So far, we have talked a lot about bowing but we haven’t actually included a real violin bow in the discussion. The measurements have used a rosin-coated rod, and the computer simulations have assumed a rigid “bow” acting at a single point on the string. We must now look to see what is different if a conventional violin bow is used in place of a rigid rod.
First, we need to understand the anatomy of bows. A carefully-shaped stick, usually made from a wood called pernambuco, has a cranked tip carved into it. This tip carries one end of the ribbon of bow-hair. The other end is carried by the “frog”, a block of wood (usually) which can slide along the stick, secured by a threaded screw within the stick which can be used to tension the hair. In violin bows of professional standard, the ribbon consists of 150—200 strands of horsehair, in a band approximately 10 mm wide.
The key figure in the development of the modern violin bow was François Tourte, working in France in the late 1780s. Before Tourte, bows for stringed instruments usually had a slight upward curve in the stick. Tourte reversed this, introducing the down-curved stick we see today. You can see a typical modern bow compared to an earlier bow (a treble viol bow) in Fig. 1. Figure 2 shows a close-up of the two bow tips.
An important consequence of the downward curvature of the bow stick was highlighted by Askenfelt and Guettler : the Tourte form allows significantly higher tension in the bow hair. As the hair is shortened, the tension tends to rotate the tip, and thus bend the stick upwards. If the curvature of the stick is already upwards, this extra curvature tends to make the stick less able to resist the axial force of the hair tension. But the Tourte stick gets straighter, not more curved, as the hair tension is increased within the usual working range.
Players are in no doubt that the bow is an important thing in its own right. They will have definite preferences for particular bows, and they may be prepared to spend a surprisingly large amount of money on a bow: a significant fraction of the value of the instrument itself. The physics behind this influence of the bow has proved quite elusive to understand. Some things are known, but if anything the question of preferences between bows remains even more mysterious than the corresponding question about preferences between violins (or cellos, or whatever). There are several possible ways that the bow can influence the sound and playing “feel” of a violin, and I will group them into three categories for this discussion.
An expert violinist requires their bow to perform all kinds of sophisticated tricks, and my first category of physical attributes of a bow contains things which affect how the bow “feels in the hand”, and which thus contribute to the ease (or otherwise) of performing these tricks. If you hold a violin bow in the normal way, then wave it around in the air in the plane of the bow, elementary mechanics tells us that just three parameters are enough to determine how it will feel and behave. One of them is the total mass, and the other two are determined by how that mass is distributed: the stick is tapered, and it has additional mass near the ends in the form of the frog and the tip.
We could use various possible parameters to characterise this mass distribution, but there are two particular ones that are commonly used by bow-makers. First, and most intuitive, is the balance point (or “centre of mass”). The second parameter is something called the “centre of percussion”, and as well as completing our characterisation of the mass distribution it also relates to an important aspect of performance with a bow. Some kinds of bowing make use of the way the bow can rebound from the string. Examples are the bowing styles known as ricochet, spiccato and sautillé. You can see demonstrations of these, and many other kinds of bowing, in this YouTube video: go to the times 0:20 for ricochet, and 1:51 for spiccato-sautillé.
The definition of the centre of percussion goes like this. Imagine hanging the bow from the point marked “pivot” in Fig. 3: it is the position where the player will usually place their thumb when holding the bow in the normal way for classical music. Now think of tapping the bow-hair with a pencil. If you tap at a position close to the pivot, there will be a reaction force at the pivot, and it will be in the opposite direction to your tap. But if you tapped right at the other end, near the tip, the reaction force would be in the same direction as your tap, because it has to restrain the bow from rotating about somewhere near the middle. Somewhere in between these two tapping positions, there will be one particular place where there is no reaction force at all. This is the “centre of percussion”. It is marked with a green arrow in Fig. 3: it is about 2/3 of the way down the bow from the pivot point. The next link gives some technical details about all this.
