A Note on Upper Bounds for the Eigenvalues ofy″ + λpy=0

Abstract

The natural modes of a small planar transversal vibration of a fixed string of unit length and tension are determined by the eigenvalues and associated eigenfunctions of the differential equation(1)subject to the boundary condition(2)where the non-negative functionpdescribes the mass distribution of the string. That the distribution of mass on the string influences the modes of vibration, may be reflected by observing that the eigenvalues determined by the system (1–2) may be considered functions of the densityp, λn(p), where λ1(p)<λ2(p)<….