Animation of Moving String Termination and Digital Waveguide Models

In the force wave simulation of Fig.4.4b,^5.3 the termination motion appears as an additive injection of a constant force $f_0=Rv_0$ at the far left. At time 0, this initiates a force step from 0 to $f_0$ traveling to the right. Since force waves are negated slope waves multiplied by tension, i.e., $f^{{+}}=-Ky'^{+}$, the slope of the string behind the traveling force step is $y'=-f_0/K$. When the traveling step-wave reaches the right termination, it reflects with no sign inversion, thus sending back a doubling-wave to the left which elevates the string force from $f_0$ to $2f_0$. Behind this wave, the slope is then $y'=-2f_0/K$. This answers the question of the previous paragraph: The string is in fact piecewise linear during the first return reflection, consising of two line segments at slope $-f_0/K$ on the left, and twice that on the right. When the return step wave reaches the left termination, it is reflected again and added to the externally injected dc force signal, sending an amplitude $2f_0$ positive step-wave to the right (overwriting the amplitude $f_0$ signal in the upper rail). This can be added to the amplitude $f_0$ samples in the lower rail to produce a net traveling force step in the string of amplitude $3f_0$ traveling to the right. The slope of the string behind this wave is $y'= -3f_0/K$, and the slope in front of this wave is still $-2f_0/K$. The force applied to the string by the termination has risen to $3f_0$ in order to keep the velocity steady at $v_0$. (We may interpret the $f_0$ input as the additional force needed each period to keep the termination moving at speed $v_0$--see the next paragraph below.) This process repeats forever, resulting in traveling wave components which grow without bound, and whose sum (which is proportional to minus the physical string slope) also grows without bound. The string is always piecewise linear, consisting of at most two linear segments having negative slopes which differ by $-f_0/K$. A sequence of string displacement snapshots is shown in Fig.4.5.

Figure 4.5: Successive snapshots of the rigidly terminated ideal string with a moving termination. For clarity, the string is plotted higher on each successive snapshot. (One can consider both endpoints to be moving at the same speed up to time 0, after which the left termination begins moving faster by a constant velocity offset.)

Note that the impedance of the terminated string, seen from one of its endpoints, is not the same thing as the wave impedance of the string itself. If the string is infinitely long, they are the same. However, when there are reflections, they must be included in the impedance calculation, giving it an imaginary part. We may say that the impedance has a ``reactive'' component. If $f(n)$ denotes the force at the driving-point of the string and $v(n)$ denotes its velocity, then the driving-point impedance is given by

$$R(\omega) = \left.\frac{F(z)}{V(z)}\right\vert _{z=e^{j\omega T}},$$

where $F(z)$ and $V(z)$ denote the $z$ transforms of $f(n)$ and $v(n)$. In the case of a rigidly terminated string above, as well as in any system in which all energy delivered to the system is ultimately reflected back to the input, the impedance is purely imaginary at every frequency (a ``pure reactance'').