Under normal, ideal bowing conditions, the bow and string interaction is referred to as a ``stick-slip'' mechanism: During the greater part of each vibration, the string is ``stuck'' to the bow and is carried with it in its motion. Then the string suddenly detaches itself and moves rapidly backward until it is caught again by the moving bow.
The beginning and end of the slipping are triggered by the arrival of the propagating bend or ``kink''.
This behavior of the bowed string is referred to as ``Helmholtz motion.''
Idealized version of the Helmholtz motion of a bowed string: (a) sketch of the string displacement at three points in the vibration cycle (with exaggerated vertical scale); (b) waveform of string velocity at the bow-string contact point; (c) waveform of transverse force exerted on the violin bridge (from Woodhouse (2014)).
The string's vertical displacement at any one point is given by an unsymmetric triangular pattern (but at the string's midpoint, it is symmetric).
The round-trip time depends only on the string length and the wave velocity.
This mechanism allows the player to add energy to the string and to sustain its vibrations.
Bowing near the string end requires greater force and produces a louder, brighter sound than bowing farther from the end.
The amplitude of vibration can be increased either by increasing the bow speed or by bowing closer to the bridge.
The violinist has a limited parameter space within which to work, governed by minimum and maximum bow forces that vary according to the position along the string, as depicted in Fig. 3.
Sketch of a Schelleng diagram (Schelleng, 1973), showing the region of the bow force-bow position plane within which it is possible to sustain a steady Helmholtz motion of the string (from Woodhouse (2014)).