Mode-locking and improved harmonicity for real strings vibrating in the presence of a curved obstacle

Abstract

We explore the dominant mechanism leading to mode-locking in real strings used in plucked stringed instruments like sitar and veena which are reported to have large number of nearly harmonic overtones. Strings used in most musical instruments have flexural rigidity and/or dissipation of energy and/or nonrigid end supports which generate inharmonic overtones. However, instruments like sitar and veena have a finite-sized bridge over which the strings wrap and unwrap during their vibrations. This wrapping generates quadratic nonlinearities in the system which facilitates locking of the higher modes to the fundamental mode, thereby countering the inharmonicity. To account for the flexural rigidity/nonrigid supports and dissipation, we introduce a frequency detuning and damping in the discretized modal equations. Method of multiple scales is used to obtain the equations governing the evolution of the modal amplitudes and frequencies. These equations are then exploited to obtain conditions for harmonic mode-locking of periodic solutions. We obtain all possible branches of harmonically resonant solutions and discuss their nature for two- and three-mode system. It has been shown that the inharmonicity introduced by different sources in a real string can be countered by mode-locking facilitated by the presence of a finite bridge in musical instruments like sitar. Analytical expression for mode-locked periodic solutions and the corresponding basins of attraction are obtained for the two-mode undamped system. For the damped system, the amplitudes are observed to decay such that the frequencies oscillate around the mean harmonic frequency. We also observe possibly chaotic modulations around the harmonically mode-locked solutions for the three-mode system.