偶數階近似解主要進行高程修正，且由結果中可知，本文近似解可求得平均靜水位上的波壓。本文的Lagrangian近似解可描述流體質點運動軌跡於波動一週期後，不會封閉而有微量前進的質量傳輸量，且可表現上層質點運動軌跡的位移範圍較下層者大的特性。 本文接著探討Lagrangian與Eulerian兩種描述方式間的轉換關係。在相同攝動參數的選定下，本文探討本文提出的Lagrangian五階解與Fenton (1985)Eulerian五階解之間的轉換方法及結果。本文透過連續泰勒展開的方式，成功的將Lagrangian五階解轉換至Fenton (1985)的Eulerian五階解。在Eulerian解轉換至Lagrangian解方面，往昔學者在進行Eulerian三階解轉換至Lagrangian三階解時，會產生含時間的不合理結果，本文主要考慮Lagrangian週波率與Eulerian週波率之不同，首先提出對週波率進行轉換的概念，透過泰勒展開，成功的將Fenton (1985)的Eulerian五階速度轉換至Lagrangian五階流體質點運動軌跡，解決往昔學者無法將三階Eulerian解轉換至Lagrangian解的問題。由本文轉換結果可知，針對同一個波浪現象的兩種描述方式可以互相轉換。 最後，本文探討Lagrangian與Eulerian兩種描述方式動力特性間的關係。本文首先由動力特性的基本定義，獲得Eulerian系統的動力特性至第四階解，並符合Longuet-Higgins (1975)動力特性間的關係。本文同時探討有限振幅波與微小振幅波能量傳遞速度的關係，發現波浪非線性交互作用會增加波浪能量傳遞速度，且較微小振幅波理論高。本文接著透過Jacobian轉換，求得Lagrangian系統的動力特性至第四階解，並將此結果與Eulerian系統的動力特性比對，而有Lagrangian與Eulerian系統的動力特性公式相同。本文主要的學術貢獻為推導出Lagrangian五階近似解，並證明Eulerian與Lagrangian兩種解析結果可以互相轉換，且其波浪的動力特性皆相同。

A set of governing equations in Lagrangian form transformed from the Eulerian system was derived for the propagating gravity waves in water of uniform depth in this paper. The technique of Lindstedt-Poincaré perturbation method was used to obtain the approximations up to fifth-order for these nonlinear governing equations. The Lagrangian frequency of the present solution consists of two parts. One part is a function of depth and increases as the water depth increases. The other part is constant for all particles and equivalent with that of Stokes’ wave theory of third-order. The present Lagrangian solution for the particle motion shows that the water particle has a drift displacement of second order, which decreases exponentially with water depth, after a period of time. The pressure of the present solution has zero value at the free surface and this exactly satisfies the dynamic boundary condition. This study investigates two-way transformations between the Eulerian and Lagrangian solutions for gravity waves propagating on a uniform depth. The Eulerian and Lagrangian fifth-order approximations with the same perturbation parameter are used to examine such possibility. Up to now, it is known that the Eulerian solution of Stokes waves up to the third order cannot be transformed into the corresponding Lagrangian solution. A key to resolve this problem is to recognize the fact that the Lagrangian frequency is not constant with water depth. With this correction, two-way transformations between the Lagrangian and Eulerian solution are shown to be possible. The dynamic properties of regular gravity waves are obtained using both Eulerian and Lagrangian approximations, based on the fundamental definition. Both obtained results are identical and satisfy some possible relationships between these dynamic properties. The energy transport velocity of high waves are demonstrated to be greater than that of low waves which can be also represented by the group velocity. The main academic contributions of the paper are that the Lagrangian approximations to the gravity wave are systematically derived, and two-way transformations between the Lagrangian and Eulerian governing equations and solutions are successfully obtained, and high order dynamic properties are first obtained.