Chladni Figures and the Tacoma Bridge: Motivating PDE Eigenvalue Problems via Vibrating Plates

Teaching linear algebra routines for computing eigenvalues of a matrix can be well motivated to students using interesting examples. We propose in this paper to use vibrating plates for two reasons: first there are interesting applications, from which we chose the Chladni figures representing sand ornaments which form on a vibrating plate, and the Tacoma bridge, one of the most spectacular bridge failures. Second, the partial differential operator that arise from vibrating plates is the biharmonic operator, which one does not encounter often in a first course on numerical partial differential equations, and which is more challenging to discretize than the standard Laplacian seen in most textbooks. In addition, the history of vibrating plates is interesting, and we will show both spectral discretizations leading to small dense matrix eigenvalue problems, and a finite difference discretization, leading to large scale sparse matrix eigenvalue problems. Hence both the QR-algorithm and Lanczos can be well illustrated.