The most ingenious experiment for investigating the friction force driving a bowed string was performed by Bob Schumacher, at Carnegie-Mellon University in Pittsburgh. A regular violin E string was mounted on a rather rigid fixture, allowing it to be tensioned with a tuning peg in the usual way. This string was “bowed” using a glass rod which had been dip-coated in rosin from solution. The rod was fixed to a movable carriage, driven by a computer-controlled system so that a desired velocity-time profile could be reliably reproduced on every individual bow stroke. The precise height of the string’s terminations could be adjusted with micrometer screws, to set the normal force between rod and string.

At both ends of the string, force-measuring sensors were built in to the holding fixture. These allow the forces exerted by the vibrating string on the terminations to be monitored, without interfering in any way with the string’s vibration. The two force signals were collected by a digital data-logger at a high sampling rate. As will now be explained, these signals can be combined to give an estimate of the velocity at the friction force at the bowed point on the string.

We can now make use of some equations from section 9.2.1. We define incoming and outgoing velocity waves on the two sides of the bowed point by $v_1^{(i)}(t), v_1^{(o)}(t), v_2^{(i)}(t)$ and $v_2^{(o)}(t)$: these are illustrated schematically in Fig. 1, in a format we have used before. The string velocity $v(t)$ at the bowed point is given by two equivalent expressions:

$$v=v_1^{(i)}+v_1^{(o)}=v_2^{(i)}+v_2^{(o)} . \tag{1}$$

We can also relate the friction force to these velocity waves. The outgoing wave on one side is given by the incoming wave from the other side, plus the new contribution arising from the friction force $f(t)$, so that

$$v_1^{(o)} = v_2^{(i)} + \dfrac{f}{2Z_0} \mathrm{~~~ and~~~} v_2^{(o)} = v_1^{(i)} + \dfrac{f}{2Z_0} . \tag{2}$$

Rearranging, we obtain two equivalent expressions

$$f=2Z_0(v_1^{(o)}-v_2^{(i)})=2Z_0(v_2^{(o)}-v_1^{(i)}). \tag{3}$$

The conclusion is that if we can estimate the current values of the four velocity waves, we can construct two different estimates of $v(t)$ and $f(t)$.

For the simplest case of an ideal string with no bending stiffness, no damping, and with rigid terminations, this is very easy. First we recall that the force exerted by the string at a rigid termination is given by $2Z_0$ times the incident velocity wave at the termination. Referring to Fig. 1, we thus see that $v_1^{(i)}$ is simply given by the measured bridge force at the point in the diagram labelled A, scaled by $1/2Z_0$ and delayed by the travel time to the bowed point. The other three velocity waves are given similarly in terms of the measured bridge force at the points labelled B, C and D. In the case of points B and D, the signal is advanced rather than delayed: we deduce the outgoing waves at the bow by looking at future values of bridge force.

Of course, the real string does not exactly satisfy the three assumptions we have just made. However, it turns out that only one we need to worry about for this particular experiment is the effect of bending stiffness. The rig is designed with rather rigid terminations, and energy loss in the string is very small during the very short travel times involved in this argument. But we do need to think about the consequences of bending stiffness, which means that different frequency components in the velocity waves travel at somewhat different speeds: higher frequencies, or shorter wavelengths, travel a little faster. High frequencies are important for this reconstruction process, because the transitions between sticking and slipping happen very fast, and that automatically means that they are influenced by very high frequencies.

We can see the effect of this bending stiffness in the results shown in Fig. 2. This shows the two measured force waveforms, not when the string is bowed but when it is set into vibration by a wire-break pluck at the position where the bow will be placed. Look at the early part of the blue curve, showing the force at the end of the string further from the bow. There is an initial flat portion , then a big downward jump marking the moment when the main “corner” from the pluck arrives at the sensor. But before that jump there is a wiggly “precursor”. This is the effect of bending stiffness: the high frequency components of the jump have arrived a little earlier because they travel faster. If you look carefully, you can see that the frequency is changing through this precursor: highest at the front, then falling a little until the main jump occurs.

The good news is that there is an approximate mathematical expression for this effect of frequency-dependent wave speed. We can use pluck responses like the ones in Fig. 2 to determine the parameters of that expression, and then use it as a basis for digital filters to remove the effect from our four velocity wave estimates. The gory details can be found in reference [1].

We can now show an example of the reconstruction procedure in action. The following plots all refer to the same section of data used to generate Figs. 2 and 3 of section 9.6. First, Fig. 3 here shows the original measured bridge forces at the two ends of the string during the bow stroke. They have been separated vertically to make the plot easier to see. Both waveforms show the characteristic Helmholtz sawtooth shape.

From these, following the procedure outlined above, we can compute two estimates of the string velocity $v(t)$ and the friction force $f(t)$. These are shown in Fig. 4 and 5. There is not perfect agreement between the two, but they are close enough to be reassured that the method is working reasonably well. The average of each pair of waveforms then gives our final best estimate, and those averages are what was used for the plots in section 9.6.

[1] J. Woodhouse, R. T. Schumacher and S. Garoff, “Reconstruction of bowing point friction force in a bowed string”; *Journal of the Acoustical Society of America* **108** 357–368 (2000).