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Coupled Oscillators
Consider two coupled crossing-frequency component modes a and b. For simplicity, assume that some external constraint increases the frequency of mode a, which in the absence of coupling leaves the frequency of mode b unchanged, as illustrated in Figure 5, dashed lines. As soon as the coupling is “switched on,” two uncoupled normal modes A and B are formed (solid lines) describing the in- and out-ofphase coupled vibrations of the a and b component modes. Well away from the crossing frequency (coincidence), the normal modes retain their characteristic component mode forms with only a small contribution from the other mode. However, as coin- cidence is approached, the normal modes acquire an increasing contribution from the other mode. This results in the illustrated veering in opposite directions of the normal mode frequencies away from those of the otherwise uncoupled component
modes. At coincidence, the normal mode frequencies are split by an amount (Δ) determined by the coupling strength, with the two com- ponent modes vibrating with equal energy in either the same or opposite phases. Well above coincidence, the normal mode A continues to acquire an increasing component b mode character at the expense of mode a. Similarly, on passing through coincidence the character of normal mode B changes from b to a, as illustrated.
The vibrational modes of the violin can be considered as independent normal modes, with resonant responses identical to those of a simple harmonic oscillator, describing the coupled modes of the component modes of vibration of the top and back plates, the ribs, the cavity air modes, the neck and fingerboard assembly and their resonance, the tailpiece, and strings.
an acoustic fingerprint for individual violins. They act as monopole sound sources radiating uniformly in all directions. Additional weak CBR, A1, and oth- er higher frequency modes are also often observed but usually only contribute weakly to the radiated sound.
(2) A transitional frequency range from around 800- 1,500 Hz, where there is a cluster of quite strong resonances that cannot so easily be characterized without detailed modal analysis measurements and analysis, such as those made by Bissinger (2008a,b) and Stoppani (2013). At these frequencies and above, the modes act as additional multipole sourc- es, with the radiated sound fluctuating strongly with both frequency and direction. This results in what Weinreich (1997) refers to as directional tone color, with the intensity of partials or the quality of sound of bowed notes varying rapidly with both direction and frequency.
(3) A high-frequency range extending to well above 4 kHz, below which there is often a rather broad peak around 2-3 kHz, originally referred to as the bridge hill (BH) feature, although no longer considered a property of the bridge alone. The density of the over- lapping damped resonances makes it increasingly difficult to identify individual resonances. Above
Figure 5. A schematic representation of the veering and splitting of normal mode frequencies describing the coupling of two component oscillators or vibrational modes.
around 3 kHz, there is a relatively rapid roll-off in the frequency response of around 12 dB/octave, as indicated by the solid line with slope −2. This is be- cause the bridge acts like a strongly damped reso- nant input filter coupling the string vibrations to the radiating modes of the body shell.
The relative contributions and acoustic importance of the signature and higher frequency components to the sound of a violin are highlighted in Audio 3, http://goo.gl/UtNOI4, which illustrates the unfiltered recorded sound of a violin, then when the hard cut-off filters are applied first above and then below 1 kHz, and then with the their combined sounds repeated.
In the high-frequency range, a statistical approach argu- ably provides a more useful way of describing the acoustic response, with a relatively broad, formantlike frequency response, with superimposed fluctuations in amplitude de- pendent on mode spacing and damping (Woodhouse and Langley, 2012 , Sect. 3.3).
At a casual glance, all fine Italian violins and many later and modern instruments have very similar acoustic responses to those shown in Figure 3. Yet players can still recognize large
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