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.

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 denotes the force applied by string-segment 1 to the mass (defined as positive in the ``up'', or positive- direction), is the force applied by string-segment 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 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,
.

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 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 force-wave variables on the string, as discussed above.

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

states that a positive slope in the string-segment 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 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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