Application of the method of lines to the wave equation for simulating vibrating strings

This paper presents simulations of continuous dynamic models of vibrating strings for several different instruments: a guitar, a piano and a violin. The basis for all three models is the wave equation and is differentiated by the initial and boundary conditions imposed. The models are implemented as part of a Graphical User Interface. The simulations are presented as animations that are conveyed in the paper as a sequence of vibrating string profiles for different times. This paper promotes the idea that using music generation as a framework for teaching dynamics makes the teaching of dynamics a more engaging process. A case is made that there is value in solving the wave equation numerically using the method of lines rather than looking for analytical solutions. The reason for this is the mathematics of the method of lines is simpler to understand than the techniques used to derive analytical solutions to partial differential equations. The numerical approach is also more easily extended to a more sophisticated model for a vibrating string that might include energy dissipation and the stiffness properties of the string.