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Simplified Impedance Analysis
The above results are quickly derived from the general
reflectioncoefficient for force waves (or voltage waves, pressure
waves, etc.):

(10.17) 
where
is the reflection coefficient of impedance
as
``seen'' from impedance
. If a force wave
traveling along in impedance
suddenly hits a new impedance
, the wave will split into a reflected wave
, and a
transmitted wave
. It therefore follows that a velocity
wave
will split into a reflected wave
and
transmitted wave
. This rule is derived in
§C.8.4 (and
implicitly above as well).
In the massstringcollision 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
, and when they hit the mass, they are hitting the series
combination of the mass impedance
and the wave impedance
of the string on the other side of the mass. Thus, in terms of
Eq.
(9.17) above,
and
.
Since, by the Ohm'slaw relations,
we have that the velocity reflectance is simply
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