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Two digital waveguide ``equivalent circuits'' are shown in
Fig.6.5. In the velocity-wave case of Fig.6.5a,
the termination motion appears as an additive injection of a constant
velocity
at the far left of the digital waveguide. At time 0
,
this initiates a velocity step from 0
to
traveling to the
right. When the traveling step-wave reaches the right termination, it
reflects with a sign inversion, thus sending back a ``canceling wave''
to the left. Behind the canceling wave, the velocity is zero, and the
string is not moving. When the canceling step-wave reaches the left
termination, it is inverted again and added to the externally injected
dc signal, thereby sending an amplitude
positive step-wave to
the right, overwriting the amplitude
signal in the upper rail.
This can be added to the amplitude
signal in the lower rail to
produce a net traveling velocity step of amplitude
traveling to
the right. This process repeats forever, resulting in traveling wave
components which grow without bound, but whose sum is always either
0
or
. Thus, at all times the string can be divided into two
segments, where the segment to the left is moving upward with speed
, and the segment to the right is motionless.
At this point, it is a good exercise to try to mentally picture the
string shape during this process: Initially, since both the left end
support and the right-going velocity step are moving with constant
velocity
, it is clear that the string shape is piece-wise linear, with
a negative-slope segment on the left adjoined to a zero-slope segment
on the right. When the velocity step reaches the right termination
and reflects to produce a canceling wave, everything to the left of
this wave remains a straight line which continues to move upward at
speed
, while all points to the right of the canceling wave's
leading edge are not moving. What is the shape of this part of the
string? (The answer is given in the next paragraph, but try to
``see'' it first.)