## ``Traveling Waves'' in Lumped Systems

One of the topics in classical network theory is the *reflection* and
*transmission,* or *scattering* formulation for lumped networks
[35]. Lumped scattering theory also serves as the starting point
for deriving *wave digital filters* (the subject of Appendix F).
In this formulation, forces (voltages) and velocities
(currents) are replaced by so-called *wave variables*

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

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

and

These are the basic relations for

*traveling waves*in an ideal medium such as an ideal vibrating string or acoustic tube. Using voltage and current gives elementary

*transmission line*theory.

### Reflectance of an Impedance

Let denote the driving-point impedance of an arbitrary
continuous-time LTI system. Then, by definition,
where and denote the Laplace transforms
of the applied force and resulting velocity, respectively.
The wave variable decomposition in Eq.(C.74) gives

We may call the

*reflectance*of impedance relative to . For example, if a transmission line with characteristic impedance were terminated in a lumped impedance , the reflection transfer function at the termination, as seen from the end of the transmission line, would be .

We are working with reflectance for *force waves.*
Using the elementary relations Eq.(C.73), *i.e.*,
and
, we immediately obtain the corresponding
*velocity-wave reflectance:*

**Next Section:**

Properties of Passive Impedances

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Digital Waveguide Filters