The Sitar String, a Vibrating String with a One-Sided Inelastic Constraint

Source abstract

The small transverse motion of a stretched string vibrating against a rigid, inelastic curved obstacle is calculated. This system models the vibration of the strings of certain Indian musical instruments, the sitar, the tanpura and the viva, where the bridge does not have a sharp edge but is smooth and forms a curved impenetrable “obstacle” around which the string wraps and unwraps during its vibration. The complete motion of the string plucked at its midpoint is calculated in closed form from the instant it is released to its asymptotic approach to equilibrium. A similarity is pointed out between the solution obtained here and the solution obtained by Helmholtz for the bowed violin string. The solution is illustrated graphically.