10.4 Optimizing transient finger damping on a driven string

Introduction

The topic of vibrating strings has long been discussed by the great minds of the Enlightenment. The study of the harmonic overtones of strings, however, seemed to have been neglected for over a hundred years. The ideas of D'Alembert, Rameau, and Rayleigh pertaining to the production of these overtones, or partials, have only recently been analyzed mathematically by Bamberger et al. [1]. We will take their lead and proceed with a string of length $\ell$ fixed at both ends. The harmonic modes can be coaxed by pressing lightly on the string with the finger at $x_c=\frac{1}{4}\ell$ and driving the string with frictional forces of a bow at $x_b=\frac{5}{8}\ell .$ The preceding will produce the fourth mode of vibration. In our formulation we assume the string to have constant linear density and uniform tension. We only consider vertical displacements in the string and assume that these displacements are small. Our goal is to achieve the best sound by optimizing the damping coefficient $c\left(t\right)$ in the following one dimensional wave equation

that induces the purest waveform–one that best resembles a sine curve. The displacement $u$ depends both on time and space in the $x$ direction. Here $\rho$ is the linear mass density, $\tau$ represents tension, $b\left(t\right)$ is the driving force simulating bow pressure, and $c\left(t\right)$ is the damping coefficient we are interested in. The $\delta$ functions are present to simulate a pointwise footprint at $x_c$ and $x_b$ . More precisely,

where $\delta \left(x\right)=0$ for all $x\ne 0$ . At $x=0$ , $\delta \left(x\right)$ is infinitely large, but for numerical purposes, we will set this to the reciprocal of our spacial step increment.

Finite difference methods

The first method of the two finite difference methods used to solve the wave equation is the forward Euler method, in which (1) is solved incrementally through both time and space given the following initial conditions

In order to solve this equation for $u$ we must approximate the partial derivatives $u_{tt}$ , $u_{xx}$ , and $u_t$ . We first approximate $u_t$ by taking the slope of $u(x,t)$ with respect to time using time step $dt$ . When approximating $u_{tt}$ , we use a similar process, where the slope of $u_t$ is taken with respect to time. The time step $dt$ must be squared in order to account for the process of taking two derivatives. The same process is performed to approximate $u_{xx}$ , where the derivative of $u$ is evaluated twice with respect to space using the space step $dx$ . The results of our approximation process are as follows: