Eliciting harmonics on strings

Cox, Steven J.; Henrot, Antoine

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

One may produce the $q$th harmonic of a string of length $\pi$ 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.

Classification : 35P10, 35P15, 74K05, 74P10
Mots clés : point-wise damping, spectral abscissa, Riesz basis

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