Kinematics of Ideal String Vibration Against a Rigid Obstacle
This paper presents a kinematic time-stepping modeling approach of the ideal string vibration against a rigid obstacle. The problem is solved in a single vibration polarisation setting, where the string’s displacement is unilaterally constrained. The proposed numerically accurate approach is based on the d’Alembert formula. It is shown that in the presence of the obstacle the lossless string vibrates in two distinct vibration regimes. In the beginning of the nonlinear kinematic interaction between the vibrating string and the obstacle the string motion is aperiodic with constantly evolving spectrum. The duration of the aperiodic regime depends on the obstacle proximity, position, and geometry. During the aperiodic regime the fractional subharmonics related to the obstacle position may be generated. After relatively short-lasting aperiodic vibration the string vibration settles in the periodic regime. The main general effect of the obstacle on the string vibration manifests in the widening of the vibration spectra caused by transfer of fundamental mode energy to upper modes. The results presented in this paper can expand our understanding of timbre evolution of numerous stringed instruments, such as, the guitar, bray harp, tambura, veena, sitar, etc. The possible applications include, e.g., real-time sound synthesis of these instruments.