# Finite element modeling of dispersive strings

Chordophones form an important group among musical instruments. In their case sound is produced by the excitation of an intense string. This can be achieved in various ways such as bowing, strucking, or plucking. This thesis focuses mainly on the latter means of excitation. It is characteristic of chordophones that sound is not produced in a direct manner from the mechanical energy of the string, yet (except electrical instruments) the string excites the body of the instrument and the body becomes the primary source of sound radiation. The first step that must be taken to be able to phisically simulate this sound generation mechanism is therefore to calculate the vibration of the string.

Calculating the vibration shapes of real strings is a complex problem, an analytic solution can only be obtained in the ideal case, applying rough estimations and neglects. If the non-ideal attributes of real strings are to be taken into consideration, analytic methods have to be given up on, as by taking inhomogeneity, non-ideal terminations, dispersion, frequency dependent damping etc. into account at once, the resulting set of equations would be too cumbersome to solve in an analytic manner.

In numerical analysis, contrary to the analytic methodology, an approximate solution of the physical system is sought; hence the achievable accuracy depends only on the precision of the model and the computational capacity at hand. In this work the problem is examined by means of the finite element method, which is a numerical technique for solving boundary value problems of differential equations in a finite domain. These boundary value problems are to be solved in the time domain herein.

An important attribute of real stings is dispersion due to their finite stiffness. This means that vibration shapes are not purely transversal, but contain bending wave components as well. As a result of dispersion, wave velocity inside the string becomes frequency dependent, yielding inharmonic string modes. The goal of this thesis is the accurate modeling of string behavior taking dispersion into account.

First, the physical model of plucked strings and important aspects of their finite element representation are described. Then differences between the ideal and real string model are examined. Afterwards, the time-stepping solution of the boundary value problem is discussed in detail. Finally, simulation results of ideal and non-ideal strings are presented and compared against analytically calculated and measured data.