9.1 On the shoulders of giants: Helmholtz and Raman

In a letter to Robert Hooke written in 1675, Isaac Newton famously wrote “If I have seen further it is by standing on the shoulders of giants.” This dictum is particularly apposite for the study of the motion of a bowed string: the subject got under way with two of the giants of 19th and early 20th century science. The first of these giants is Hermann von Helmholtz (1821—1894). Helmholtz made significant contributions to several scientific fields ranging from physics to physiology: at different times in his academic career he held professorships in both disciplines.

Figure 1. Hermann von Helmholtz in about 1894. Image: Heliogravüre by Meisenbach, Riffarth & Co. Berlin. Scanned, image processed and uploaded by Kuebi = Armin Kübelbeck, Public domain, via Wikimedia Commons

In 1862 Helmholtz wrote a book, later translated into English as “The sensations of tone as a physiological basis for the theory of music” [1]. The book ranges over what we would today regard as many different disciplines: musicology, perception and psychoacoustics, the physiology of hearing, but also mathematical theories of vibration and waves, and experimental work on vibration and acoustics. In among all this, he describes an ingenious experiment to observe the motion of a bowed violin string. This led him to discover and describe mathematically the unexpected way that a bowed string normally vibrates; something now known as “Helmholtz motion” in his honour. The details appear in one of 19 Appendices to the book, covering topics as diverse as the constructional details for what we now call Helmholtz resonators (see section 4.2.1), experiments on the production of vowel sounds (see section 5.3), the mathematical theory of the motion of plucked strings and struck piano strings, and a design of a keyboard instrument able to play in “just intonation”.

Helmholtz’s experimental setup made use of his own improvement to the “vibration microscope” first developed by Lissajous. This involved a lens system, attached to the vibrating tine of an electrically-maintained tuning fork. The violin string was blackened, and then a bright white spot of starch was added at the point to be observed. The string was bowed, and arranged so that the vibration was perpendicular to the direction of motion of the lens. By synchronising the pitches of the bowed string and the tuning fork, and then observing the path traced by the white dot when seen through the vibrating lens, a closed loop called a “Lissajous figure” is seen. A bit of careful thought can then reveal how the marked point on the string must be vibrating.

What Helmholtz discovered is illustrated in the top animation of Fig. 2. At any given moment, the string has a triangular shape: two straight portions separated by the “Helmholtz corner”. This corner shuttles back and forth between the bridge and the player’s finger. If you look carefully at the point where the string crosses the moving bow, you can see that all the time the corner is making the long journey to the finger and back, the string is moving at the same speed as the bow. In other words, during that part of each cycle of vibration there is sticking friction between the bowhair and the string. But when the corner travels to the bridge and back, the string is slipping across the bow hairs: slipping friction. So this motion of the string is a stick-slip vibration, but the frequency is governed by the time-keeping role of the travelling corner. Since that corner travels at the normal wave speed on the string, this explains why the frequency of a bowed note is the same as the frequency of a plucked note on the same string.

Figure 2. Animation of Helmholtz motion of a bowed string, in idealised form. Top panel shows the string motion, and the moving bow. Lower panel shows the transverse force exerted on the bridge of the instrument, which is the source of body vibration.

We will see in later sections that many aspects of the description just given are not exactly correct. There are small but important differences between real string motion and the idealised form described by Helmholtz. The corner is not really perfectly sharp; the bow does not really act at a single point on the string because the ribbon of hair on a violin bow is a few millimetres wide; the frequency of a bowed note is not always exactly the same as that of the plucked string. But we will also see that Helmholtz motion gives an excellent first approximation to what real bowed strings do.

