Simulations of string vibrations with boundary conditions of third kind using the functional transformation method

The functional transformation method (FTM) is an established mathematical method for accurate simulation of multidimensional physical systems from various fields of science, including optics, heat and mass transfer, electrical engineering, and acoustics. It is a frequency-domain method based on the decomposition into eigenvectors and eigenfrequencies of the underlying physical problem. In this article, the FTM is applied to real-time simulations of vibrating strings which are ideally fixed at one end while the fixing at the other end is modeled by a frequency-dependent input impedance. Thus, boundary conditions of third kind are applied to the model at the end fixed with the input impedance. It is shown that accurate and stable simulations are achieved with nearly the same computational cost as with strings ideally fixed at both ends.