Eliciting harmonics on strings

Steven J. Cox; Antoine Henrot

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 4, page 657-677
  • ISSN: 1292-8119

Abstract

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

How to cite

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Cox, Steven J., and Henrot, Antoine. "Eliciting harmonics on strings." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 657-677. <http://eudml.org/doc/250319>.

@article{Cox2008,
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 was made precise by a model of Bamberger, Rauch and Taylor. Their 'touch' is a damper of magnitude b concentrated at $\pi/q$. The 'correct touch' is that b for which the modes, that do not vanish at $\pi/q$, are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree $q-1$. We establish lower and upper bounds on the spectral abscissa and show that the set of associated root vectors constitutes a Riesz basis and so identify 'correct touch' with the b that minimizes the spectral abscissa. },
author = {Cox, Steven J., Henrot, Antoine},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Point-wise damping; spectral abscissa; Riesz basis; pointwise damping},
language = {eng},
month = {1},
number = {4},
pages = {657-677},
publisher = {EDP Sciences},
title = {Eliciting harmonics on strings},
url = {http://eudml.org/doc/250319},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Cox, Steven J.
AU - Henrot, Antoine
TI - Eliciting harmonics on strings
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/1//
PB - EDP Sciences
VL - 14
IS - 4
SP - 657
EP - 677
AB - 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 was made precise by a model of Bamberger, Rauch and Taylor. Their 'touch' is a damper of magnitude b concentrated at $\pi/q$. The 'correct touch' is that b for which the modes, that do not vanish at $\pi/q$, are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree $q-1$. We establish lower and upper bounds on the spectral abscissa and show that the set of associated root vectors constitutes a Riesz basis and so identify 'correct touch' with the b that minimizes the spectral abscissa.
LA - eng
KW - Point-wise damping; spectral abscissa; Riesz basis; pointwise damping
UR - http://eudml.org/doc/250319
ER -

References

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