A simple model of the bouncing frequency of a bow was presented by Askenfelt and Guettler [1]. We used this model to plot Fig. 4 of section 9.7, to show how the bouncing frequency varies depending on the bow-string contact position. The model can easily be derived: the arrangement is shown schematically in Fig. 1. The bow stick with its tip and frog is treated as a rigid frame, holding the bowhair which has tension $T$. This frame is pivoted at one end, indicated by the blue circle in Fig. 1, and it is assumed to have moment of inertia $I$ about this point.
At a given moment, the frame has rotated by a small angle $\theta$ about the pivot. At a distance $x$ along the bowhair from the frog, the hair is in contact with a string. The string is assumed to have a much higher tension than the bowhair, so that for a first approximation it can be treated as rigid. This rigid string has pressed into the bowhair by a distance $y \approx x \theta$. This generates a restoring force
$$F \approx T \frac{y}{x} +T \frac{y}{L-x} \approx T \theta + T \frac{x}{L-x} \theta = T \theta \frac{L}{L-x} . \tag{1}$$
Now taking moments about the pivot we obtain the governing equation:
$$Fx \approx T \theta \frac{Lx}{L-x} =-I \ddot{\theta} . \tag{2}$$
This is the simple harmonic equation, so we deduce immediately that the frequency $\omega$ of bouncing motion satisfies
$$\omega^2=\frac{T}{I}~\frac{Lx}{L-x} . \tag{3}$$
Of course, the string is not really rigid, so that in practice the frequency will be a little lower than this estimate.
This very simple model of bow bouncing has been extended by Gough [2], using a combination of finite-element computation and measurements on real bows. Needless to say, the full pattern of behaviour is revealed to be more complicated. However, for the purposes of qualitative description it is reassuring to find that the plot shown in Fig. 4 of section 9.7 is changed only very slightly for bow-string contact positions up to the point indicated as the “typical spiccato position”.
But there is a more important effect for actual ricochet or spiccato playing. This model has assumed that the bow remains in contact with the string throughout the motion, but the whole point of these bowing techniques is that the bow bounces off the string for a short time in each cycle, then makes contact again to start the next note. This “flying time” will add to the period of the bouncing as we have just calculated it, by an amount depending on details of the player’s gesture.
[1] Anders Askenfelt and Knut Guettler, “The bouncing bow: an experimental study”, Catgut Acoustical Society Journal 3, 6, 3—8 (1998)
[2] Colin E. Gough, “Violin bow vibrations”, Journal of the Acoustical Society of America 131, 4152–4163 (2012).