An Inverse Eigenvalue Problem for a Vibrating String with Two Dirichlet Spectra

Handle URI:
http://hdl.handle.net/10754/597535
Title:
An Inverse Eigenvalue Problem for a Vibrating String with Two Dirichlet Spectra
Authors:
Rundell, William; Sacks, Paul
Abstract:
A classical inverse problem is "can you hear the density of a string clamped at both ends?" The mathematical model gives rise to an inverse Sturm-Liouville problem for the unknown density ñ, and it is well known that the answer is negative: the Dirichlet spectrum from the clamped end-point conditions is insufficient. There are many known ways to add additional information to gain a positive answer, and these include changing one of the boundary conditions and recomputing the spectrum or giving the energy in each eigenmode-the so-called norming constants. We make the assumption that neither of these changes are possible. Instead we will add known mass-densities to the string in a way we can prescribe and remeasure the Dirichlet spectrum. We will not be able to answer the uniqueness question in its most general form, but will give some insight to what "added masses" should be chosen and how this can lead to a reconstruction of the original string density. © 2013 Society for Industrial and Applied Mathematics.
Citation:
Rundell W, Sacks P (2013) An Inverse Eigenvalue Problem for a Vibrating String with Two Dirichlet Spectra. SIAM Journal on Applied Mathematics 73: 1020–1037. Available: http://dx.doi.org/10.1137/120896426.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Applied Mathematics
KAUST Grant Number:
KUS-CI-016-04
Issue Date:
23-Apr-2013
DOI:
10.1137/120896426
Type:
Article
ISSN:
0036-1399; 1095-712X
Sponsors:
This research was supported by the National Science Foundation under grant DMS-0715060 and by KAUST award KUS-CI-016-04.
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorRundell, Williamen
dc.contributor.authorSacks, Paulen
dc.date.accessioned2016-02-25T12:41:37Zen
dc.date.available2016-02-25T12:41:37Zen
dc.date.issued2013-04-23en
dc.identifier.citationRundell W, Sacks P (2013) An Inverse Eigenvalue Problem for a Vibrating String with Two Dirichlet Spectra. SIAM Journal on Applied Mathematics 73: 1020–1037. Available: http://dx.doi.org/10.1137/120896426.en
dc.identifier.issn0036-1399en
dc.identifier.issn1095-712Xen
dc.identifier.doi10.1137/120896426en
dc.identifier.urihttp://hdl.handle.net/10754/597535en
dc.description.abstractA classical inverse problem is "can you hear the density of a string clamped at both ends?" The mathematical model gives rise to an inverse Sturm-Liouville problem for the unknown density ñ, and it is well known that the answer is negative: the Dirichlet spectrum from the clamped end-point conditions is insufficient. There are many known ways to add additional information to gain a positive answer, and these include changing one of the boundary conditions and recomputing the spectrum or giving the energy in each eigenmode-the so-called norming constants. We make the assumption that neither of these changes are possible. Instead we will add known mass-densities to the string in a way we can prescribe and remeasure the Dirichlet spectrum. We will not be able to answer the uniqueness question in its most general form, but will give some insight to what "added masses" should be chosen and how this can lead to a reconstruction of the original string density. © 2013 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipThis research was supported by the National Science Foundation under grant DMS-0715060 and by KAUST award KUS-CI-016-04.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectInverse eigenvalue problemen
dc.titleAn Inverse Eigenvalue Problem for a Vibrating String with Two Dirichlet Spectraen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Applied Mathematicsen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionIowa State University, Ames, United Statesen
kaust.grant.numberKUS-CI-016-04en
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