Simplified Impedance Analysis
The above results are quickly derived from the general
reflection-coefficient 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 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

, 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'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