There is a problem with Helmholtz motion, in the idealised form we described in the previous section. There is apparently nothing a player can do to vary the details of the sawtooth waveform, apart from adjusting the amplitude. But surely violinists can make tonal variations by changing their bowing, or by playing the same note on different strings? That intuition is perfectly correct, as can readily be verified by measurement. Figure 1 shows two examples of the Helmholtz sawtooth waveform: the red curve is measured on the open E string of a violin, while the blue curve shows the same note played high on the G string.
Perhaps the difference between these two waveforms looks rather subtle and unimportant? By no means: look at Fig. 2, which shows the frequency spectra of these two waveforms. The blue curve, for the G string, shows only 6 harmonics with significant amplitudes, confined to the frequency range below about 5 kHz. But the red curve, for the E string, shows strong peaks extending way beyond the range of human hearing, up to some 50 kHz! It is hardly surprising that we hear a strong difference of “brightness” between the sounds of these two notes. Indeed, your dog or your cat might complain about the sound of the E-string note.
We can see something else interesting in Fig. 2. Notice that the first few harmonics show peaks of essentially the same height in both red and blue curves. These heights fall with frequency in a smooth and regular pattern. This pattern is dictated by the Fourier series of an ideal sawtooth wave, for which the $n$th harmonic has an amplitude proportional to $1/n$ (we proved that in section 2.2.1). The blue curve starts to fall away from this pattern by the 5th harmonic, but the red curve continues to follow it for a lot longer. Above about 10 kHz the pattern of peaks looks different, for reasons we will come back to in section 9.3. We showed the progression of the ideal sawtooth Fourier series in Fig. 1 of section 2.2, but it is worth repeating it here: see Fig. 3.
This plot reminds us that the sharpness of the “flyback” portion of the sawtooth depends upon how many harmonics are included. This is the clue to understanding the difference between the two waveforms in Fig. 1: the Helmholtz corner is rather rounded in the note on the G string, but much sharper in the note on the E string. This insight, and its consequences for a violinist, were first explored by Lothar Cremer in the late 1960s. Cremer (1905–1990) was a leading acoustician of the mid-20th century, famous particularly for his work in architectural acoustics (he was the acoustical consultant for the Berliner Philharmonie concert hall). He was also a keen amateur viola player, and had a lifelong interest in the acoustics of the violin and its relatives. Figure 4 shows him in 1974, talking to another luminary of the violin acoustics world, Carleen Hutchins.
Cremer thought about what happens to the Helmholtz corner during a cycle of Helmholtz motion. Starting from the moment when the corner leaves the bow heading towards the player’s finger, the effects of propagation along the string, reflection from the finger, and propagation back again all tend to make an initially sharp corner more rounded. Particularly for a stopped note, the mechanical properties of the player’s fingertip will result in significant energy loss during wave reflection: the difference in sound between an open string and a stopped note when played pizzicato demonstrates this effect very directly.
The returning Helmholtz corner then passes the bow, and triggers a transition from sticking to slipping friction. If we ignore for the moment the finite width of the ribbon of bow-hair, this transition must happen at a definite moment: the string sticks to the hair until limiting friction is reached, then releases rather abruptly. That abruptness has the effect of sharpening up the rounded corner. The corner then travels to the bridge and back, becoming somewhat more rounded, then passes the bow again triggering the opposite frictional transition and getting sharpened up. We will shortly illustrate this process in graphical form, once we have covered a bit more background material.
Competing effects of corner-rounding and corner-sharpening happen during each cycle, and the shape of the periodic waveform is determined by a balance of the two effects. The rounding effects depend on the physical properties of the string and the details of its terminations. For a thin, flexible, lightly-damped string like a violin E string these rounding effects are rather weak, especially for an open string without the influence of a finger. But for a thicker string like the G string, the bending stiffness is higher and the string damping is higher, both contributing to extra rounding. If the string is also stopped by a finger, the rounding effect is even stronger. These are the reasons for the contrast seen in Figs. 1 and 2.
The corner-sharpening effect is determined by the frictional interaction between the string and the rosin-coated bow hair. It is crucially dependent on how hard the player presses down with the bow, so-called “bow force”: pressing harder makes the effect stronger. Thus higher bow force shifts the balance in favour of corner-sharpening, and this provides the player with a way to vary the sound. We will see some computer-simulated examples of this effect shortly, and more detailed results on the effect in the next section.
