Computational Improvements to a Vibrating String Simulation

Jim Roberts & Jesse Davis

I. Introduction

Our objective is to create a vibrating string simulation which has intuitive appeal both from a scientific perspective and for the purpose a simple user interface. This means that our existing program must be organized so that the input parameters have a simple physical significance, and also have simple arithmetic relationships to a set of orthogonal, dimensionless computational parameters. Our simulation begins by mapping the fundamental physical properties of the system to a stable numerical solution. Regrouping allows for a dimensional reduction which uncouples the parameters of the physical system and makes for an easy computational analysis. The computational parameters are then mapped back to closely related physical parameters which have an intuitive range of values, and which can be input from an initialization file.

II. Physics Background

We are currently working with the standard acoustic model for a stiff, lossy string. The model includes terms for tension and bending forces, and also a two term approximation for energy dissipation. The following PDE describes our system:

here c is the ideal wave speed, E is Young’s modulus, S is the cross-sectional area, and K is the radius of gyration. This second term determines the dispersion due to stiffness and bending forces in the string. In the third term, b₁ is the first order linear approximation for energy dissipation, and b₃ is the second order approximation term, which damps vibrational eigenstates with linear dependence on the energy of the eigenstate.

III. Computational Method and Parameter Reduction

Our next step is to convert this equation into Finite Difference form, which is essential Euler’s method of solution to the ordinary differential equations, modified slightly for implementation in our partial differential equation solution. The method simple involves an approximation of slope that is recursively substituted for successive derivatives. For example, the following approximations were made for our discretization:

The same approximations can be made for spatial derivatives. Substitution into equation [1], followed by a lot of tedious algebra and regrouping, yields the following relation:

where

is our stiffness parameter,

is our sampling parameter,

is our linear damping parameter, and

is our frequency dependant damping parameter.

This equation relates the future state of the system to known states, and includes only dimensionless parameters. Because these parameters are dimensionless, their impact is easily understood according to their magnitude relative to unity. Our set of parameters can also be easily mapped back to an intuitive set of user inputs. The sampling parameter can be eliminated from the parameter set by a simple stability relation:

This requires that the sampling rate be at least as fast as the fastest wave speed, which is determined by the amount of stiffness or dispersion. Leaving initial conditions aside for the moment, the only other required parameter is the number of spatial steps, which is related to the length of the string, and thus its fundamental frequency. We prefer to have the frequency initialized by the user, rather than the number of steps, because we feel it is a more physically significant value:

This final relation reduces our parameter set to the following user-defined values: m, b_1, b_3, f, and f_s, the sampling frequency. We have shown that this set represents the fundamental characteristic parameters for the problem.