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From Lumped to Distributed Modeling

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

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

$\displaystyle \zbox{ Ky''= \epsilon {\ddot y}}

where, for transverse displacement $ y(t,x)$ , we have

K& \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \hbox{string tension} & \qquad y & \mathrel{\stackrel{\mathrm{\Delta}}{=}}& y(t,x) \nonumber \\ [10pt]
\epsilon & \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \hbox{linear mass density} & {\dot y}& \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{\partial}{\partial t}y(t,x) \nonumber \\ [10pt]
y & \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \hbox{transverse displacement} & y'& \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{\partial}{\partial x}y(t,x) \nonumber

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

Ky''&=& \mbox{\emph{vertical force density} on the element}\\
\epsilon {\ddot y}&=& \mbox{\emph{vertical inertial reaction force density} from the element}\\
&=& \mbox{\emph{mass-density} times \emph{vertical acceleration}}

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``Computational Acoustic Modeling with Digital Delay'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-03-29 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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