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:

• Conservation of Mass:
  $$\begin{eqnarray*} \frac{\partial \rho}{\partial t}=-\frac{\partial (\rho u)}{\partial x} \end{eqnarray*}$$
  

• Conservation of Momentum:
  $$\begin{eqnarray*} \frac{\partial (\rho u)}{\partial t}=-c^{2}\frac{\partial \rho}{\partial x} \end{eqnarray*}$$
  

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:

$$\begin{eqnarray*} \frac{\partial i_{1}}{\partial t}=-\frac{\partial i_{2}}{\part... ...rtial i_{2}}{\partial t}=-c^{2}\frac{\partial i_{1}}{\partial x} \end{eqnarray*}$$

and finally:

$$\begin{eqnarray*} \frac{\partial i_{1}}{\partial t}+\frac{\partial i_{2}}{\parti... ...tial i_{2}}{\partial t}+c^{2}\frac{\partial i_{1}}{\partial x}=0 \end{eqnarray*}$$

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:

• After a coordinate change, and impedance transformations (to remove negative inductances), we are left with a circuit that can be discretized according to WDF principles outlined above.

• MD-WDF one-ports are simple generalizations of their 1-D equivalents, except that internal delays may include spatial steps as well.

• Scattering is still memoryless (KCL equations depend only on voltage and current, not on the independent variuables).