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 traveling-wave) 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.
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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
where




To derive the traveling-wave 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,
.
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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 mass-string junction to the force-wave 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
or, substituting the definitions of these forces,
The inertial force of the mass is

The force relations can be checked individually. For string 1,


Now that we have expressed the string forces in terms of string
force-wave variables, we can derive digital waveguide models by
performing the traveling-wave decompositions
and
and using the Ohm's law relations
and
for
(introduced above near
Eq.
(6.6)).
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Mass Reflectance from Either String
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Matrix Bridge Impedance