Cremer used simple analytical calculations to explore the balance of corner-rounding and corner-sharpening, but he was a scientist from the pre-computer age. Reading his paper , it became apparent to scientists of the next generation that he had in fact described the essential stages of a time-stepping numerical simulation of the bowed string. The result was the development of the first “digital waveguide” simulations of a bowed string. The method is closely related to the description we gave in section 8.5, of a model for self-excited oscillation of a simplified clarinet. Just like the clarinet model, it involves two variables linked by a feedback loop, as sketched in Fig. 5. This time the variables are the sliding speed $v(t)$ of the string at the bowed point, and the friction force $f(t)$ acting at that point.
Simulations based on this model started to be performed in the late 1970s, allowing Cremer’s mechanism for the influence of bow force on tone quality to be investigated in quantitative detail. However, the simulation model does much more than that because it opens the door to the investigation of transient motions of the string: these include initial transients from a given bow gesture, transitions between oscillation regimes when bifurcation events occur, and the curious transient interaction that sometimes occurs between bowed-string vibration and the motion of the instrument body, known as a “wolf note”. We will look at all these topics in subsequent sections.
This early model was based on the same model of friction that we have already met: what we can call the “friction curve model” because it assumes that the friction force during sliding is determined by a nonlinear function of the sliding speed. Figure 6 shows the results of some measurements of friction force during steady sliding of a surface coated with violin rosin, together with a curve fit to those measurements that can be used in the computer simulation model. The plot also shows the vertical portion where the string and the bow are sticking, and the mirror-image portion that would apply if the string were to slip in the opposite direction across the bow. This red curve plays the same role in the model as the nonlinear mouthpiece characteristic in the clarinet model, as shown in Fig. 7 of section 8.5. We will assume the Amontons-Coulomb “law” that the friction force is proportional to the bow force. The plot in Fig. 6 takes advantage of that assumption by showing the result in the form of the coefficient of friction, the friction force divided by the bow force. Coefficient of friction is normally described using the Greek letter $\mu$ (“mu”).
The details of the bowed-string model are described in the next link. Just as in the clarinet model, the new values of $v(t)$ and $f(t)$ at each time step are found where the friction-force curve intersects with a straight line. The curve for friction force is obtained by taking the curve from Fig. 6, and scaling it by the bow force. Figure 7 shows the result, for two different values of the bow force: the solid line has a higher bow force while the dashed line has a lower one. The slope of the straight line is determined by the properties of the string. The horizontal position of the line is determined by the sum total of reflected waves arriving back at the bowed point from the two sides of the string, which can be computed using a pair of “reflection functions” like Fig. 10 of section 8.5: one for the length of string between bow and bridge, the other for the length between bow and finger.
Figure 7 shows three possible positions of this line. All three positions correspond to slipping friction with the lower value of bow force (the dashed curve). The position marked ‘a’ corresponds to slipping friction for the higher bow force (solid curve), but the position ‘b’ indicates sticking friction, and we can see that the vertical portion of the friction curve causes no difficulty: there is a perfectly well-defined intersection with the sloping line. The position ‘c’ is somewhat problematic for the higher bow force: there are three intersections with the friction curve. However, for the lower bow force this ambiguity cannot arise, because the slope of the straight line is lower than the maximum slope of the friction curve. When the ambiguity arises, how do we know which of the three intersections to choose?
The answer is the same as we found when facing a similar problem with Duffing’s equation, back in section 8.2 and illustrated in Fig. 4 there. The string chooses to stay as long as it can on the branch of the curve it is already following, until it is forced to make a jump. The result is a hysteresis loop, illustrated in Fig. 8 for the case when the bow force is the same as the higher value in Fig. 7. If the string is slipping, and the line enters the ambiguous range from the left, it continues to slip until the sloping line reaches the right-hand position in the figure, whereupon the system jumps (following the dark green arrow) to the sticking branch. Once it is sticking, it continues on that branch until such time as the sloping line reaches the left-hand position in the figure, whereupon it is forced to make a jump to the slipping branch following the longer dark green arrow.
To see the significance of this hysteresis behaviour, we can look at a first example of computer simulation. We will choose a deliberately simplified model, ignoring many aspects of the behaviour of real strings, in order to bring out one particular phenomenon in the clearest possible way. (Remember the “physics agenda”? This is a good example. We will gradually add in the complicating factors ignored here, in the course of subsequent sections.) We will assume for the moment that the two reflection functions for the two sections of the string are the same, with a simple symmetrical shape like the one used in the clarinet model (see Fig. 10 of section 8.5).
