On the eigenvalue ratio for vibrating strings

Abstract

For vibrating strings with concave densities or symmetric single-barrier densities, the ratio λ 2 / λ 1 \lambda _2/\lambda _1 of the first two eigenvalues is minimized when the density is constant; while, for vibrating strings with symmetric single-well densities, the ratio λ 2 / λ 1 \lambda _2/\lambda _1 is maximized when the density is constant.