Many bowed-string instruments, particularly cellos, exhibit a phenomenon called a “wolf note”. When a cellist tries to play a particular note they find that it is particularly prone to “surface sound”, and they may get a kind of “stuttering” or “warbling” effect instead of the steady note they were expecting. This effect is fascinating to a physicist, but it makes life difficult for cellists and they may go to some lengths to control and moderate the problem. You can hear an example in Sound 1, where the player has tried to produce the effect, rather than trying to suppress it as they would usually do. The bow speed is steady and the player’s left-hand finger is not moving, but you can hear the intermittent warbling sound of the wolf, and you can also hear instability of the pitch.
Figure 1 shows an extract of a bridge force recording during a warbling wolf like this, and Fig. 2 shows a zoomed view of the portion outlined in black in Fig. 1. What these waveforms reveal is that the cycle of the wolf involves an alternation of Helmholtz motion, with the familiar sawtooth waveform, and double-slipping motion.
The physics behind the wolf note is yet another thing that was first explained correctly by Raman . Testing Raman’s explanation was an early target for the computer model described in section 9.2, and we can give a clear description of his argument using a simple simulated example. The great advantage of simulation for this purpose is that we have access to complete information about what is going on, including some things that are hard to see clearly by measurement.
The first clue about the explanation comes from the fact that a classic cello wolf note is always associated with a particularly strong resonance of the cello body. To incorporate body resonances into the synthesis model, we need to modify the reflection function describing how a wave is modifying during travel from the bow to the bridge and back. The details of how to do this are described in the next link. For the purposes of demonstration, we need only include a single body resonance, with a frequency very close to the fundamental frequency of the bowed note.
Figure 3 shows the resulting simulated bridge force, from one cycle of the wolf note’s “warble”. The same alternation of Helmholtz motion and double slipping can be seen, although the details are not exactly the same as Figs. 2 and 3. This simulation does not claim to reproduce the exact details of that particular cello, string and performance, but it captures the essence of the phenomenon: a cyclical warbling waveform, when bowing a note with a fundamental frequency matching a strong resonance of the instrument body.
We can use the simulated results to generate a plot corresponding to Raman’s original measurement. He made simultaneous observations of the string motion and the motion of the cello body, and this gave the basis for his explanation of the wolf phenomenon. Figure 4 shows a corresponding plot from our simulated example: the red curve shows the string velocity, while the blue curve shows the “body” motion. The alternation of Helmholtz motion and double-slipping motion is very clear in the red curve.
The single resonator representing the “body” shows a cyclical growth and decay during the “warble” cycle. The explanation of the behaviour we see in Fig. 4 is intimately connected to the idea of minimum bow force, explored in the previous section. We found that the minimum bow force depends upon the level of energy dissipation into the body of the instrument: with no dissipation, the minimum bow force would go to zero. For this energy loss, the more the body moves, the greater is the energy dissipation rate, and the higher the resulting minimum bow force.
Now think about what happens when you start to bow a note matching a strong body resonance. Initially the body is not vibrating, and it may be possible to start the Helmholtz motion with a relatively low bow force. The resonant response of the body will then start to grow, but as we can see in Fig. 4 this growth takes several cycles: the timescale is determined by the damping factor of the body mode. As the motion at the bridge gradually increases, the effective minimum bow force rises with it. If the player maintains the low initial bow force, it may happen that the minimum level overtakes the actual bow force. Helmholtz motion must then give way to double-slipping motion. But such motion, especially if the two slips are rather symmetrical, produces far less excitation at the fundamental frequency, and this allows the body vibration to die away again. The effective minimum bow force will fall back until it is below the actual force, and Helmholtz motion may then be able to re-establish. The cycle repeats, and the result is the “warbling wolf”. All these stages can be followed in the two traces of Fig. 4.
Having seen that the wolf behaviour is intimately linked to minimum bow force, this raises the question of whether we can extend Raman’s and Schelleng’s argument to predict the note-by-note variation in minimum bow force for a particular instrument. The original argument was explained in detail in section 9.3.1. The essential steps went like this. First, we assumed a perfect Helmholtz motion on the string, which tells us the waveform of force at the bridge. That force was used to predict the motion of the bridge. Then, assuming that the bowing point is close to the bridge, we estimated the extra force at the bow caused by that bridge motion. Finally, the minimum bow force criterion was obtained from the requirement that this extra force must not cause a second slip during the supposed sticking interval of the Helmholtz motion.
Raman and Schelleng found the bridge motion by approximating the “body” by a dashpot. But we can do something better than that, if we have a measurement of the bridge admittance of the instrument. It is straightforward to make use of this admittance to compute the actual bridge motion in response to the Helmholtz sawtooth wave of bridge force, at any chosen frequency. The rest of the Raman—Schelleng argument then carries through without any change. The details are explained in the next link.
