Modeling Vibrating String Wave Forms

Abstract

This project is aimed to construct a simplified, but justifiable model, for vibrating strings so that we are able to explore questions related to musical instruments. We begin with solutions to the 1-dimensional wave equation with fixed, and later, mixed boundary conditions given by Fourier coefficients. For the mixed boundary conditions, the eigen-values are unsolvable, so instead we find a numerical approximation for the solution. Then, we investigate the energy and distribution of energies of each solution. Finally, we construct a mathematical model for the musical performance technique, vibrato. The exact solution of this mathematical model is discussed by Gaffour in his papers ”Analytical Method for Solving the One Dimensional Wave Equation with Moving Boundary" [3] and “Vibrating String with a Variable Length" [7]. In our paper, we adopted an adiabatic approximation [15] in order to obtain an approximate solution with a simpler form than the exact solution given by Gaffour. Additionally, justification is provided that our adiabatic approximation is within good reason of the exact solution given by Gaffour.