Torsional Waves in a Bowed String

Bowing a string with a non-zero radius exerts a torque, which excites torsional waves. In general, torsional standing waves have higher fundamental frequencies than do transverse standing waves, and there is generally no harmonic relationship between them. Although torsional waves have little direct acoustic effect, the motion of the bow-string contact depends on the sum of the transverse speed v of the string plus the radius times the angular velocity (rω). Consequently, in some bowing regimes, torsional waves could introduce non-periodicity or jitter to the transverse wave. The ear is sensitive to jitter so, while quite small amounts of jitter are important in the sounds of (real) bowed strings, modest amounts of jitter can be perceived as unpleasant or unmusical. It follows that, for a well bowed string, aperiodicities produced in the transverse motion by torsional waves (and other effects) must be small. Is this because the torsional waves are of small amplitude or because of strong coupling between the torsional and transverse waves? We measure the torsional and transverse motion for a string bowed by an experienced player over a range of tunings. The torsional wave spectrum shows a series of harmonics of the translational fundamental, with strong formants near the natural frequencies for torsion. The peaks in rω, which occur near the start and end of the 'stick' phase in which the bow and string move together, are only several times smaller than v during this phase. We present sound files of the transverse velocity and the rotational velocity due to the torsional wave. Because the torsional waves occur at exact harmonics of the translational fundamental and because of similarities in the temporal envelope, the sound of the torsional signal alone clearly suggests the sound of a bowed string with the pitch of the translational fundamental. However, the harmonics that fall near the torsional resonances are so strong that they may be heard as distinct notes.