Next  |  Prev  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Finite Difference Approximation (FDA)

Consider the simple differential equation relating velocity and force for an ideal mass:


\epsfig{file=eps/lmass.eps,width=6in}


\begin{eqnarray*} f(t)=m\frac{dv}{dt} \end{eqnarray*}

Finite Difference Approximation:

\begin{eqnarray*} \frac{dv}{dt} &\approx& \frac{v_n-v_{n-1}}{T} \qquad\quad\hb... ...{v_{n+1}-v_{n-1}}{2T} \qquad\hbox{(\lq\lq centered difference'')}\\ \end{eqnarray*}

E.g.,

$\displaystyle v_n=v_{n-1} + \frac{T}{m} f_n, \quad n=0,1,2,\ldots\,. $

(FDA for a force-driven mass)


Subsections
Next  |  Prev  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search
Download SMAC03S.pdf
Download SMAC03S_2up.pdf
``Recent Developments in Musical Sound Synthesis Based on a Physical Model'', by Julius O. Smith III, (Stockholm Musical Acoustics Conference (SMAC-03), August 6--9, 2003).
Copyright © 2006-02-19 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]