Computed eigenmodes of planar regions by Trefethen, L, Betcke, T

“…Recently developed numerical methods make possible the high-accuracy computation of eigenmodes of the Laplacian for a variety of "drums" in two dimensions. A number of computed examples are presented together with a discussion of their implications concerning bound and continuum states, isospectrality, symmetry and degeneracy, eigenvalue avoidance, resonance, localization, eigenvalue optimization, perturbation of eigenvalues and eigenvectors, and other matters. …”

Eigenvalues and Musical Instruments by Howle, V, Trefethen, L

“…This paper attempts to fill that gap by plotting eigenvalues for a guitar string, a flute, a clarinet, a kettledrum, and a musical bell. For the drum and the bell, simple idealized models have eigenvalues that are irrationally related, but as the actual instruments have evolved over the generations, the leading five or six eigenvalues have moved around the complex plane so that their relative positions are musically pleasing.…”