The eigenvalue ratio for a class of densities

We investigate the nature of the eigenvalues for vibrating strings with the density functionρ=ρ(x,t)={−xif −1≤x≤0txif 0≤x≤1 where t>0t>0. The n  th eigenvalue λn(t)λn(t) has a monotonicity property when t   is changed. By means of Bessel functions, we obtain the limits of λn(t)λn(t) as t→0t→0 and as t→∞t→∞. We also prove that the minimum of the ratio λ2(t)/λ1(t)λ2(t)/λ1(t) for t>0t>0 occurs at t=1t=1.