## ``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*

*reference impedance.*Since the above wave variables have dimensions of force, they are specifically

*force waves.*The corresponding

*velocity waves*are

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) givesWe 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

**Previous Section:**

Digital Waveguide Filters