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

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 : https://doi.org/10.1051/cocv:2008004
Classification : 35P10,  35P15,  74K05,  74P10
Mots clés : point-wise damping, spectral abscissa, Riesz basis
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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. http://www.numdam.org/articles/10.1051/cocv:2008004/

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