The key to testing that claim lies in the lower panel of the animation. Fortunately, modern electronic and computer-based measurement methods mean that we can observe string motion in a less complicated way than with Helmholtz’s vibration microscope. One way to do this is to embed a small force-measuring sensor in the top of the violin bridge, at the point where the string sits: the next link explains how that works. Such a sensor doesn’t interfere (much) with the player’s ability to perform in a normal way, so it is possible to capture waveforms of “bridge force”. We will see many examples in the course of this chapter. The signature of Helmholtz motion in such a measurement is a sawtooth waveform, as shown in the lower panel of Fig. 2. The force ramps steadily upwards (or downwards if the direction of bowing is reversed), then it jumps abruptly down (or up) at the moment when the Helmholtz corner reflects from the bridge.


A measured example is shown in Fig. 3. The sawtooth shape is clearly recognisable, but we can see some details that are different from the idealised version. There are small wiggles during the “ramp” phase of each cycle, and if you look very carefully you can see that these wiggles are not exactly the same in every cycle, although they are very similar. We will find out in section 9.2 what causes these wiggles, and also about other details of the string motion, and what the player can do to influence them.

Figure 3. Measured bridge force waveform showing Helmholtz motion, for the note D$_4$ played on the G string of a violin

In the decades following Helmholtz’s book, there was a surprising amount of published research on the vibration of violin strings. We get a clue about why this apparently obscure topic was being studied, from a story about Lord Rayleigh (1842—1919), who overlapped for much of his career with Helmholtz. According to a biography written by his son [2], Rayleigh sometimes worried that he might run out of physics problems of the kind he liked to study. It seems extraordinary to us today that anyone in 1900 or so might have thought that “physics was nearly all done”, because we know that just a few years later the revolutionary developments of relativity and quantum mechanics would open up new vistas of unimagined science. But, of course, they didn’t know that at the time.

We can try to glimpse the mindset of a 19th century scientist. We get an interesting hint from Rayleigh’s classic book “The theory of sound” [3]. Volume 1, dealing with mechanical vibration, came out in 1877. Rayleigh summarised the state of knowledge in his day, including a lot of his own work. Mechanics was a high-profile science in those days, as is clear from Rayleigh’s discussion and his footnote references. All the big names of European science come into the discussion: Laplace, Lagrange, Euler, Kirchhoff. But then Rayleigh produced a second edition of the book in 1894, and the material he added is very revealing. As well as new material on mechanics (such as shell vibration: recall section 3.2), there are several new sections applying his analysis methods to the new-fangled electrical circuits that were just starting to become important. Networks of capacitors and inductances obey the same mathematical equations as networks of masses and springs, so results from mechanics can be carried over directly: but the theory was developed first in the mechanical context. This age saw the very beginnings of electronics as a discipline.

So perhaps it is not so surprising that physics researchers shortly after Helmholtz, in search of unsolved problems in mechanics, chose to investigate the relatively obscure, but intriguing, question of the motion of a bowed violin string. Some of the published work is experimental: ingenious rigs were designed to allow strings to be bowed in a controlled and repeatable way, and the string motion was observed using photographic methods which allowed waveforms of string displacement to be captured for the first time. These observations revealed that a bowed string can vibrate in a great variety of ways, of which Helmholtz motion is just the simplest. Other researchers followed Helmholtz in giving mathematical form to these various observations.

These threads of research were pulled together, developed and given their definitive form by our second giant, C. V. Raman. He would later become India’s first Nobel prize winner, for discovering an optical scattering effect leading to the important technique still known as Raman spectroscopy. But earlier in his scientific career he studied several problems involving musical instruments. We have already met Raman back in section 3.6, because as well as his work on bowed strings he studied the acoustics behind the characteristic sound of Indian drums. But the work that interests us here is a “blockbuster” monograph, over 150 pages long, that Raman published in 1918 [4]. It has the snappy title “On the mechanical theory of the vibrations of bowed strings and of musical instruments of the violin family, with experimental verification of the results: Part 1” (there never was a “Part 2”).

