A literature review:
The acoustics of the violin
Lei Fu April, 2015 Instructor: Prof. Gary P. Scavone
Final project of Seminar MUMT 618 Schulich School of Music, McGill University
2.1 Early work
The first breakthrough of the motion of a bowed string was made by Helmoltz. This 'Helmholtz motion' is illustrated in its most idealized form in figure 2. At any given instant the string has a triangular shape, two straight portions being separated by a sharp kink, the 'Helmholtz corner'.
This corner travels back and forth along the string. As the Helmohotz corner passes the bow it triggers transitions from slipping to sticking friction between the string and the bow-hair, and back again. The string sticks to the bow for the majority of each period, while the corner travels the long way to the player's finger and back. During the shorter trip to the bridge and back, the string slips rapidly across the bow-hair.
Indeed, the Helmhotz motion is a possible free motion of an undamped string.
Schelleng combined and extended earlier results of Raman to produce the now-famous diagram shown in figure 3. Ingoring transients and concentrating for the moment on the bowing of steady notes, the violinist has only a rather small number of parameters to control: the speed of the bow, the normal force with which the bow is pressed against the string, the position of the bowed point along the length of the string, and the angle of tilt of the bow-hair ribbon relative to the string.
Ignoring tilt at this stage, Schelleng chose to fix the bow speed,and look at the region in the force-position plane within which steady Helmholtz motion is possible. For a given position of the bowing point, usually described by a parameter β which is the distance from the bridge divided by the total vibrating length of the string. Below the minimum bow force, the string is not able to stick to the bow-hair as long as it is supposed to. Above the maximum force the opposite problem arises: the arrival of the Helmholtz corner at the bow is insufficient to guarantee a transition from sticking to slipping. Approximate analysis by Raman and Schelleng [13] of these two force limits suggested that they both vary with β: based on the simplest theory, the minimum force varies proportional to β−2 while the maximum force is proportional to β −1 . Plotting these results on log–log axes gives Schelleng’s diagram, figure 3.
In the idealized form of Helmoholtz motion, the sawtooth waveform of force driving the violin body is completely independent of anything the player does, except for changing the amplitude of the motion. But it is well known to every violinist that changes in bowing can influence the tone colour of a steady note on a violin.
Cremer and Lazarus used an electromagnetic sensor to observe the string velocity at the position of bowing, and they saw waveforms like those illustrated in figure 4. While both waveforms are recognizably similar to the idealized version of figure 2(b), no abrupt jumps of velocity are seen. Instead there is smooth variation, and crucially the detailed shape varies with the normal bow force: the lower the bow force, the more rounded the waveform becomes, and this translates into a sound containing less energy in higher harmonics.
Equilibrium periodic waveform is determined by the balance of the two effects: roundness and sharpness of the corner. But the rounding effects are independent of what the player does with their bow, whereas the sharpening effect is crucially dependent on bow force: pressing harder makes the effect stronger.
