非線性孤立波之數值解析及其動力特性探討

Abstract

由往昔地震引致海嘯資料顯示，尤其2004 年之印尼外海強烈的地震，能造成近十 萬人罹難及海岸極嚴重的災害，因南亞海嘯再被科學家及學者以撰書及論著來更解析 海嘯特性。解析海嘯傳遞過程多視海嘯為單一長波(long wave) 處理，或者以孤立波 (solitary wave)解析之。本計畫分析往昔研究成果發現仍有些問題如(1)有限振幅孤 立波級數解析之收斂問題。(2)以簡單之顯示公式表示小至大孤立波之數值結果。(3) 描述孤立波軌跡之方法及結果。(4)最大孤立波波峰分岐奇異點之夾角是90o 或120o， 及最大孤立波之整體波場解。 本計劃欲以2 年時間有系統分析探討符合精度且實用之非線性孤立波之數值解析，可 精準計算小至大孤立波之動力特性。第一年:探討有限振幅孤立波之解析，主要推導孤 立波基本解之tanh 形式(即tanh 孤立波)，分析低階孤立波之運動軌跡及漂移速度； 以數值collocation 方法計算極座標形式之孤立波解，解析小至大孤立波之動力特性， 並表示成顯示公式；第二年:探討近最大波及極限孤立波最大孤立波之數值解析。

Tsunamis induced by seabed earthquake can cause thousands of people death and coastal disasters. The vigorous Indian ocean tsunami happened in Dec. 2004 makes scientists and scholars to pay more attentions and researches on the mechanics and process of tsunamis. Tsunami is generally taken as a long wave or a solitary wave. From detailed review on previous works on tsunamis, some drawbacks in the methodology and results would be solved to accurately calculate the dynamic properties of tsunamis for coastal engineering. These are (1) convergence of series solution; (2) unavailable explicit expresses for dynamic properties of solitary waves; (3) unavailable description for particle trajectory of a solitary wave; (4) controversial issue of the crest angle of the highest solitary wave for the branch-point singularity, 90o or 120o, and unavailable description for whole highest solitary wave. The proposed two-year project is to investigate a numerical method with high accuracy to calculate the dynamic properties of a nonlinear solitary wave from drawf solitary wave to the extreme solitary wave. In first-year project, basic tanh-form solution for a solitary wave will be first derived for low-order perturbation approximation and high-order numerical approximation in the next-year work. Drift velocity and particle trajectory will be determined to estimate the movement of pollutants in the sea due to solitary waves. In the second-year project, basic expression for the solution in polar form will be transformed from the z-plane form to fit the collocation method for numerical computation. Dynamic properties of solitary waves from small-amplitude to the extreme waves will be computed and expressed by empirical explicit forms for engineering applications.