The linear and nonlinear vibrations of a stretched string and the relevance to stringed musical instruments

Abstract

For small amplitudes of string vibration, the modes of a stretched string are shown to be equivalent to the normal modes of a coupled oscillator problem, the coupled oscillators being the string itself and the vibrational modes of the body on which the string is supported. The coupling at the bridge splits the degenerate modes of the ideal string, with a degree of perturbation that depends on the strength of the coupling and the damping of the various modes involved. Under extreme conditions this can lead to a splitting of the modes resulting in the famous wolf-note on bowed instruments, which was first studied by Raman and later by Schelling. At large amplitudes, the nonlinear coupling of the orthogonal transverse modes will be shown to result in a precession in the plane of polarization of the now elliptically polarized transverse modes, which could lead to variations in amplitude of emitted radiation of any stringed instrument. In this talk, the theoretical predictions for both linear and nonlinear string vibrations will be briefly outlined and measurements will be described for idealized experimental situations and for instruments of the violin family.