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Distributed Modeling Example: 1-D wave equation

Let's look at the continuity equations for the 1-D acoustic tube, in the linearized, adiabatic case:

Now, let us rename density $ \rho$ by a variable $ i_{1}$ , and momentum density $ \rho u$ by a variable $ i_{2}$ , which we will associate with currents:

\begin{eqnarray*}
\frac{\partial i_{1}}{\partial t}=-\frac{\partial i_{2}}{\partial x},
\frac{\partial i_{2}}{\partial t}=-c^{2}\frac{\partial i_{1}}{\partial x}
\end{eqnarray*}

and finally:

\begin{eqnarray*}
\frac{\partial i_{1}}{\partial t}+\frac{\partial i_{2}}{\partial x}=0,
\frac{\partial i_{2}}{\partial t}+c^{2}\frac{\partial i_{1}}{\partial x}=0
\end{eqnarray*}

Now each $ \frac{\partial}{dt}$ or $ \frac{\partial}{dx}$ applied to a ``current'' such as $ i_{1}$ or $ i_{2}$ might be thought of as some sort of voltage corresponding to a generalized inductor (i.e., temporal or spatial). This is merely a way of making the jump to the following circuit representation of the equations:

\epsfig{file=eps/MDcircuit.eps}


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``Wave Digital Filters'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-25 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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