Abstract
This chapter focuses on the interaction between a bow and a string, and on the resulting self-sustained oscillations. It starts with the presentation of the friction phenomena and tribology of the rosin, which play a major role in the dynamics of the bowed string. Among the variety of possible dynamical regimes induced by the stick-slip mechanism on a string, the so-called Helmholtz motion (HM) deserves particular attention because it is the most widely used in the musical context. First, the kinematical and dynamical characteristics of the ideal HM are described. It is followed by the presentation of some more realistic features of the HM observed on real strings, such as the flattening effect, the wolf note, and the anomalous low frequency (ALF) tones.
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Notes
- 1.
The formal analogy between strings and pipes presented in Chap. 1 can be extended by considering a reed instrument. The bow velocity v b corresponds to the pressure p m inside the mouth, the string velocity v s to the pressure p in the mouthpiece, and the force F b/s to the air flow u input in the mouthpiece. The pressing force of the bow onto the string, a control parameter of the player, may even be associated with the maximum flow u A that can enter in the pipe [Eq. (9.22)].
- 2.
Bow hair are made from horse tails.
- 3.
Buckling refers to the deformation of beams subject to an axial compressing force.
- 4.
- 5.
Later on, Raman received the Nobel Prize in physics for his work in spectroscopy.
- 6.
This result may look strange since the force exerted by the string on its ends is not identically zero in time. In reality, the force exerted at x = L is exerted by the left part of the string on the right part of the string whereas the force impulse at stake here is local and results from the left and the right parts of the string.
- 7.
This conclusion applies in the particular case β = 1∕2 which corresponds to that of the clarinet: since the slope of the sticking branch is infinite, the stability condition (9.64) cannot be met. For other values of β, finding a stability condition is much more difficult, as mentioned for the cylindrical saxophones.
- 8.
One may think that the corresponding condition is \(Z_{\mbox{ c,T}}\varDelta \dot{\xi } = (\mu _{\mbox{ s}} -\mu _{\mbox{ d}})F_{\mbox{ P}}\), but this is not the correct result (see below).
- 9.
This equation can also be obtained from (4.64) with the following equivalence: in the case of the string, the source and the receptor are located at the same place (x = x S), with p H being the sum of all terms generated by reflections, in other words by convolutions with r 0(t) and r ℓ (t). Equation (4.65) is written for quantities pertaining to a string, in the Raman model.
- 10.
An early scientific observation of the effect has been reported by H. Bouasse [3] in an experiment which he had designed for demonstrating the stability of the playing frequency of a bowed string.
- 11.
A first approximation of the stiff string case is treated in [4].
- 12.
This quantity is labeled in contrast to the active power \(F_{n}\ V _{n}\ \cos \varphi _{n}\). The principle of this computation has been used in Chap. 9 (Sect. 9.4.5) with a different objective: when there is no hysteresis, the area A is zero, which yields a relationship between the imaginary parts of the admittances (or mobilities) and the amplitudes of the harmonics.
- 13.
Michèle Castellengo, personal communication.
- 14.
The reader may consult her website for examples of musical usage.
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Boutillon, X. (2016). Bowed String Instruments. In: Acoustics of Musical Instruments. Modern Acoustics and Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3679-3_11
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