This page is an appendix to the scientific paper: Torsional waves in a bowed string. The research question: why is there so little jitter in the sound of a bowed string? The bow drives the surface of the string, not its centre of mass. And the resonances for torsional motion are unrelated to those from transverse motion. So how big is the rω term compared with v, and why don't we notice inharmonic resonances?
The figure below, reproduced from the paper, shows the velocity, in time and frequency domain representation, of the transverse and torsional waves in a bass E string, bowed by an experienced string player. For comparison, the torsional wave is represented by rω, the product of the string radius and the angular velocity. Of course v drives the bridge more effectively than ω. But that doesn't answer our question, because the stick-slip action of the bow doesn't drive v, it drives v+rω. Human hearing is very sensitive to jitter and rω/v is not negligible!
Sound of the transverse wave
(2.7 M)
Sound of the torsional wave
(2.7 M)
What you are hearing is not the sound made by an instrument, but the sounds of v and rω of the string respectively, captured electromagnetically as described in the paper. Notice that both sounds have the clear pitch associated with a highly periodic signal and that, in both cases, the pitch is that of the transverse fundamental. This pitch corresponds to the period (26 ms) that is clearly visible in the time domain graphs. The pitch frequency (40 Hz) is the strongest harmonic in the transverse wave, and is also of course the spacing between adjacent harmonics. In the torsional wave, it is also the spacing of harmonics, but the fundamental at this frequency is very weak. The transverse velocity wave sounds rather like a bowed bass string. The similarity is not surprising: both are bowed strings. (The spectra are very different, of course. Although the force exerted on the bridge increases with frequency when compared with velocity, this is in part offset by the lack of filtering by the radiativity of the instrument.) The angular velocity wave also sounds somewhat like a bowed bass that has been filtered in an odd way; however, the rich harmonic content and the initial transients suggest a bowed string. The formants around 225 and 450 Hz are near the frequencies of the natural resonances of the torsional wave, in the absence of a bow. These are so strong, however, that one or both of the harmonics may be heard individually in the torsional sound file.
The animation (made by Heidi Hereth) shows idealised Helmholtz motion of a transverse wave.
A brief introduction and summary
The main function of a violin or bass bow is to induce a sideways or transverse motion of the string. Rosin placed on a bow ensures that static friction with the string may be much greater than kinetic friction. Consequently, in a cycle of normal playing, the string at the position of the bow travels with the bow at a nearly constant, low velocity in one direction (the stick phase), then slides rapidly past the bow in the opposite direction (the slip phase), as shown in the animation.
However the bow acts on the surface of the string, rather than at its centre, and so also must exert a twisting or torsional force. This torque excites additional torsional or twisting waves that travel up and down the string. These torsional waves exert only a small torque on the bridge and so produce little sound by themselves. Nevertheless, they can have an important effect on the overall sound produced.
The motion of the point of contact between bow and string depends on both the transverse speed v of the string, and on the torsional velocity ω (its speed is v+rω, where r is the radius of the string). During the stick phase, v+r? must equal the bow speed. The component waves of the familiar transverse modes of the string are in harmonic ratios and so produce a periodic wave: one that repeats exactly after one period. However, there is no a priori harmonic relationship between the torsional and transverse waves. Consequently, the torsional waves may produce non-periodic motion or jitter at the bow-string contact. Because the ear is very sensitive to jitter, this can have a considerable effect on the perceived sound. So we set out to answer the question: in normal playing, why don't we hear jitter associated with the inharmonic resonances of the torsional motion and its contribution to the speed of the bow-string contact point?
The bowed string has been studied for centuries by scientists, including Helmholtz and Raman. It is thus a little surprising to discover that the relative magnitudes and phases of the torsional and transverse motion had not been measured. We did this electromechanically by attaching tiny sensing coils, using a low bass string to minimise perturbation.
The magnitude of the torsional waves was surprising: they may contribute as much as tens of percent of the speed at the contact point with the bow, as shown in the figure above. In the first experiments, the strings were bowed by experienced players. In musically acceptable bowing regimes, the torsional motion was always phase-locked to the transverse waves, producing highly periodic motion. The spectrum of the torsional motion includes the fundamental and harmonics of the transverse wave, with strong formants at the natural frequencies of the torsional standing waves in the whole string. Volunteers with no experience on bowed string instruments, on the other hand, often produced non-periodic motion. This suggests that experienced players have learnt how to find quickly the subtle combination of bow force and speed that phase locks the fundamental torsional wave to a nearby harmonic of the translational wave.
Eric Bavu worked on this project as an undergraduate research project. Other undergraduates who worked on earlier projects on the bowed string, and who therefore contributed to this project, are Pierre-Yves Placais and Manfred Yew.
