利用Sinc函數為基底分析二維本徵模態

Abstract

特徵值問題（Eigenvalve problem）普遍存在於自然界中，然而，在求解的過程中，只有少數情形能得到解析解，如何利用數值方法解出各種情形的的特徵值問題是一個重要的課題。在這篇論文中，我們利用特徵函數展開法，以Sinc函數為基底，分析物理系統中相當重要的特徵值問題：二維赫姆霍茲方程式（Helmholtz equation）。 首先為了證實數值方法的可行性，我們以此數值方法解出邊界為二維方形及圓形的赫姆霍茲方程式，並將模擬結果和解析解比較，發現在同定態下的圖形是相當接近的；比對特徵能量值（Energy），能量值誤差並不大。接著我們利用此數值方法解任意邊界的赫姆霍茲方程式，選取的邊界為金門形狀及小提琴形狀。我們更用此方法推廣研究振盪平板的節線圖騰，並推算微擾對節線圖騰的影響，比較數值模擬與實際震砂實驗的結果，兩者之間有良好的對應關係。

Eigenvalue problem has been widely existed in the nature. Unfortunately, due to the complexities of the physical system, analytical solutions to eigenvalue problem can be obtained only in few cases. Some numerical methods must be used to solve the eigenvalue problem. In this paper, we used expansion method based on Sinc function as basis to analyze the famous eigenvalue problem in physical system: 2D Helmholtz equation.

To consider the feasibility of numerical method, we numerically calculated the eigenstates and eigenenergy in rectangular and circular membrane and compared the numerical results with the analytical solutions. We also solved the two dimensional (2D) membranes problems with arbitrary shapes of Kinmen and violin by using the numerical method.

We further used the numerical method to simulate Chladni Nodal line pattern and investigate the influence of perturbation on Chladni Nodal line pattern. It can be seen that there is a good agreement between numerical results and experiment results.