In this work we write and solve a first principles model for the
motion of a bowed string. This model quantitatively incorporates the
stick-slip friction at the outset. The motion of the string is governed by
the wave equation, while stick-slip friction at the bow point introduces a
nonlinearity. By piecewise solution of the wave equation in the stick and
slip phases, we obtain the solution of the nonlinear system. In the most
favourable regions of parameter space we find stable limit cycle
oscillations whose shape is in accordance with the Helmholtz-Rayleigh
motion. We observe that when bow force, bow speed and other parameters are
varied, the stable limit cycle occurs in a narrow region of parameter
space. This explains why it is difficult for amateurs to produce musically
acceptable sounds from the instrument.