Vol. 20, No. 3, 1967

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ISSN: 0030-8730
Lower bounds for the eigenvalues of a vibrating string whose density satisfies a Lipschitz condition

Dallas O. Banks

Vol. 20 (1967), No. 3, 393–410
Abstract

If a string has a density given by a nonnegative integrable function ρ defined on the interval [0,a] and is fixed at its end points under unit tension, then the natural frequencies of vibration of the string are determined by the eigenvalues of the differential system

u′′ + λρ(x)u = 0, u(0) = u(a) = 0.
(1)

As is well known, the eigenvalues of (1) form a positive strictly increasing sequence of numbers which depend on the density ρ(x). We denote them accordingly by

0 < λ1[ρ] < λ2[ρ] < ⋅⋅⋅ < λn[ρ] < ⋅⋅⋅ .

In this paper we find lower bounds for these eigenvalues when the density ρ satisfies a Lipschitz condition with Lipschitz constant H and 0aρdx = M. The bounds will be in terms of M and H.

Mathematical Subject Classification
Primary: 34.30
Milestones
Received: 16 August 1965
Published: 1 March 1967
Authors
Dallas O. Banks