Vol. 29, No. 1, 1969

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 297: 1
Vol. 296: 1  2
Vol. 295: 1  2
Vol. 294: 1  2
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Some isoperimetric inequalities for the eigenvalues of vibrating strings

David Clarence Barnes

Vol. 29 (1969), No. 1, 43–61
Abstract

If a string with integrable density function p(x) is fixed at the points x = 0, x = a then the natural frequencies of vibration are determined by the eigenvalues of the Sturm-Liouville System

y′′ + λp(x)y = 0 y(0) = y(a) = 0.
(1)

These eigenvalues depend on the density function p(x) and we denote them accordingly by λn(p),

0 < λ1(p) < λ2(p) < ⋅⋅⋅ .

In this work we investigate the nature of the density functions which yield the largest and smallest possible value for λn(p) assuming that the average value of the density p(x) defined by

       1 ∫ x
P (x) = --  p(ζ)dζ
x  0

is restricted in some manner.

Mathematical Subject Classification
Primary: 34.30
Secondary: 35.00
Milestones
Received: 22 April 1968
Published: 1 April 1969
Authors
David Clarence Barnes