Another way to describe the effect would be to think of lying the bow on its side on an ice ring so that it could slide around any way it wanted, and then tapping the hair in a horizontal direction with your pencil. When you tap exactly at the centre of percussion, the bow will start to move by rotating about the “pivot” point, even though there is no pivot present this time. If you tap anywhere else, the “pivot” point will move: in one direction if you are closer to the frog, and in the opposite direction if you are further from the frog.
When a player wants to perform ricochet or spiccato notes, they need the natural bouncing behaviour of the bow to do some of the work for them. This in turn depends on the position of the bowing point along the bow, partly through the influence of the centre of percussion. It seems reasonable that very rapid ricochet playing might be easier if the bowing point is fairly near the centre of percussion, so that the bow naturally wants to rotate around the player’s thumb during the bouncing action. Askenfelt and Guettler did an interesting study of “the perfect spiccato” , and they suggest that such strokes are “always played well inside the centre of percussion (about 10 cm)”. For the bow in Fig. 3, that position would lie halfway between the balance point and the centre of percussion. They also point out that such bowing techniques only became possible with the higher hair tension available from a Tourte bow. Viol players do not do such flashy bowings!
There is another aspect of the dynamics of a bouncing bow that is very important for a player aiming to perform rapid spiccato or ricochet. When the bow is in contact with a string, it has a resonant bouncing frequency determined by the inertia of the bow and a stiffness coming mainly from the bow-hair tension. This resonance frequency varies strongly with position along the bow, and also with bow-hair tension. Askenfelt and Guettler  gave a simplified theoretical expression for this: Fig. 4 shows a version of their plot. The next link describes their model. They tested a professional violinist with a wide variety of bows, asking them to play the same rapid spiccato with each one. They found that the player did indeed adjust the hair tension and bowing point for each bow so as to create essentially the same bouncing frequency every time.
We see a hint of a different aspect of how a bow “feels in the hand” if we make a link with the discussion from section 9.6. Figure 5 shows a measured Guettler diagram using a real bow in the Galluzzo experimental rig, compared with the one we have seen before measured with the rigid rod (repeated here as Fig. 6). The green and magenta lines will be explained in a moment. At first glance, these two plots look fairly similar. They both show a vaguely wedge-shaped region of transients that led to successful Helmholtz motion, and they both include quite a few white or yellow pixels indicating very short transient length. Both show some “speckly” texture, suggesting a degree of “sensitive dependence”.
One clear difference between the two figures is in the position and shape of the lower boundary of the region of coloured pixels. Figure 5 shows more black pixels than Fig. 6 in the lower left corner, but the pattern is opposite on the right-hand side: Fig. 6 shows a boundary curve that rises more steeply than the one in Fig. 5. We can suggest a tentative explanation for at least part of this difference, and it will tell us something interesting about the “feel” of a bow.
Back in section 9.6, we looked at alternative models for friction. But one aspect of that discussion was deferred until later: I said we should review the Amontons-Coulomb “law” that friction force is proportional to the normal force (i.e. to the bow force in our case). The time has now come to look at this issue. The Amontons “law” is so familiar to anyone who has studied mechanics or physics at any level that many people are surprised to learn that there is any question about it. But the result is a purely empirical finding, and the generally accepted explanation involves something unexpectedly subtle.
Amontons published his “laws of friction” in 1699 (although in fact this was a re-discovery: they were stated in the notebooks of Leonardo da Vinci some 200 years earlier, but Leonardo never published them). The first two of these laws state that the force of friction is proportional to the applied load, and independent of the apparent area of contact between the sliding surfaces. The original experimental evidence gathered by Amontons and others involved the classic experiment of an object like a brick resting on an inclined plane (such as a wooden plank). The angle of the plane can be slowly increased, until the brick starts to slide.
However, if you were to examine with a microscope the surfaces of the brick and the wooden plank, neither of them would be smooth. Instead, they would be covered with small lumps and bumps known in the jargon as asperities. The result would be something like what is sketched in Fig. 7: actual contact between the two surfaces only occurs near the tips of these asperities. The real area of contact will be much smaller than the apparent area of contact.
