Page 26 - 2016Summer

Violin Acoustics
Figure 2. A schematic representation of the excitation of the vibra- tional modes of the body shell and Helmholtz f-hole resonance by the bowed string via the asymmetric rocking of the bridge.
modes of the body shell, which are only weakly perturbed by their coupling to other attached parts of the violin such as the neck, fingerboard, tailpiece supporting the strings, and even the player holding the instrument.
String Vibrations
Hermann von Helmholtz (1821-1894) was the first to both observe and explain the transverse vibrations of the bowed string. His measurements and their interpretation were de- scribed in The Sensation of Tone (Helmholtz, 1863), which laid the foundation for the discipline of psychoacoustics and our understanding of the perception of sound. Although a strongly bowed string appears to be vibrating as a simple half-wavelength sinusoidal standing wave, what we observe is only the time-averaged parabolic envelope of the much more interesting Helmholtz wave.
For an ideally flexible string with rigid end supports, Helm- holtz showed that the waveform consists of two straight sec- tions of the tensioned string rotating in opposite direction about its ends, with a propagating “kink” or discontinuity in the slope at their moving point of intersection. The kink traverses backward and forward at the speed of transverse waves, √T/μ , reversing its sign on reflection at both ends, where T is the tension and μ is the mass per unit length of the string. The Helmholtz wave is therefore periodic with the same repetition frequency or pitch as the fundamental sinu- soidal mode of vibration.
Such a wave can be considered as the Fourier sum of the sinusoidal eigenstates of an ideal string with rigid end sup- ports, with “harmonic” partials (fn = nf1), and amplitudes varying as 1/n, where n is an integer and f1 is the frequency of the fundamental component. On an ideal string, such a wave would propagate without damping or change in shape.
In practice, the Helmholtz string vibrations are excited and controlled in amplitude by the high nonlinear frictional “slip- stick” forces between the moving bow hair and string similar to the forces giving rise to the squeal of car tires under heavy breaking. Video 1 (http://goo.gl/UtNOI4) (Wolfe, 2016) il- lustrates the bowed waveform as it sticks to and then slips past the steadily moving bow.
To produce sound, the string vibrations clearly have to transfer energy to the radiating shell modes via the asym- metrically rocking bridge. As a result, each mode of the string contributing to the component partials of the Helm- holtz wave will be selectively damped and changed in frequency by its coupling to the individual shell modes (Gough, 1981). Nevertheless, provided the coupling of the lowest partials is not too strong, the highly nonlinear, slipstick, frictional forces between the string and rosined bow can still excite a repetitive Helmholtz wave. Cremer (1984) showed that the kink is then broadened with addi- tional ripples that are also excited by secondary reflections of the kink at the point of contact between string and bow.
If the fundamental string mode contributing to the Helm- holtz wave is too strongly coupled to a prominent body res- onance, even the highly nonlinear frictional force between bow and string is unable to sustain a repetitive wave at the intended pitch. The pitch then rises an octave or leads to a warbling or croaking sound, the infamous “wolfnote,” which frequently haunts even the finest instruments, especially on fine cellos. This is an extreme example of the way the string- shell mode coupling affects the “playability” of an instru- ment (Woodhouse, 2014, Sect. 5), which is almost as impor- tant to the player as its sound.
The excitation and properties of Helmholtz waves on the bowed string are so important that Cremer (1984) devotes almost half his seminal monograph on The Physics of the Vi- olin to a discussion of string vibrations. In Cambridge, UK, McIntyre and Woodhouse (1979) developed elegant com- puter simulations to investigate the physics involved, with more recent advances described by Woodhouse (2014, Sect. 2) in his recent comprehensive review of violin acoustics.
Major advances in our understanding of how the player and the properties of the bow determine the time evolution and shape of the circulating kink, hence the sound of the bowed string, were made by the late Knutt Guettler (2010), a vir- tuoso soloist and teacher of the double bass. The rapid ex- citation of regular Helmholtz waves on short, low-pitched,
24 | Acoustics Today | Summer 2016