Kinematics of the Bowed String
As the bow is drawn across the strings of a violin, the string appears to vibrate back and forth smoothly between two curved boundaries, much like a string vibrating in its fundamental mode. However, this appearance of simplicity is deceiving. Over 100 years ago, von Helmholtz (1877) showed that the string more nearly forms two straight lines with a sharp bend at the point of intersection. This bend races around the curved path that we see, making one round trip each period of the vibration.
The science behind bowed strings
Because of its speed, our eye ordinarily sees only the curved envelope. During the greater part of each vibration, the string is carried along by the bow. Then it suddenly detaches itself from the bow and moves rapidly back until it is caught by the moving bow again. Up to the point of release a and again from c to i, the string moves at the constant speed of the bow. From a to c, the string makes a rapid return until it is caught by a different point on the bow.
The oscillogram on the left, which shows the displacement versus time, is almost Displacement of bow and string at the point of contact with the bow. Note that the midpoint on the vibration is displaced slightly from the rest position of the string, whereas the graph of velocity versus time (on the right) shows rather narrow spikes, which represent the large negative velocity of the string when it slides rapidly along the bow.
The letters of the frames correspond to the letters.
At the position of the bow, the arrows in the sequence at the right correspond to the velocity shown in Fig. 12.2b. At the moment of release, shown in Fig. 12.3a, the bend has just passed the bow. In (b), the bend has reached the bridge, from which it will reflect back down the string in (c), (d), and (e), until it reaches the nut in (f) and is again reflected. At point c in Fig. 12.1 and also in frame (c) in Fig. 12.3, the string is captured by the bow, and once again moves upward at the speed of the bow.
The stick and slip of the string relative to the moving bow are determined partly by the friction between the string and the horsehair of the bow. It is well-known that the force of friction between two objects is less when they are sliding past each other than when they move together without slippage. After the string begins to slip, it moves rather freely until it is once again captured by the bow. It is important to note, however, that the beginning and the end of the slipping are triggered by the arrival of the bend (slipping begins when the bend arrives from the nut and ends when it arrives again from the bridge).
Because the time required for one round trip depends on the string length and wave velocity (which, in turn, depends on tension and string mass), the vibration frequency of the string remains the same under rather widely varying bowing conditions. If only friction and the restoring force of the displaced string determined the beginning and the end of slipping, the vibrations would be irregular rather than regular.
The limits on the bowing conditions are the limits on the conditions at which the bend can trigger the beginning and the end of slippage between bow and string. For each position of the bow, there is a maximum and minimum bowing force, as shown in Fig. 12.4. The closer to the bridge the string is bowed, the less leeway the violinist has between minimum and maximum bowing force. Bowing close to the bridge (sul ponticello) gives a loud, bright tone, but requires considerable bowing force and the steady hand of an experienced player.
Players are well aware of how the amount of bow hair in contact with the string affects bowing. A physical model of the bowed string by Pitteroff (1994) takes into account the width of the bow, the angular motion of the string, bow-hair elasticity, and string bending stiffness. The frictional force for the edge of the bow facing the nut is lower than that of the edge facing the bridge through the entire sticking time.