Numerical Analysis of Chladni figures

Square Region

• Masashi Katsurada and Yuuki Hirano, Finite difference equations for a biharmonic eigenvalue problem in a square region, the abstract of our talk in MSJ Autumn Meeting 2012 (2012/9/21). (in Japanese), a short abstract in English, presentation slides (in Japanese)
• nodal lines in the case Poisson's ratio=0.3

Reports by my students

• Yuuki Hirano, Eigenvalue problems of the biharmonic operator in a square region --- finite difference analysis of Chladni figures, master thesis of Meiji University, 2012 March. (in Japanese) ,
  MATLAB programs, MATLAB programs tarball, how to use programs
  Brief instruction: Download plate_f1.m and plot_n.m. Try

         N=160
         A=plate_f1(N,0.3);
         [v,d]=eigs(A,200,0);
         plot_n(v(:,201-4),N,N)
         plot_n(1.535*v(:,201-7)+v(:,201-8),N,N)
         

  N=160, Poisson's ratio=0.3, compute 200 eigenvalues and eigenfunctions, draw the fourth eigenfunction v4 and 1.535*v7+v8

Links

• Ernst Chladni in Wikipedia
• Martin J. Gander and Gerhard Wanner, From Euler, Ritz and Galerkin to modern computing.
• Martin J. Gander and Felix Kwok, Chladni figures and the Tacoma bridge: Motivating PDE eigenvalue problems via vibrating plates.

Links (in Japanese)

• Dancing sand patterns, an exhibition in the NAGOYA CITY CENTER MUSEUM