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 traveling-wave components of the force on the mass to sum to instead of 0:
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 step-by-step 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.
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 wave-variable reflectance, as we did in §F.1.1 to derive Eq.(F.1).
Figure F.13 depicts a next intermediate equivalent circuit in which the mass has been replaced by its reflectance (using ``'' to denote the continuous-time reflectance , as derived in §F.1.1). The infinitesimal transmission-line 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.
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 two-port 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
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 Kelly-Lochbaum 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.
Force Driving a Spring against a Wall
``Piano hammer in flight''