本文由相似轉換法之觀點,探討振動薄膜膜態之特徵值問題,並藉以設計出不同形狀之邊界和其對應薄膜密度之改變,而求得變密度薄膜膜態特徵值問題之相似確解,以供數值計算之參考和結果之核對。 本文就螺線、同心圓、偏心圓及上半平面缺圓等區域,而設計出其對應薄膜密度之改變,而求得變係數之亥姆霍茲(Helmholtz)方程式特徵值問題中,其特徵函數之相似確解及其對應之特徵值。 The vibrating membrane problem with variable density is discussed. Designing the weight function (density distribution of the membrane) of Helmholtz equation by the method of group invariance is exhibited. The exact similarity solutions of the eigenvalue problem of Helmholtz equation under the spiral and inversion groups are constructed in the regions bounded by concentric circles, eccentric circles, and a half-plane with circle removed. The eigenvalues of the problem are also obtained by the numerical and variational methods and they agree well with the exact solutions.
中華民國力學學會第二十屆全國力學會議論文集(二)= Proceedings of the 20th National Conference on Theoretical and Applied Mechanics (II)，頁108-115