Figure 4. C. V. Raman pictured in 1930. Image: Nobel Foundation, Public domain, via Wikimedia Commons

In this work (and other papers published around the same date), Raman advanced the subject to the limit of what was possible before the development of electronic measurement equipment, and long before the age of computers. First, he brought order to the bewildering array of bowed-string waveforms that had been revealed by measurements. He used an ingenious argument, explained in the next link, to suggest that the velocity of the string under the bow should always alternate between two fixed values, with sudden jumps between them. He was thus able to describe all the observed waveforms based on travelling “corners” on the string. Helmholtz motion is the simplest of the family, with a single corner, but there are many other possibilities with multiple corners. The number of corners gave Raman the basis for a classification scheme for what he called “higher types” of bowed-string motion. This work of Raman’s was a very early example of detailed study of a non-linear dynamical system of sufficient complexity to show multiple solutions that are qualitatively different: probably the only such system about which more was known at this early date was the motion of the planets.


We can see some examples of Raman’s “higher types”. They divide into two families: one family involves more than one stick-slip alternation in each cycle, while the other family is more like Helmholtz motion in that it involves only a single slipping episode per cycle. The simplest member of the first family will be very important to us in the next section. Figure 5 shows an example: for obvious reasons, it is known as “double-slipping motion”. In place of the Helmholtz sawtooth wave, we see a kind of double sawtooth shape with two abrupt “flyback” episodes. There are two travelling corners, and each generates a flyback jump when it hits the bridge during its round trip. Every time a corner passes the bow, it triggers a transition between sticking and slipping, so there are two slip episodes in every cycle. If you are interested, you can see an animation of the string motion during this kind of double-slipping motion in the previous link.

Figure 5. Measured bridge force for the open G string of a violin, showing “double-slipping” motion

Figure 6 shows three examples from the second family of “Raman higher types”. They each involve an underlying sawtooth shape, but superimposed on this they have a regular pattern of large wiggles. Raman explained such waveforms in terms of a large number of travelling corners (the exact number being related to the number of the wiggles in each cycle). But most of these corners do NOT trigger a transition between sticking and slipping at the bow, because the pattern involves a pair of corners reaching the bridge simultaneously from opposite sides in such a way that their effects cancel out. In each cycle there are just two corners that pass the bow without being cancelled like this, with the result that there is just a single episode of slipping in every cycle. It turns out to be shorter than the corresponding slip for Helmholtz motion, because the corner causing slipping to start is not the same one that causes it to stop: it is a different corner, arriving at the bow before the other one has had time to travel to the bridge and back.

Figure 6. Three examples of measured bridge force for the open D string of cello, showing different cases of the family of Raman “higher types” also known as “S-motion”

This family was later rediscovered by Bo Lawergren in 1980 [5], using a completely different style of analysis from Raman’s. Lawergren called this kind of waveform “S-motion”, and we will use that convenient label. You can see an animation of the string motion during an example of S-motion in the previous link. We will find out more about S-motion, and see how these particular measured bridge-force waveforms were obtained, in section 9.3.

But before that we need to see what happened in the next burst of scientific activity concerning bowed strings. This came decades after Raman’s time, but many of the key issues are pre-figured in his 1918 monograph. So we aren’t by any means done with Raman yet: his name will crop up several times during the next few sections.

[1] H. von Helmholtz: On the sensations of tone as a physiological basis for the theory of music (1862, reprinted by Dover, New York 1954)

[2] R. J. Strutt: The Life of John William Strutt, 3rd Baron Rayleigh, Edward Arnold (1924)

[3] J. W. S. Rayleigh:The Theory of Sound (1877, reprinted by Dover, New York 1945)

[4] C. V. Raman: “On the mechanical theory of the vibrations of bowed strings and of musical instruments of the violin family, with experimental verification of the results. Part I.” Indian Association for the Cultivation of Science Bulletin 15, 1–158 (1918).

[5] B. Lawergren: “On the motion of bowed violin strings.” Acustica 44, 194–206 (1980).