### Mass and Dashpot in Series

This is our first example illustrating a *series* connection of
wave digital elements. Figure F.26 gives the physical scenario of
a simple mass-dashpot system, and Fig.F.27 shows the
equivalent circuit. Replacing element voltages and currents in the
equivalent circuit by wave variables in an infinitesimal waveguides
produces Fig.F.28.

The system can be described as an ideal force source connected
in *parallel* with the *series* connection of mass and
dashpot .
Figure F.29 illustrates the resulting wave digital filter.
Note that the ports are now numbered for reference. Two more symbols
are introduced in this figure: (1) the horizontal line with a dot in
the middle indicating a series adaptor, and (2) the indication of a
*reflection-free port* on input 1 of the series adaptor (signal
). Recall that a reflection-free port is always necessary
when connecting two adaptors together, to avoid creating a delay-free
loop.

Let's first calculate the impedance necessary to make input 1 of the series adaptor reflection free. From Eq.(F.37), we require

The parallel adaptor, viewed alone, is equivalent to a force source driving impedance . It is therefore realizable as in Fig.F.20 with the wave digital spring replaced by the mass-dashpot assembly in Fig.F.29. However, we can also carry out a quick analysis to verify this: The alpha parameters are

Therefore, the reflection coefficient seen at port 1 of the parallel adaptor is , and the Kelly-Lochbaum scattering junction depicted in Fig.F.20 is verified.

Let's now calculate the internals of the series adaptor in Fig.F.29. From Eq.(F.26), the beta parameters are

Following Eq.(F.30), the series adaptor computes

We do not need to explicitly compute because it goes into a purely resistive impedance and produces no return wave. For the same reason, . Figure F.30 shows a wave flow diagram of the computations derived, together with the result of elementary simplifications.

Because the difference of the two coefficients in Fig.F.30 is 1, we can easily derive the one-multiply form in Fig.F.31.

#### Checking the WDF against the Analog Equivalent Circuit

Let's check our result by comparing the transfer function from the input force to the force on the mass in both the discrete- and continuous-time cases.

For the discrete-time case, we have

*adaptor*, so that the signs are swapped relative to element-centric notation.)

We now need . To simplify notation, define the two coefficients as

From Figure F.30, we can write

Thus, the desired transfer function is

We now wish to compare this result to the bilinear transform of the
corresponding analog transfer function. From Figure F.27, we
can recognize the mass and dashpot as
*voltage divider*:

Thus, we have verified that the force transfer-function from the driving force to the mass is identical in the discrete- and continuous-time models, except for the bilinear transform frequency warping in the discrete-time case.

**Next Section:**

Wave Digital Mass-Spring Oscillator

**Previous Section:**

Spring and Free Mass