The accepted explanation for Amontons’ laws then goes like this. The friction force is proportional to the real area of contact. When the applied load is increased so that the surfaces are pressed together more firmly, there is a bit of deformation near the asperity tips, with the result that each individual contact gets a bit bigger, and also additional asperities may come into contact. In a classic paper by Greenwood and Williamson , it was shown that under reasonable assumptions about the statistical description of the surface roughness, the real area of contact ends up being proportional to the applied load. Putting these two things together, the friction force is proportional to the load, as Leonardo and Amontons found.
What does all this have to do with our two Guettler diagrams? Well, in the experiment with the rosin-coated rod we would probably not expect this argument based on rough surfaces to work. When a string is bowed by a rod, the geometry is like two cylinders, crossing at right angles. The apparent area of contact is very small, so the contact pressure (force per unit area) will be very large. Rosin is not a very hard material, especially when it has been warmed up by friction. With this large contact pressure, any asperities that may have originally been present on the surfaces of the rosin-coated string and rod will be squashed flat. The real area of contact is then the same as the apparent area of contact.
Under those circumstances, the argument based on rough surfaces goes out of the window. Instead, we would expect the material near the contact to behave in a way first described in 1882 by Heinrich Hertz (the same scientist that our unit of frequency is named after). You can find a description of this “Hertzian contact”, and of the Greenwood-Williamson model of rough surfaces, on this Wikipedia page. For a Hertzian contact, the friction force is not proportional to the applied load: instead, it is proportional to the 2/3 power of that load.
So now look again at the Guettler diagram in Fig. 6. Guettler’s original argument suggested that the wedge of short transients would be bounded by a straight line — but he was assuming the Amontons-Coulomb law. What his criterion really calls for is that the friction force is proportional to the acceleration. If the rod-string contact is Hertzian, the boundary should show bow force proportional to the acceleration raised to the power 3/2: a rising curved line, not a straight line. The magenta line in Fig. 6 shows what that looks like: it doesn’t do a bad job of tracking the lower boundary of the coloured pixels.
Now what about the Guettler diagram measured with a real bow? We can make a guess. The ribbon of bow-hairs must produce an effect which is rather like the rough surfaces of Greenwood and Williamson. Each individual contact between the string and a single hair might play the role of an asperity contact. As the bow force is increased, more hairs might come into contact with the string. The result might, perhaps, be to resurrect the Amontons-Coulomb law. If that were so, the boundary would be a radial straight line in the Guettler diagram. The green line in Fig. 5 shows an example. It doesn’t track the boundary very well at low values of bow force, where perhaps things are more complicated, but with the eye of faith it does a reasonable job of tracking the boundary at higher forces. It certainly goes some way towards explaining the contrast with the magenta line in Fig. 6.
This effect might, just possibly, be the main reason for using the complicated arrangement of a ribbon of horsehairs to make a bow. A bow could be strung with something like a large-diameter gut string (rather in the style of an archery bow). The string could be chosen to have the same total mass as the horsehair bundle, so that if it was adjusted to the same tension it would have the same bouncing frequency. Such a string will accept a coating of rosin just as well as horsehair does. But, despite matching all aspects of the mechanical behaviour we have talked about so far in this section, it is virtually impossible to play an instrument with such bow! I have tried the experiment, and there is an overwhelming impression that you can’t press hard enough to get the string response you are expecting. That impression would be a natural consequence of the contrast between Hertzian contact and Amontons’ law.
 Anders Askenfelt and Knut Guettler, “The bouncing bow: an experimental study”, Catgut Acoustical Society Journal 3, 6, 3—8 (1998)
 Knut Guettler and Anders Askenfelt, “On the kinematics of spiccato and ricochet bowing”, Catgut Acoustical Society Journal 3, 6, 9—15 (1998)
 J. A. Greenwood and J. B. P. Williamson, “Contact of nominally flat surfaces”, Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 295, 300-319 (1966)