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Eliciting harmonics on strings

Published online by Cambridge University Press:  18 January 2008

Steven J. Cox and
Antoine Henrot
Steven J. Cox
Affiliation:
Computational and Applied Mathematics, Rice University, Houston, TX, USA; cox@caam.rice.edu
Antoine Henrot
Affiliation:
Institut Élie Cartan, UMR 7502, Nancy Université - CNRS - INRIA, Nancy, France.
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Abstract

One may produce the qth harmonic of a string of length π by applying the 'correct touch' at the node $\pi/q$ during a simultaneous pluck or bow. This notion wasmade precise by a model of Bamberger, Rauch and Taylor. Their 'touch' isa damper of magnitude b concentrated at $\pi/q$ . The 'correct touch' is that b for which the modes, that do not vanishat $\pi/q$ , are maximally damped. We here examine the associated spectralproblem. We find the spectrum to be periodic and determined by a polynomialof degree $q-1$ . We establish lower and upper bounds on the spectral abscissaand show that the set of associated root vectors constitutes a Riesz basisand so identify 'correct touch' with the b that minimizes the spectral abscissa.

Keywords

Point-wise dampingspectral abscissaRiesz basis

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008, pp. 657 - 677
Copyright
© EDP Sciences, SMAI, 2008

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