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spectral abscissa

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Eliciting harmonics on strings
Steven J. Cox, Antoine Henrot
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 14 / Issue 4 / October 2008

Published online by Cambridge University Press:

18 January 2008, pp. 657-677

Print publication:

October 2008
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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.