Suppose now that we wish to
drive the
mass along a frictionless
surface using a variable
force . This is similar to the
previous example, except that we now want the
travelingwave
components of the force on the mass to sum to
instead of 0:
Since
and
are given,
must be computed as
. This is shown in Fig.
F.9.
The simplified form in Fig.
F.9b can be interpreted as a
wave
digital spring with applied force
delivered from an infinite
source
impedance. That is, when the applied force goes to zero, the
termination remains rigid at the current
displacement.
Above we derived how to handle the external
force by direct physical
reasoning. In this section, we'll derive it using a more general
stepbystep procedure which can be applied systematically to more
complicated situations.
Figure
F.10 gives the physical picture of a free
mass driven by
an external force in one dimension. Figure
F.11 shows the
electrical
equivalent circuit for this scenario in which the external
force is represented by a voltage source emitting
volts,
and the mass is modeled by an
inductor having the value
Henrys.
Figure F.10:
Physical diagram of an external force driving a mass
sliding on a frictionless surface.

Figure:
Electrical equivalent circuit of the forcedriven mass in Fig.F.10.

The next step is to convert the voltages and currents in the
electrical equivalent circuit to
wave variables.
Figure
F.12 gives an intermediate equivalent circuit in which
an infinitesimal transmission line section with real
impedance
has been inserted to facilitate the computation of the
wavevariable
reflectance, as we did in §
F.1.1 to derive Eq.
(
F.1).
Figure F.12:
Intermediate equivalent circuit for the
forcedriven mass in which an infinitesimal transmission line section
has been inserted to facilitate conversion of the mass impedance
into a wavevariable reflectance.

Figure:
Intermediate wavevariable model of the
forcedriven mass of Fig.F.11.

Figure
F.13 depicts a next intermediate equivalent circuit in
which the mass has been replaced by its reflectance (using ``
''
to denote the continuoustime reflectance
, as derived in
§
F.1.1). The infinitesimal transmissionline section is now represented
by a ``resistor'' since, when the voltage source is initially
``switched on'', it only ``sees'' a real resistance having the value
Ohms (the
waveguide interface). After a short
period of time
determined by the reflectance of the mass,
^{F.4} ``return waves'' from the mass result in an ultimately
reactive impedance. This of course must be the case because the
mass does not dissipate energy. Therefore, the ``resistor'' of
Ohms is not a resistor in the usual sense since it does not convert
potential energy (the voltage drop across it) into
heat. Instead, it
converts potential energy into propagating waves with 100%
efficiency. Since all of this wave energy is ultimately reflected by
the terminating element (mass,
spring, or any combination of masses
and springs), the net effect is a purely reactive impedance, as we
know it must be.
Figure F.14:
Interconnection of the wave digital mass
with an ideal force source by means of a twoport parallel adaptor.
The symbol ``'' is used in the WDF literature to signify a
parallel adaptor.

To complete the
wave digital model, we need to connect our wave
digital mass to an ideal force source which asserts the value
each sample time. Since an ideal force source has a zero internal
impedance, we desire a parallel twoport junction which connects the
impedances
(
) and
(
), as
shown in Fig.
F.14. From
Eq.
(
F.18) we have that the common junction force is equal to
from which we conclude that
The outgoing waves are, by Eq.
(
F.19),
Since
and
for this model, the
reflection
coefficient seen on port 1 is
. The
transmission coefficient from port 1 is
. In the opposite
direction, the
reflection coefficient on port 2 is
, and
the transmission coefficient from port 2 is
. The final
result, drawn in
KellyLochbaum form (see §
F.2.1), is
diagrammed in Fig.
F.15, as well as the result of some
elementary simplifications. The final model is the same as in
Fig.
F.9, as it should be.
Figure F.15:
Wave digital mass driven by external force .

Next Section: Force Driving a Spring against a WallPrevious Section: ``Piano hammer in flight''