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Reflectance of a General Lumped Waveguide Termination
Calculate the reflectance of the terminated waveguide.
That is, find the Laplace transform of the return wave divided by the
Laplace transform of the input wave going into the waveguide. In general,
the reflectance of an impedance step for force waves (voltage waves in
the electrical case) is
 |
(F.1) |
This is easily derived from continuity constraints across the
junction. Specifically, referring to Fig.
F.1b, let

denote the physical
force and its
traveling-wave
components within the ``pseudo-infinitesimal-generalized-waveguide''
defined by the element
impedance 
, with the `

' superscript
denoting a right-going wave.
F.1 Similarly, let

denote the
velocity and its component
wave variables on
the side of the junction at impedance

, and let

denote the corresponding quantities on the
element-side of the junction at impedance

. Again, the `

'
superscript denotes travel to the right. Then the physical continuity
constraints imply
By the definition of wave impedance in a waveguide, we have
Thus,
Defining
and
, we have
 |
(F.2) |
Now that we've solved for the junction force

, the outgoing
waves are simply obtained from the force continuity constraint,

:
Finally, the force-wave reflectance of an impedance step from

to

can be found by solving Eq.

(
F.3) and (
F.2) for

with

set to zero:
as claimed.
Previous: A Physical Derivation of Wave Digital ElementsNext: Reflectances of Elementary Impedances
About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at
Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See
http://ccrma.stanford.edu/~jos/ for details.