One may produce the th harmonic of a string of length by applying the ’correct touch’ at the node 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 concentrated at . The ’correct touch’ is that for which the modes, that do not vanish at , are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree . 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 that minimizes the spectral abscissa.
Keywords: point-wise damping, spectral abscissa, Riesz basis
@article{COCV_2008__14_4_657_0, author = {Cox, Steven J. and Henrot, Antoine}, 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://archive.numdam.org/articles/10.1051/cocv:2008004/} }
TY - JOUR AU - Cox, Steven J. AU - Henrot, Antoine TI - Eliciting harmonics on strings JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 657 EP - 677 VL - 14 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008004/ DO - 10.1051/cocv:2008004 LA - en ID - COCV_2008__14_4_657_0 ER -
%0 Journal Article %A Cox, Steven J. %A Henrot, Antoine %T Eliciting harmonics on strings %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 657-677 %V 14 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008004/ %R 10.1051/cocv:2008004 %G en %F COCV_2008__14_4_657_0
Cox, Steven J.; Henrot, Antoine. Eliciting harmonics on strings. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 4, pp. 657-677. doi : 10.1051/cocv:2008004. http://archive.numdam.org/articles/10.1051/cocv:2008004/
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