Vibrational modes of circular free plates under tension

The vibrational frequencies of a plate under tension are given by the eigenvalues $\omega$ of the equation ${\mathrm{\Delta }}^{2}u-\tau \mathrm{\Delta }u=\omega u$. This paper determines the eigenfunctions and eigenvalues of this bi-Laplace problem on the ball under natural (free) boundary conditions. In particular, the fundamental modes — the eigenfunctions of the lowest nonzero eigenvalue — are identified and found to have simple angular dependence.