Java Appletsによる振動と波動の可視化  [in Japanese] Simulations of Classical Vibrations and Waves in terms of the Java Applets  [in Japanese]

理系基礎としての古典力学の中から連続体の振動と波動をとりあげ,基準振動,固有振動モードとともに差分による数値解法にも焦点をあてた教材としてJava Appletsを開発した.具体的に,3質点とばねからなる連成振動の数値解法やノーマルモードとともに1次元の弦の振動・波動のFourier級数展開法や数値計算法を取り上げた.そこで,Fourier展開の有限項近似による誤差や数値積分の不安定条件なども取りあげた. また, 波動方程式のダランベール解も取り上げるとともに,両端が固定された弦の振動も方向の異なる2つの波動の重ね合わせで表現されることを示した。2次元膜の振動についても,同様なFourier級数展開や数値積分による解法を議論した. これらのJava Appletsを使った可視化により,連続媒体の振動と波動の現象を統一的に波動方程式を通して理解することができるようになると期待される.

We have developed some Java applets to simulate the motion of three coupled oscillators, and the vibrational displacements for a string and a membrane: The motion of three coupled-oscillators is briefly studied by employing numerical methods of the Euler and the Runge-Kutta methods, which are compared with the normal mode methods. The vibration of a string is also studied by employing two methods, i.e. the Fourier expansion and the numerical integration in solving the wave equation. The convergence of the Fourier series and the numerical instability condition are studied. As demonstrated by the d'Alembert solution, the vibration of a string bounded on both ends are well described as the superposition of the waves propagating to the opposite ways. The vibrational displacement for 2-dimensional case of a square membrane is studied in the same way as in the 1-dimensional case. These applets may facilitate students to learn the basics of vibrations and waves on an equal basis of solving the wave equation.