``Traveling Waves'' in Lumped Systems

One of the topics in classical network theory is the reflection and transmission, or scattering formulation for lumped networks [33]. In this formulation, forces (voltages) and velocities (currents) are replaced by so-called wave variables

$$\begin{eqnarray*} f^{{+}}(t) &\isdef & \frac{f(t) + R_0v(t)}{2} \\ f^{{-}}(t) &\isdef & \frac{f(t) - R_0v(t)}{2} \end{eqnarray*}$$

where $R_0$ is an arbitrary reference impedance. Since the above wave variables have dimensions of force, they are specifically force waves. The corresponding velocity waves are

$$\begin{eqnarray*} v^{+}(t) &\isdef & \frac{1}{2}[v(t) + f(t)/R_0] \\ v^{-}(t) &\isdef & \frac{1}{2}[v(t) - f(t)/R_0] \end{eqnarray*}$$

Dropping the time argument, since it is always `(t)', we see that

$$\begin{eqnarray*} f^{{+}}&=& R_0v^{+}\\ f^{{-}}&=& -R_0v^{-} \end{eqnarray*}$$

and

$$\begin{eqnarray*} f &=& f^{{+}}+ f^{{-}}\\ v &=& v^{+}+ v^{-} \end{eqnarray*}$$

These are the basic relations for traveling waves in an ideal medium such as an ideal vibrating string [297]. Replacing force by pressure, we obtain the traveling-wave relations for acoustic tubes [275]. Using voltage and current gives elementary transmission line theory.