Mass Termination Model
The previous discussion solved for the motion of an
ideal mass
striking an ideal string of infinite length. We now investigate the
same model from the string's point of view. As before, we will be
interested in a
digital waveguide (
sampled travelingwave) model of
the string, for efficiency's sake (Chapter
6), and we
therefore will need to know what the mass ``looks like'' at the end of
each string segment. For this we will find that the
impedance
description (§
7.1) is especially convenient.
Figure 9.15:
Physical model of
massstring collision after time 0. The mass is drawn as having a
finite diameter for conceptual clarity. However, the model is
formulated for the limit as the diameter approaches zero in the
figure (bringing all three forces together to act on a single
massstring junction point). In other words, we assume a point
mass.

Let's number the string segments to the left and right of the mass by
1 and 2, respectively, as shown in Fig.
9.15. Then
Eq.
(
9.8) above may be written

(10.11) 
where
denotes the force applied by stringsegment 1 to the
mass (defined as positive in the ``up'', or positive
direction),
is the force applied by stringsegment 2 to the mass
(again positive upwards), and
denotes the
inertial force applied by
the mass to both string endpoints (where again, a positive force
points up).
To derive the
travelingwave relations in a
digital waveguide model,
we want to use the force
wave variables
and
that we defined for
vibrating strings in
§
6.1.5;
i.e., we defined
, where
is the string tension and
is the string
slope,
.
Figure 9.16:
Depiction of a string
segment with negative slope (center), pulling up to the right and down
to the left. (Horizontal force components are neglected.)

As shown in Fig.
9.16, a negative string slope pulls ``up''
to the right. Therefore, at the mass point we have
, where
denotes the position of the
mass along the string. On the other hand, the figure also shows that
a negative string slope pulls ``down'' to the left, so that
. In summary, relating the forces we
have defined for the massstring junction to the
forcewave variables
in the string, we have
where
denotes the position of the mass along the string.
Thus, we can rewrite Eq.
(
9.11) in terms of string
wave variables as

(10.12) 
or, substituting the definitions of these forces,

(10.13) 
The inertial force of the mass is
because the mass
must be accelerated downward in order to produce an upward reaction
force. The signs of the two string forces follow from the definition
of forcewave variables on the string, as discussed above.
The force relations can be checked individually. For string 1,
states that a positive slope in the stringsegment to the left of the
mass corresponds to a negative acceleration of the mass by the endpoint
of that string segment. Similarly, for string 2,
says that a positive slope on the right accelerates the mass upwards.
Similarly, a negative slope pulls ``up'' to the right and ``down'' to
the left, as shown in Fig.
9.16 above.
Now that we have expressed the string forces in terms of string
forcewave variables, we can derive digital
waveguide models by
performing the
travelingwave decompositions
and
and using the
Ohm's law relations
and
for
(introduced above near
Eq.
(
6.6)).
Next Section: Mass Reflectance
from Either StringPrevious Section: Matrix Bridge Impedance