Energy bounds on point-wise damped wave operators
Hardesty, Sean S.
Master of Arts
One plays a harmonic by gently touching a finger to a vibrating string so as to divide it into two segments whose lengths form the ratio of small integers. The only prominent frequencies in the sound thereby produced correspond to those modes of the undamped string that have nodes at the contact point; the others are damped by the action of the finger. Bamberger, Rauch, and Taylor  modeled the phenomenon with point-wise damping and suggested that the "correct touch" (force applied by the finger) is that which causes the damped modes to decay most rapidly. Cox and Henrot  investigated the spectral properties of the associated operator, and identified the correct touch as that which minimizes its spectral abscissa. We give bounds on the total energy associated with the damped modes, and assess their utility in helping us understand the correct touch.