Helmholtz Motion Research Articles

Effects of bow‐hair compliance on bowed‐string motion

Bow–hair compliance is likely to have a strong effect on bow–string interaction. The first line of evidence comes from analyzing experimental results presented by Cremer [L. Cremer, The Physics of the Violin (MIT, Cambridge, MA, 1985)] of the reflection and transmission behavior of transverse waves on a string incident on a bow at rest. The second line of evidence comes from simulations of the bowed string taking into account the finite width of the bow. The nature of the ‘‘differential slipping’’ which may arise due to the kinematical incompatibility of uniform bow velocity across the width of the bow and the string velocity in standard bowed‐string motion (Helmholtz motion) is strongly dependent on bow–hair compliance. Simulations also demonstrate that suitable tilting of the bow can reduce the extent to which ‘‘differential slipping’’ is detrimental to the establishment of the desired string motion. Preliminary results of an experimental investigation of finite bow‐width effects will be discussed.

Some aspects of bow resonances—Conditions for spectral influence on the bowed string

The resonant bow seems to some extent capable of modifying the power spectrum of the bowed string, even during a steady‐state Helmholtz motion, where only one slipping interval occurs during a fundamental period. Longitudinal resonances in the bow hair, strongly coupled to the transverse vibrations of the stick, are being excited by the changes of the frictional force which occur during each individual period. In general, these velocity fluctuations—superimposed on the steady bowing velocity—show small amplitudes compared to those of the string under the bow. In certain frequency regions however, dependent on the bow/bridge distance, impedances, etc., their energy content is sometimes great enough to cause noticeable modification of the string velocity. Such spectral changes would at any rate be small, but may still bear some acoustic significance. The present analysis is based on measurements on a violin bowed by a computer‐controlled bowing machine, and supported by computer simulations. [Thanks are extended to Dr. Anders Askenfelt for sharing his experience in measuring bows, and to The Swedish Royal Institute of Technology for the use of their bowing machine and other laboratory facilities.]

Quasiperiodicity and bifurcations in wolf tones

The wolf tone, often obtained on the lower bowed stringed instrument, is studied with an experimental setup that mimics the true instrument with the help of a digital bow [Caussé and Weinreich, Proceed. 13th ICA, Belgrade 1989]. In this experiment, the resonant frequency of the bridge can be adjusted so that it is low enough for a good coupling with the fundamental frequency of the string, while keeping the bridge as rigid as possible. With this experimental setup, verification of the well‐known result that the wolf tone depends strongly on the pressure and speed of the bow and on the bowing point of the string has been made. By changing bow speed and pressure, by bifurcations after the normal periodic sound (Helmholtz motion), various quasiperiodic tones built on two or three basic frequencies have easily been obtained. Other possible scenarios related to bifurcation theory are indicated by observations of more complex signals.

Elementary stability considerations for bowed-string motion

Using the approach first pioneered by Raman, the Helmholtz motion of a bowed string is discussed as a special case of “two-velocity motions,” in which a given point (at which the bow is located) alternates, in the course of a cycle, between two constant velocities. The fact that the bow typically presents a negative resistance to the string during the “slipping” part of the cycle is adduced as a reason for the “duty cycle,” that is, the fraction of the period that corresponds to slipping, to try to become as short as possible. It is shown that, for a string without dissipation or stiffness, this duty cycle can be arbitrarily low for general bow positions; data obtained with the “digital bow” illustrate this behavior. It is shown theoretically, and confirmed with computer simulations, that instabilities arising from the negative slipping resistance cannot be eliminated by assigning a finite positive value to the sticking resistance. The apparent stability of Helmholtz motion observed in real playing situations remains a puzzle.

Kinematical and dynamical factors for deviation from Helmholtz motion

In the ideal Helmholtz motion, the string velocity at the bow alternates between two constant values, the ratio of whose magnitudes is equal to the ratio of lengths of the two segments into which the bow divides the string. Deviations from this ideal can be of two types: (a) motions for which the string behavior at the bow maintains its “two‐velocity” nature but with a different ratio; and (b) motions that deviate from the “two‐velocity” pattern. Kinematically, two‐velocity motions of arbitrary ratio are possible if the bowing point is not a rational subdivision of the string; otherwise, if it is defined as a fraction m:n of two integers in its lowest terms, the shortest possible duty cycle (that is, fraction of period for which “slipping” occurs) is 1/n. Small ratios may be favored by stability considerations. Deviations from the two‐velocity pattern must occur when dissipation is present, and involve the response of the string to near‐resonant periodic forces. Examples observed with the “digital bow” will be shown. [Work supported by NSF.]

Theoretical relationship between bow pressure and vibration amplitude, derived from a sticking‐sliding force‐velocity characteristic

The Helmholtz motion for bowed strings may be described with the help of a characteristic between the bow's frictional force and the string velocity at the bowing point. Although this characteristic has never been dependably measured, some of its features are implied by players' common experience:. In general, the vibration amplitude changes with bow pressure or bow velocity. The relationships between these changes and the local evolution of the characteristic will be established and discussed. The case of perfect stickiness, in which the amplitude becomes independent of bow pressure, follows as a special case. [Work supported in part by NSF and the French Ministry of Culture.]

The digital bow

The “digital bow” is a device that senses the instantaneous velocity of a chosen point on a string and, by looking up an equivalent frictional characteristic stored in a computer table, exerts an electrodynamic force on the string corresponding to what a physical bow would do under the same circumstances. The process is repeated at a high sampling rate in real time. Under such conditions, we observe the string breaking into spontaneous Helmholtz motion. The arrangement is of great interest to the physicist because it provides exact control of the frictional characteristic of the “bow,” allowing one to investigate details of the bowing process to a degree hitherto unattainable. It may also have considerable musical potential, since the behavior of this “bow” is not limited by the properties of Siberian horse hair or of the secretions of dripping evergreens. Video and audio tapes will be presented. [Work supported in part by NSF.]

Violin sound synthesis from first principles

A simple computer model of a violin has been constructed. It consists of two arrays carrying the forward and backward waves on the string, appropriate reflection rules at the bridge and finger, a bow of finite width, and a combination of parallel two-pole filters to simulate the radiativity of the violin body. The string is assumed to be nondispersive, although an equivalent amount of dispersion can be included in one or both of the boundary conditions. The instantaneous force exerted by the bow on the string is formulated as a single-valued function of their relative velocity. The output signal can be converted into sound. The model has been used to study a number of phenomena, including the dynamic changes in the Helmholtz motion when the bow velocity changes, and in particular the starting transients.

On the motion of the bowed violin string

Oscillations with nodes at the bow point have been observed. For a typical violin string such oscillations exist for β > 0.08 (here β is the relative bow position along the string). For β > 0.18 the amplitude of this motion may become comparable to the Helmholtz motion if the bow force exceeds a critical value (2–40 g/wt. depending on the value of β). At the critical force and with β ≠ n−1 (n = 2, 3, 4...), the waveform can be well described analytically. The analytic expression consists of two terms, a sinusoid and the usual Helmholtz expression. It is used to calculate consecutive string shapes. At the critical bow force, the slip period of the string at the bow point is observably reduced to half the value given by the classical Helmholtz theory. However, this slip duration is given by the analytic expression. For bow forces much larger than the critical value the sinusoids of the waveforms (including the slip duration at the bow point) become narrower than the “predictions.” It is shown that this type of motion does not exist for β < 0.18 because it demands too high a bow force. The relationship between this motion and Raman's “higher types of motion” will be explained.