Ideal String with a Moving Termination

Perhaps the simplest example of a digital waveguide string model arises for the case of an ideal string, rigidly terminated, wherein one of the terminations is set in motion at a uniform velocity, as shown in Fig. 4.

Figure 4: Moving rigid termination for an ideal string at time $0<t_0<L/c$.

The left endpoint is moved at velocity $v_0$ by an external force $f_0 = R v_0$, where $R=\sqrt{K\epsilon}$ is the wave impedance for transverse waves on the string [162].⁶

The moving-termination simulation is highly relevant to bowed strings, since, when the bow pulls the string, the string can be modeled, to first order, as two strings terminated by the (moving) bow.

Figure 5: Digital waveguide simulation using a) velocity waves, b) force waves.

Figure 5 illustrates two ``waveguide equivalent circuits'' for the uniformly moving rigid string termination, for velocity and force waves, respectively (typically chosen wave variables). The upper delay line holds samples of traveling waves propagating to the right (increasing $x$ direction), while the lower delay line holds the left-going traveling-wave components of the string vibration. Because delay lines are very inexpensive to implement, this is where the digital waveguide method saves much computation relative to a FDA. The rigid string terminations reflect with a sign inversion for velocity waves, and no sign inversion for force waves. Note that the physical transverse velocity distribution along the string may be obtained by simply adding the contents of the upper and lower delay lines, and similarly for the force distribution [156]. (The string force at any point is defined as minus the tension times the slope of the string.) Also, a velocity wave may be converted to a force wave by multiplying it by the wave impedance $R=\sqrt{K\epsilon}$, and flipping the sign in the left-going case.

Figure 6: String displacement snapshots for a moving termination.

Figure 6 shows successive snapshots of the string displacement, plotted with an increasing vertical offset for clarity.⁷String displacement can be computed from Fig. 5a by summing the delay lines to get physical string velocity, and then adding that to an ``accumulator'' buffer each time instant to approximate an integral with respect to time. Note that the string shape is always piecewise linear, with only one or two straight segments in existence at any time. A ``Helmholtz corner'' (slope discontinuity) shuttles back and forth at speed $c$. While the string velocity takes on only two values ($v_0$ or 0) at each point, the string slope increases without bound as the left termination proceeds to $y=\infty$ with the right termination fixed. The magnitude of the applied force at the termination, which is proportional to string slope, similarly steps to infinity as time increases.⁸ As a result, a velocity-wave simulation is better behaved numerically than a force-wave simulation in this instance.