Role of curvatures in determining the characteristics of a string vibrating against a doubly curved obstacle

The motion of a string in the presence of a doubly curved obstacle is investigated. A mathematical model has been developed for a general shape of the obstacle. However, detailed analysis has been performed for a shape relevant to the Indian stringed musical instruments like Tanpura and Sitar. In particular, we explore the effect of obstacle's curvature in the plane perpendicular to the string axis on its motion. This geometrical feature of the obstacle introduces a coupling between motions in mutually perpendicular directions over and above the coupling due to the stretching nonlinearity. We find that only one planar motion is possible for our system. Small amplitude planar motions are stable to perturbations in the perpendicular direction resulting in non-whirling motions while large amplitude oscillations lead to whirling motions. The critical amplitude of oscillations, across which there is a transition in the qualitative behavior of the non-planar trajectories, is determined using Floquet theory. Our analysis reveals that a small obstacle curvature in a direction perpendicular to the string axis leads to a considerable reduction in the critical amplitudes required for initiation of whirling motions. Hence, this obstacle curvature has a destabilizing effect on the planar motions in contrast to the curvature along the string axis which stabilizes planar motions.