A literature review:
The acoustics of the violin
Lei Fu April, 2015 Instructor: Prof. Gary P. Scavone
Final project of Seminar MUMT 618 Schulich School of Music, McGill University
3.3 Higher frequencies, statistical and hybrid methods
3.3.1 Frequency responses and bridge admittance
A small internal microphone, positioned on the nodal lines of A1 and the first transverse standing-wave mode, can be used to make the measurement, a trick which is especially valuable for the larger instruments where the problem of getting far enough away to be in the external far field becomes acute.
For some of the earlier published measurements, it is not entirely clear what the direction of the applied force was. Some researchers now routinely make separate measurements with force applied parallel and perpendicular to the top plate, while others are careful to specify the bowing direction as described above. The issue has also been raised of the extent to which artificially applied force at the bridge produces the same result as excitation by the vibrating string . For an instrument equipped with a bridge-force sensor, as discussed in section 2, it is possible to measure a bridge admittance using actual playing as the input signal. Admittance is defined as the velocity response per unit applied force, so if an accelerometer is used for the measurement, the result is integrated in the computer.
Examples of this impedance ratio are shown in figure 19, for five typical violins, and also for five typical classical and flamenco guitars. Understanding the mechanism and significance of some of the characteristic features of violin response will be the main goal of this section.
3.3.2 Stastical and hybrid theories of structural vibration
The most well-known theoretical framework is called statistical energy analysis (SEA), first developed in the 1960s in the context of such problems as noise and vibration prediction for space vehicles during launch.
In recent years, an important extension has been made to SEA that opens up a much wider range of potential applications. A feature of many complex structures is a range of intermediate frequencies where some parts of the structure need to be treated by a statistical method like SEA, while others retain fairly low modal and statistical overlap factors, so that deterministic methods are needed. A consistent hybrid theory has been developed to apply to such problems, and a number of example studies have shown that the method can be very effective.
3.3.3 The 'Bridge hill' and other hybrid features of violin response
Many excellent violins show a broad peak of response in the vicinity of 2.5 kHz, a feature which has been called the “bridge hill”. The hump was originally called the ‘bridge hill’ by Jansson, because its frequency is generally in the same region as the first in- plane bending vibration mode of a typical modern violin bridge, first explained by Reinicke and Cremer and illustrated in figure 20(a).
Y(ω) = Ydir(ω) + Yrev(ω).where Ydir and Yrev denote the direct and reverberant contributions to the bridge admittance Y.
Ydir (ω) ≈ ⟨Y (ω)⟩,where ⟨· · ·⟩ represents an average taken over a frequency band of width , centred on ω.
A simulated example is shown, in amplitude and phase, in figure 22(a) (for more detail, see Woodhouse [139]). Corresponding results for the bridge hill of a typical violin are shown in figure 22(b). These plots show the full response with all the detailed resonances, and also, as dashed lines, the computed Ydir(ω).
Figure 23 shows plots of Ydir(ω) for four violins, now including one with significantly different behaviour, and for four typical guitars. Although the bridge hill is the most-studied ‘hybrid’ feature of the violin, it is almost certainly not the only one. One of the violins shown in figure 23, an experimental ultra- light instrument by American violin maker Joseph Curtin, shows a very clear double peak, suggesting the presence of a second similar feature. There is an intriguing hint in the detailed results of Bissinger (for example ) that in this frequency range the soundpost appears to show significant extensional deformation, so perhaps a soundpost resonance (modified by added inertia of the top and back plates) is producing a ‘soundpost hill’? In addition, several researchers such as Jansson and Curtin have recently been looking at the bridge admittance in the perpendicular direction and have commented on another ‘hill’ feature seen in the 5–6 kHz range.
By keeping the material properties, total area and shell thickness all fixed, it is possible to make fair comparisons of total radiated sound power from the various structures in response to point driving at a ‘typical’ position (see caption for details).
An example of the results is reproduced in figure 24. The flat-plate critical frequency is indicated by a vertical dashed line. The result for the baffled flat plate shows the behaviour described earlier. The complete sphere shows, in this case, a strong peak close to the critical frequency.
