Raman’s argument applies strictly to bowed-string motion based on two assumptions: that the string is an ideal, textbook string, and that the friction force is governed by a “Stribeck law” in which the friction force varies with relative sliding speed according to a curve like the example shown schematically in Fig. 1. Neither of these assumptions is exactly true for a real bowed string, as we will see later in this chapter, but they give a good first shot at explaining a lot of bowed-string waveforms.
The periodic string motions in which Raman was interested were all observed to involve the bowed string oscillating at the same frequency as the free string. As a consequence, he argued, the friction force at the bow must remain essentially constant: the modes of a string have very low damping, so any significant variation of the friction force at the frequency of string modes would produce a resonant response to high amplitude. For the extreme case of the undamped textbook string, the predicted amplitude would be infinite. Helmholtz motion is an example: it is a possible free motion of an ideal string. So one way to think about Helmholtz motion and all the other regimes we will find in this section is that they are free motions of the string which happen to be able to tolerate the presence of the bow.
Figure 1 then tells us that the string velocity at the bowed point must alternate, in some pattern, between two fixed values. During an episode of sticking, the string velocity must match the bow speed and lie somewhere on the vertical portion of the friction curve. But the velocity must integrate to zero over a complete cycle, in order that there is no mean sideways motion of the string. So episodes of sticking must be balanced by episodes of slipping with a negative string velocity. If the friction force is to remain constant during this process, the point on the curve corresponding to this slipping speed must lie on a horizontal line with the relevant sticking point, as indicated by the two black stars in the figure.
This argument shows that any possible motion of the string at the bowed point must involve spells of constant velocity interrupted by jumps, all of the same magnitude. Now we can use D’Alembert’s solution for the general motion of an ideal string (section 5.4.2) to deduce what must be happening along the rest of the string. The string displacement $w(x,t)$ at position $x$ and time $t$ must be the sum of a right-travelling wave $f$ and a left-travelling wave $g$, both with fixed shape:
$$w(x,t)=f(t-x/c)+g(t+x/c) \tag{1}$$
where $c$ is the wave speed on the string. The string has fixed ends at $x=0$ and $x=L$, so
$$f(t)+g(t)=0 \tag{2}$$
and
$$f(t-L/c)+g(t+L/c)=0 , \tag{3}$$
so
$$g=-f \tag{4}$$
and
$$f(t)=f(t+2L/c) . \tag{5}$$
The string velocity $v(x,t)$ can also be written in travelling-wave form:
$$v(x,t)=f^\prime(t-x/c) – f^\prime(t+x/c) . \tag{6}$$
from eqs. (1) and (4), where primes denote the derivative.
Our measurements are of the force exerted on the bridge. If the string tension is $T$, this is given by
$$T \left.\dfrac{\partial w}{\partial x} \right|_{x=0} = \dfrac{T}{c} \left[ g^\prime(t) – f^\prime(t) \right]=-\dfrac{2T}{c}f^\prime(t) . \tag{7}$$
So, apart from a constant multiplier, the bridge force wavefrom directly reveals the form of the travelling wave $f^\prime(t)$.
For the particular case of Helmholtz motion we already know that the bridge force is a sawtooth wave. So this sawtooth also gives the form of the travelling velocity waves. The result is animated in Fig. 2. The two travelling waves are shown at the top, over a range that is three times as long as the physical string, indicated by the vertical lines. The black shape at the bottom shows $v(x,t)$ for the string.
This may not be immediately recognisable as corresponding to Helmholtz motion. To understand the plot, try laying a vertical ruler against the screen, selecting a particular position on the string. This is your chosen bowing point. Now look at how the velocity at that point varies through the cycle of vibration. If your chosen point is near the left-hand end, where you would normally bow a violin string, you will see that it spends most of the cycle with a relatively small positive velocity, and the remainder with a larger negative velocity. This is exactly what were were expecting: the amplitude of motion will be scaled so that the positive velocity matches the bow speed. The negative sliding speed has the correct value so that the integrated velocity over the cycle is zero. This description will hold whatever point you select as your bowed point: the only thing that will change is the amplitude scaling in order to match the positive velocity to the bow speed.
We can see the string vibration much more clearly if we integrate the sawtooth waveforms to obtain the corresponding travelling wave contributions to the string displacement. Each linear ramp in the sawtooth integrates to a parabolic section of curve, and where the sawtooth had jumps, the integrated function has a sharp corner, in other words a slope discontinuity. Using these travelling waves, the corresponding animation for string displacement is shown in Fig. 3, and now it is clear that it does indeed reproduce Helmholtz motion.
Raman used this approach, via travelling waves of velocity, to catalogue all the possible idealised bowed-string waveforms. The two waves from eq. (6) must always add up to give a velocity waveform at the bowed point consisting of constant velocity segments, interrupted by jumps. He showed that this can only occur if the travelling waves of velocity take the form of a linear ramp, interrupted by jumps. The ramp segments always have the same slope, and the only distinction between different regimes of vibration comes in the number and disposition of the jumps. We will show some examples, to illustrate the two types of bridge force waveform shown in section 9.1.
Helmholtz motion is the only possible solution with a single velocity jump. The next simplest possibility has two jumps. Figure 4 shows an animation corresponding to Fig. 2, for a typical example of such motion. We know from eq. (7) that the blue curve, the right-travelling velocity wave, mirrors the waveform of bridge force, and we can recognise that waveform as corresponding to Fig. 5 of section 9.1, describing a typical case of double-slipping motion. If you do the ruler trick again, selecting a bowing point and watching how the string velocity varies at that point, you will quickly see that the motion does indeed involve two slipping episodes in every cycle. Figure 5 shows the corresponding animation for string displacement.
Such motion is classified as Raman’s “second type”. There is a special case of this motion that has direct relevance to violinists. If the two velocity jumps are arranged in a regular and symmetrical manner, the resulting motion is illustrated in Fig. 6 and 7. This is still double-slipping motion, but because the two slips in each cycle occur with equal spacing, the result is a sound with half the period: in other words, a note that plays an octave higher. Now look at the animation of the string motion in Fig. 7: it consists of two “Helmholtz motions”, going on simultaneously in the two halves of the string. The string remains stationary at its midpoint. This is the motion that arises when a violinist plays a “harmonic”, by lightly touching a finger at the mid-point of the string.
For a final example, we look at a case of “S-motion”. The particular case shown in Figs. 8 and 9 would be classified by Raman as of 7th type, based on the number of velocity jumps. The blue velocity waveform in Fig. 8 can be seen to have the same general form as the measured bridge-force waveforms in Fig. 6 of section 9.1. If you do the “ruler trick” with Fig. 8, you may be able to see that some possible bowing positions have a single slip per cycle in this vibration regime, while others show more than one slip. In practice, S-motion usually appears with a single slip: the particular bowing position then governs which Raman higher type is excited. We will see more about this in section 9.3.