We will start the system off with an ideal, sharp Helmholtz corner, and then track it through the first few stages of corner-rounding and corner-sharpening. These are shown in Fig. 9, but you will have to look backwards and forwards between the two halves of this plot in order to follow the story. The dashed line in the left-hand plot shows the original ideal Helmholtz corner. We are plotting waveforms of string velocity, so this appears as an abrupt jump downwards from the bow speed (assumed here to be 0.1 m/s) to the slipping speed (assumed here to be $-1$ m/s). The jump is downwards, so this plot corresponds to the moment of release, when the string starts to slip.
The corner then travels to the violin bridge and back, and it returns rounded and inverted. This new shape appears in the right-hand plot, as the blue curve. Now it interacts with the friction force at the bow. We have chosen a bow force high enough that the hysteresis effect of Fig. 8 operates. The string is initially slipping, so we are on the lower branch of the hysteresis cycle. The string follows round the curve until it reaches the point where it must make a relatively small jump across to the sticking part of the curve: this is the moment of capture of the string by the bow. The result is the waveform shown in green in the right-hand panel of Fig. 9. You can see that this follows the blue curve quite closely for a while, but then rises above it and commences sticking with a sharp corner in the waveform. This is “corner-sharpening” in action.
The new shape of Helmholtz corner then travels down to the player’s finger and back, which results in more rounding and another inversion. It arrives at the bow as the blue curve in the left-hand panel of Fig. 9. Now it interacts again with the friction force, and this time we are following the upper part of the hysteresis cycle. The result is the green curve in this left-hand panel: the string continues to stick for a long time, then makes a big jump down to rejoin the blue curve: this is the moment of release of the string by the bow.
That is the end of one complete cycle, and the process then repeats in the same sequence. The Helmholtz corner travels to the bridge and back, then interacts with friction again, to produce the red curve in the right-hand panel. Then to the finger and back and another frictional interaction, to produce the red curve in the left-hand panel. These two red curves look very similar to the two corresponding green curves, except for one thing: there is a time shift. The effect of the hysteresis cycle is that the round-trip time is a little longer than we expected.
The period of the motion is systematically lengthened relative to the natural period of the free string. In musical terms, hysteresis means that the note plays flat. The extent of hysteresis increases when bow force increases, as Fig. 7 showed. The result is that if bow force is slowly increased after a Helmholtz motion has been established, the note may play progressively flat. This is an effect that can be readily demonstrated on a real bowed instrument. The scope for flattening is determined by how rounded the Helmholtz corner is: each stick-slip transition must occur somewhere within the range of the rounded corner, and so the corner width puts a limit on the possible extent of flattening. As a result, the flattening effect occurs most strongly when playing in a high position on a thicker string, as in the blue curve in Fig. 1. From the perspective of a player, the effect may feel more like pitch instability rather than flattening as such: violinists are used to fine-tuning the pitch of each note, but when the flattening effect is strong the pitch may “waver” in response to changes in bow force, for example if the bow is bouncing slightly on the string.
Figure 10 shows the hysteresis cycle during the simulation shown in Fig. 9. This plot shows the regions of the friction curve traversed during the capture process in red, and during the release process in blue. It should be clear how this plot relates to Fig. 8.
Figures 11 and 12 show plots corresponding to Fig. 9 and 10, but using a much lower bow force. It is low enough that hysteresis does not occur, as can be seen directly in Fig. 12. Comparing Figs. 9 and 11, we can see several important differences. This time, the two red curves do not show a time delay compared to the two green curves. Without hysteresis, the pitch-flattening effect does not occur. However, this time the shapes of the red and green curves are quite different from each other. The effects of corner-sharpening due to the frictional interaction are now much weaker, so, exactly as Cremer’s original argument told us, the effects of corner-rounding become more significant. We have only followed the Helmholtz corner for two cycles, starting from an ideal sharp jump. That is not enough time for the effects of corner-rounding to take full effect: the Helmholtz corner would continue to get more and more rounded for a few more cycles, but we haven’t included them in this plot.
