A bow induces sideways or transverse motion of the string. Rosin on the bow hair ensures that static friction with the string may be much greater than kinetic. Consequently, in a cycle of normal playing, the string travels with the bow at a nearly constant, low velocity (the stick phase), then slides rapidly past the bow in the opposite direction (the slip phase).

However the bow acts on the surface of the string, not at its centre, and so exerts a twisting or torsional force. This excites additional torsional waves that travel along 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 ?. During the stick phase, v+r? must equal the bow speed, where r is the radius of the string. The familiar transverse modes of the string are in harmonic ratios and so produce a periodic wave. 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. This can have a considerable effect on the perceived sound.

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 the large strings of a double bass.

The magnitude of the torsional waves was surprisingly large – see figure. When the strings were bowed by experienced players 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. Finding (quickly) the subtle combination of force and speed that controls the non-harmonic torsional waves seems to be a skill that string players must learn.

Our web site has sound files of both the transverse and torsional velocity signals of the phase-locked signals.

Eric Bavu, John Smith and Joe Wolfe

Simultaneous measurements of the transverse and torsional velocity of a bowed bass string (sound files available).