Geodesic
Eliciting harmonics on strings
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 657-677

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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.

DOI : 10.1051/cocv:2008004
Classification : 35P10, 35P15, 74K05, 74P10
Keywords: point-wise damping, spectral abscissa, Riesz basis 
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     title = {Eliciting harmonics on strings},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {657--677},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {4},
     year = {2008},
     doi = {10.1051/cocv:2008004},
     mrnumber = {2451789},
     zbl = {1154.35411},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2008004/}
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Cox, Steven J.; Henrot, Antoine. Eliciting harmonics on strings. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 657-677. doi: 10.1051/cocv:2008004

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