There is another consequence of rounding of the Helmholtz corner, which we haven’t mentioned yet. With a rounded corner, each velocity transition from sticking to slipping, or the reverse, takes a finite time, as illustrated in Figs. 9 and 11. During that time, if the friction-velocity characteristic curve is still being followed, the force cannot be constant as in Raman’s original argument. Instead, there will be a pulse of extra friction force, and this will excite some additional motion of the string. There will be two of these pulses per cycle, generated by the processes of capture and release. John Schelleng (who we will meet in the next section) called the resulting features in the waveform of friction force “rabbit ears”.
These force pulses will send outgoing waves along the string in both directions. The wave travelling in the same direction as the Helmholtz corner creates the effect we have already discussed: this is precisely the origin of the corner-sharpening effect we have already seen. But the other outgoing wave will travel off in the opposite direction. Figure 13 might help to visualise what then happens. This diagram shows time running horizontally, and distance along the string running vertically.
The Helmholtz corner, bouncing back and forth along the string, traces out the zig-zag line shown in red. The position of the bow is indicated by the horizontal line. This is shown solid during the sticking intervals, and dotted during the slipping intervals. This change of line type is meant to indicate an important effect. While the string is sticking to the bow, if any perturbing velocity hits the bow it will be reflected: the sticking bow acts like a fixed end of the string. But during slipping, the bow is almost transparent to any such additional perturbation (this is a consequence of the friction curve being almost flat near the slipping velocity).
The blue lines indicate what happens to the additional waves generated by Schelleng’s “rabbit ears”. Each force pulse produces a corresponding pulse of velocity, which travels along the string. The solid blue zig-zag shows the first three bounces of one of these pulses, generated at a slip-to-stick transition (where the dotted line turns into a solid line). It is confined between the bow and the bridge, because it hits the bow during sticking and so is reflected. The dashed blue zig-zag shows the corresponding set of three bounces of a velocity pulse generated at a stick-to-slip transition. This time it is trapped between the bow and the player’s finger.
The solid blue zig-zag shows extra activity accompanying the Helmholtz motion, and because it is on the bridge side of the bow it will show up in a measurement of bridge force. Look back at Fig. 1: this effect accounts for the fact that neither waveform is a perfect sawtooth. The path in Fig. 13 shows that pulses at the bridge are expected, with a spacing determined by the position of the bow on the string: bowing closer to the bridge makes the spacing tighter. The result was described as “ripples” by Schelleng, and “secondary waves” by Cremer. The pattern is particularly clear in the red curve of Fig. 1: it shows up as a series of peaks following the flyback of the sawtooth, with a magnitude that steadily decreases because the secondary waves are affected by the same corner-rounding effects we have already discussed, but they do not have any compensating corner-sharpening mechanism because they always hit the bow during an episode of sticking.
To see this effect in action, we can again use the simulation model. Figure 14 shows the resulting bridge force for three cases with different values of bow force. The red curve has the lowest force, the black one the highest. The pattern of “Schelleng ripples” is clear: as expected it gets more marked as the bow force increases. The spacing of the ripples is governed by the assumed position of the bow on the string. That position is usually described by a parameter $\beta$ (“beta”), which is the bow-bridge distance as a fraction of the total vibrating length of the string. From Fig. 13, we expect to see roughly $1/\beta$ ripples in each cycle. For these particular simulations, the value of $\beta$ was 0.083, so $1/\beta \approx 12$. You can indeed count about 12 ripples in each period. Some other key parameter values for these simulations are as follows. The string corresponds to the open D string of a cello, with fundamental frequency 147 Hz and string impedance $Z=0.55$ Ns/m. It is bowed at a speed of 5 cm/s, with the three values of bow force 0.03 N, 0.1 N and 0.6 N.
Figure 15 shows the corresponding waveforms of string velocity, which can be compared with the shapes seen in Figs. 9 and 11. The red curve, with the lowest bow force, shows a very rounded shape, and as the force increases the velocity pulse gets progressively more square. If you look carefully at the black curve, you can see that the shape of the pulse is not quite symmetrical, as a result of the hysteresis effect. In fact, this black curve shows a bow force that is close to the limit, beyond which Helmholtz motion ceases to be possible. This gives us a neat link to the next section, where the limits on bow force will be discussed in some detail.
 Lothar Cremer; “Influence of bow pressure on self-excited vibrations of stringed instruments”, Acustica, 30, 119–136 (1974).