Examples and counter-examples for a string density formula in the case of a discontinuity

Abstract

In a recent paper, Pranger [l] proved a formula for the density function of a vibrating string with fixed ends in terms of its eigenvalues and their spatial rates of change (l.la) where s(x) = 1 [w;(x)] ~ ‘; n = 1, 2, . (l.lb) Here, w,,(L) are the eigenfrequencies of u” + wf,p(x)u = 0, OdsdL (1.2a) subject to u(O)=O=u(L). (1.2b) This formula demonstrates explicitly that the density function is deter- mined for a single set of boundary conditions by both the mode-number dependence of the eigenvalues and their functional dependence on length, in contrast to an earlier formula of Barcilon [2] which required spectral sequences for two suitable boundary configurations. As observed by Pranger [ 11, if p = 1 then o,(L)=nx/L and S(X) = (.u’/lr*) C( l/n*) = +u2/6, and one recovers p from (1.1). There is one continuous iso-spectral partner of the constant density case (as far as mode-number dependence is concerned) [3,4], viz. p(x) = (1+.X))” with o,(L) = (1 + L)(m/L), i.e., with different L-dependence. 363