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From Lumped to Distributed Modeling

\epsfig{file=eps/SpringMassChain.eps,width=\textwidth }

As mass-spring1density approaches infinity, we obtain an ideal string, governed by ``wave equation'' PDEs such as

$\displaystyle \zbox{ Yd''= \rho\,\ddot{d}}

where, for longitudinal displacement $ d(t,x)$ , we have

Y & \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \hbox{Young's Modulus} & \qquad d & \mathrel{\stackrel{\mathrm{\Delta}}{=}}& d(t,x) \nonumber \\ [10pt]
\rho & \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \hbox{mass density} & \dot{d}& \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{\partial}{\partial t}d(t,x) \nonumber \\ [10pt]
d & \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \hbox{longitudinal displacement} & d'& \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{\partial}{\partial x}d(t,x) \nonumber

The wave equation is once again Newton's $ f=ma$ ,
but now for each differential string element:

Yd''&=& \mbox{\emph{force density} on the element}\\
\rho\ddot{d}&=& \mbox{\emph{inertial reaction force density}}\\
&=& \mbox{\emph{mass-density} times \emph{acceleration}}

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``Computational Acoustic Modeling with Digital Delay'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2020-02-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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