Numerical investigation into sound synthesis by physical modeling

Abstract

A linear and a non-linear partial di erential equation is used as a phenomenological model to describe the synthesis of a vibrating string. The models are numerically simulated using nite di erence method and an exact solution of the linear partial di erential equation is derived using a functional transformation method. The functional transformation method is based on the transformation to the frequency domain in time and in space by applying the Laplace transformation and the Finite Sine transformation. The string is excited through the initial conditions, initial displacement and an initial velocity, of the system. Another excitation mode that is considered is the force density, where the initial conditions are set to zero. The force density is added as a Source term to the partial di erential equation and the initial excitation enters the system by the use of this term. To analyse the model di erent observation positions are considered, the observation points are the positions of the string where the vibrations are observed. The exact solution is used to verify the Finite di erence approximation and a spectrogram analysis of the di erent excitation points is used to asses the methods. The nite di erence method is then used to analysis the vibrations of the string. The nonlinear model include the e ects of tension modulation, which is de ned by the Kirchho -Carrier equation. The results are analysed using a quantitative analysis and a Fourier analysis.