We can show an example: it relates to the cello responsible for the wolf note we heard in Sound 1. Figure 5 shows the measured bridge admittance of this cello. Frequency scales are given in semitones (lower) and in Hz (upper). Actual values of the admittance are shown on the right-hand scale; the left-hand scale shows the result scaled by the impedance of the C string of the cello, giving the string-to-body impedance ratio for that string. You may recall that we used a similar scaled admittance when we talked about plucked strings, back in Chapter 5: see for example Fig. 8 of section 5.1.
Figure 6 shows the predicted minimum bow force, computed by the procedure described in the previous link. The frequency axis here denotes the fundamental frequency of a played note. For each of the four strings, a range of 3 octaves from the open string is shown. This is rather more than you can in fact play on a cello with a fingerboard of conventional length, but it ensures that we have covered all playable notes. Vertical lines indicate equal-tempered semitones: Cs are shown in yellow, other notes in grey. The absolute numbers on the vertical axis should not be taken too seriously: the values depend on the particular chosen values of bow speed, bowing position $\beta$, and the friction coefficient of rosin. But all these things only contribute to an overall scale factor: the extent of variation from note to note and between the four strings should be reliably represented in the plot.
The curves for the different strings are identical apart from a scaling factor relating to the impedance of the string. It is immediately clear that the minimum bow force for a given note played on the C string is significantly higher than for the other strings. The highest peak occurs at F$_3$ (174.6 Hz), and this is indeed the wolf note played (on the C string of the cello) in Sound 1. A plot like Fig. 6 reveals the location of the wolf note from a simple physical measurement (of the bridge admittance). It can do more than that: there are other peaks in the plot, and these serve to indicate to a luthier where other potential “problem notes” may occur on the cello.
This brings us to a question you may have been wondering about: why do cellos suffer from wolf notes so much more than violins do? Plots like Fig. 5 give an immediate clue, because this scaled admittance is a direct measure of how strongly the string is coupled to the instrument body. One view of the wolf note is that it is what can happen if that coupling gets a bit too strong. Figure 7 shows the same scaled cello admittance as in Fig. 5, and it also shows the corresponding scaled admittance for two violins (in blue curves).
The frequency scale based on semitones above the lowest tuned note of the instrument allows these different instruments to be compared directly. This comparison shows that the violins also have strong resonances at about the same place in their playing range as the peak that caused the wolf in the cello. The “signature” mode shapes responsible for these peaks were shown back in section 5.3 (see Figs. 5c and 5d in that section). But we see in Fig. 7 here that the peak in the cello is higher than the highest peak for either violin, by about 5 dB. That may not look a very big difference in the plot, but it corresponds to nearly a factor of 2 increase in the string-to-body impedance ratio. In case you think I have cheated by choosing violins with rather low peaks, I should say that the dashed curve in Fig. 7 is for a violin that most players find loud to the point of “crudeness”.
There is a simple reason for this difference between the cello and the violin. If you take a violin, and you scale everything in the right way to reflect the difference of tuning of the two instruments, you obtain an instrument that is significantly bigger than a conventional cello. This causes ergonomic problems for the player: the string length is uncomfortably long, and the body size is rather unwieldy. No doubt in response to comments of this kind from players, instrument makers adjusted the design, to give the smaller body and shorter string length that we are used to.
But there is a price for this. We want the “signature mode” resonances to fall in roughly the same place in both instruments, in order that the musical qualities of the cello are rather like a “big violin”. We have already seen in Fig. 7 that makers have indeed evolved a design for the cello that comes quite close to achieving this objective. But in order to do so, they have had to change two things relative to the “scaled violin”. The smaller body would tend to have higher resonance frequencies, so that body (especially the top plate) is made thinner. On the other hand, the shorter strings still need to be tuned to the correct scaled notes, so the strings need to be heavier than the “scaled violin” would suggest. So we have heavier strings on a lighter body: exactly the recipe we talked about in section 5.2 to increase the loudness of an instrument by making the scaled bridge admittance bigger. But we now see that there is a down-side: this design change increases the likelihood of wolf notes.
The idea of the “scaled violin” is not purely hypothetical: such instruments have been designed and made. The theory underlying the scaling procedure was another thing explained by John Schelleng , and the driving force behind building the first instruments was American luthier Carleen Hutchins, who we saw in Fig. 4 of section 9.2. They didn’t just build a cello-sized instrument: they made an octet of instruments, of which the conventional violin is number 3 in terms of size. You can read more, and see pictures of them, on this web site.
