Simplified Impedance Analysis
The above results are quickly derived from the general
reflection-coefficient for force waves (or voltage waves, pressure
waves, etc.):
![$\displaystyle \zbox {\rho = \frac{R_2-R_1}{R_2+R_1} = \frac{\mbox{Impedance Step}}{\mbox{Impedance Sum}}} \protect$](http://www.dsprelated.com/josimages_new/pasp/img2118.png) |
(10.17) |
where
![$ \rho$](http://www.dsprelated.com/josimages_new/pasp/img1197.png)
is the
reflection coefficient of
impedance ![$ R_2$](http://www.dsprelated.com/josimages_new/pasp/img35.png)
as
``seen'' from impedance
![$ R_1$](http://www.dsprelated.com/josimages_new/pasp/img34.png)
. If a
force wave
![$ f^{{+}}$](http://www.dsprelated.com/josimages_new/pasp/img2119.png)
traveling along in impedance
![$ R_1$](http://www.dsprelated.com/josimages_new/pasp/img34.png)
suddenly hits a new impedance
![$ R_2$](http://www.dsprelated.com/josimages_new/pasp/img35.png)
, the wave will split into a reflected wave
![$ f^{{-}}=\rho f^{{+}}$](http://www.dsprelated.com/josimages_new/pasp/img2120.png)
, and a
transmitted wave
![$ (1+\rho)f^{{+}}$](http://www.dsprelated.com/josimages_new/pasp/img2121.png)
. It therefore follows that a
velocity
wave
![$ v^{+}$](http://www.dsprelated.com/josimages_new/pasp/img2122.png)
will split into a reflected wave
![$ v^{-}= - \rho v^{+}$](http://www.dsprelated.com/josimages_new/pasp/img2123.png)
and
transmitted wave
![$ (1-\rho)v^{+}$](http://www.dsprelated.com/josimages_new/pasp/img2124.png)
. This rule is derived in
§
C.8.4 (and
implicitly above as well).
In the mass-string-collision problem, we can immediately write down
the force reflectance of the mass as seen from either string:
That is, waves in the string are traveling through
wave impedance
![$ R$](http://www.dsprelated.com/josimages_new/pasp/img9.png)
, and when they hit the mass, they are hitting the
series
combination of the mass impedance
![$ ms$](http://www.dsprelated.com/josimages_new/pasp/img92.png)
and the
wave impedance
![$ R$](http://www.dsprelated.com/josimages_new/pasp/img9.png)
of the string on the other side of the mass. Thus, in terms of
Eq.
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
(
9.17) above,
![$ R_1=R$](http://www.dsprelated.com/josimages_new/pasp/img2126.png)
and
![$ R_2=ms+R$](http://www.dsprelated.com/josimages_new/pasp/img2127.png)
.
Since, by the Ohm's-law relations,
we have that the
velocity reflectance is simply
Next Section: Mass
Transmittance from String to StringPrevious Section: Mass Reflectance
from Either String