The choice of body sizes for these octet instruments was arrived at by a compromise between complete geometric scaling and ergonomic considerations for the player . The instruments larger than the violin are bigger than the usual viola, cello and bass, but a fully-scaled bass would be impossibly large for a human performer. Having chosen these body sizes, the plate thicknesses and rib heights were adjusted based on Schelleng’s scaling theory, to place the “signature modes” at corresponding frequencies to those of the violin, relative to the tuning of each instrument. The choice of strings was then made with an eye to controlling the string-to-body impedance ratio, to keep the wolf under control.
Returning to the wolf note, there is one more topic we need to address. What can a player or an instrument maker do to reduce the impact of a wolf note? Well, there are some simple things. Fitting lighter-gauge strings to the cello will reduce the string impedance, and thus reduce the height of the peak in the scaled admittance as in Fig. 5 or 7. That peak could also be reduced in height by increasing the damping of the body mode: for example, players sometimes wedge something like a sock under the tailpiece. The trouble with both those “solutions” to the wolf problem is that they will influence all the notes, not just the wolf note. Players will very often regard this as too high a price to pay: skilled players often learn to live with their wolf, by careful control of bowing (and probably choosing to play that F on the G string, not high on the C string).
But there is one kind of treatment that can add damping selectively to the “wolf” peak, without having much effect on the rest of the playing range of the instrument. Somewhat unusually in the world of musical instrument acoustics, this is a familiar approach to vibration problems in many other areas of engineering: it is usually called a “tuned mass damper” or a “tuned absorber”.
The trick is to fix another oscillator of some kind to the structure you want to control, with a resonance frequency tuned so that it exactly matches the resonance that is causing a problem, in our case the wolf note. When that problem mode starts to vibrate, the additional oscillator will also vibrate strongly. If you have designed your additional oscillator so that it has quite high damping, the result will be that it will suck some energy out of the original vibrating structure: exactly the result we were hoping to achieve. The details of how such a tuned mass damper works are given in the next link.
We will look at some non-musical examples first. Figure 8 shows the famous “wobbly bridge” in London. The Millennium Bridge was opened with much fanfare to celebrate the new millennium in 2000, but it soon had to be closed again because pedestrians became alarmed by the way the bridge vibrated in response to their walking rhythm. The problem was fixed, at considerable expense, and the bridge is now a major tourist attraction in London. Part of the fix involved installing tuned mass dampers underneath the bridge deck: you can see one in Fig. 9, looking very obviously like a spring-mass oscillator.
Another rather neat example of the use of tuned mass dampers can be seen in hi-tech archery bows used in competition shooting. When an archer draws their bow back and then releases the arrow, they will excite a lot of vibration in the bow, particularly in the fundamental mode because that mode shape is very similar to the static shape of the drawn bow. This residual vibration will be felt through the archer’s hand and arm: it will contribute to fatigue and potentially lead to less accurate shooting.
The archer would like to reduce this vibration, but they absolutely do not want to slow down the initial response of the bow when it is released, because that would slow down the flight of the arrow. Adding a tuned-mass damper is a good solution. Resonant vibration of the additional oscillator will take a little time to build up, so it will not change anything during the critical early time before the arrow is released. But the damper will suck energy out of the residual vibration, exactly what is wanted.
You can see two competition archery bows in Fig. 10. It may not be immediately obvious where to look to see the tuned mass dampers. Each bow has a pair of rods, projecting from just below the hand-grip. In the picture the archer is holding the bows back-to-front: in normal use these rods would lie at an angle on either side of the archer’s left hand and arm. The thing you can’t see in the picture is that each rod is attached to the bow by a flexible rubber mount. This rubber serves as both the spring and the damping of the oscillator.
How does this idea apply to the cello? Several types of commercial “wolf note eliminator” are available. The most familiar is shown in Fig. 11. A brass weight is attached to one of the “afterlengths” of string, between the bridge and the tailpiece. By adjusting its position carefully, the first resonance frequency of this weighted length of string can be tuned to the wolf frequency. With luck, it will then work as a tuned mass damper exactly as we described above, adding additional damping to the body mode and thus reducing the height of the peak in the scaled admittance without making significant difference to the body response at other frequencies.
There are also devices that work on the same principle but are glued to the underside of the cello soundboard to make a permanent installation. A virtue of these is that the player does not have to adjust something carefully every time they change a string. Alternatively, an instrument maker can sometimes make use of something that is already there on the cello. The tailpiece (which holds the strings, visible at the bottom of Fig. 11) has several resonances. By choosing the right tailpiece and then adjusting details of its installation carefully, one of those resonant frequencies can sometimes be adjusted to match the wolf note so that it can act as a tuned mass damper.
 C.V. Raman; “On the ‘Wolf-note’ of the Violin and ‘Cello”, Nature 2435, 362–363 (1916)
 J. C. Schelleng; “The violin as a circuit”, Journal of the Acoustical Society of America 35, 326–338 (1963).