Monopole Radiation at Low Frequencies in the Violin
J. E. McLennan
At 1 kHz, the wavelength of the fundamental is about equal to the linear dimension of the violin body. At lower frequencies, especially below about 500 Hz (~C5), monopole radiation is important. The monopole component can be thought of as a net surface area of the body moving in phase.
This brief note uses the data of K. D. Marshall [1, 2] for a violin named SUS 295 to estimate the monopole component for four main resonances below 1 kHz, namely, A0 (278 Hz), B1 (466 Hz), B1+ (555 Hz) and B1+ (574 Hz). His experimental data show the Operating Deflection Shape (ODS) and the amplitude, in microns, at many locations for the resonances listed in his paper.
For the calculations given here, the area of the top and back plates moving in phase was determined by identifying regions in his original data plots, cutting them out on paper with area density 80 g.m^{−2} and weighing them. The areas thus obtained were multiplied by the average amplitude obtained by adding the local deflections and dividing by their number for each area, to give the volume amplitude displaced. The total change in volume is taken as twice this value and is shown in table 1. The outline of the plates was followed and no attempt was made to allow for any edge area that would be immobile or any contribution from the sides. In table 1 the volume amplitude is also shown. This is defined by Weinreich [3] as volume change multiplied by 2πf(0).
Table 1. Monopole components for four resonances from SUS 295 as determined from body deformations alone. This is an underestimate for A0: it is the component due to the finite compliance of the body and does not include the
motion of the air through the fholes.
ODS 
f(0) Hz 
Volume change cc 
Volume amplitude cc/s 
A0 (body only) 
278 
0.98 
1860 
B1 
466 
2.6 
7610 
B1+ (neck torsion) 
555 
0.4 
1510 
B1+ (neck bending) 
574 
4.6 
18000 
It can be seen that B1 and B1+ (neck bending) have higher monopole components than B1+ (neck torsion). The value listed for A0 in table 1is a considerable underestimate of the monopole component for A0, because the A0 monopole has a relatively small component due to body motion and a rather larger one from the air in the body. A0 is a modified Helmholtz resonance, with a compliant container.
The body modes rely on the monopole component for their radiating ability at the fundamental of notes played in this range. B1 and B1+(neck bending) radiate well as shown in table 1.
There is a discrepancy in the reporting of the B1 radiation in the literature. In Marshall’s [1] paper, figure 4 shows B1 (466 Hz), B1+(neck torsion) (555 Hz) and B1+(neck bending) (574 Hz) as prominent peaks (A0 is not shown and is expected to be small). Hutchins [4] in figure 2 for the same violin, shows “monopole” peaks at A0 (280 Hz), an “air” peak at 455 Hz and a body peak at 555 Hz. These seem to be in conflict with Marshall who has B1 at 466 Hz, B1+ at 555 Hz and an “air” peak at 478 Hz. The two papers appear to be at variance in assigning the peaks at ~460 Hz and at 555 Hz. Marshall has B1 at 466 Hz and no “air” peak until 478 Hz (A1) where it is usually found. Hutchins shows only one peak, 555 Hz, whereas Marshall has three peaks close together at 555Hz, 574 Hz and 599 Hz, of which the peak at 574 Hz is the strongest.
Since modal analysis is not possible on a routine basis, the determination of effective mass and stiffness of body ODS may give a measure of the strength of the monopole component of resonances at frequencies below 1 kHz where fundamentals are important. R is defined as the ratio of mechanical impedance, Z, measured near the bass foot of the bridge, to the quality factor, Q. A low value for R, the radiation constant, i.e. a low value for Z and a large Q, would indicate good radiation. The determination of these parameters is easily made with inexpensive equipment. As described in my thesis [5], with the violin set up to show the peak of interest, the dependence of the peak frequency on added mass can be obtained and from this the values for effective mass, m, effective stiffness, s, and Z are calculated. Q can be found from the bandwidth knowing the peak height.
The effective mass and stiffness for the ODS associated with A0 is shown in table 2 for the modern version of the violin that featured in my thesis [5].
Table 2 Effective mass and stiffness for the body ODS associated with A0.
f(0) Hz 
df/dm Hz/kg 
m kg 
s MN/m 
Z kg/s 
Q (f(0)/Δf) 
R kg/s 
285 
450 
0.32 
1.04 
577 
14 
41 
274 
500 
0.27 
0.81 
468 
14 
33 
278 
800 
0.17 
0.53 
300 
12 
25 
The variation in these experimental results could be due to minor modifications in setup including change of strings, soundpost position, etc.
References
1. K. D. Marshall, “Modal analysis of a violin” J. Acoust. Soc. Am. 77 (2) 1985, p695.
2. K. D. Marshall, private communication.
3. G. Weinreich, “Violin Radiativity: Concepts and Measurements” SMAC 83 Vol II, p99.
4. C. M. Hutchins, “The future of violin research” Catgut Acoustical Society J. vol 2, (Series II) May 1992 p1.
5. J. E. McLennan, Doctoral thesis “The violin: music acoustic from Baroque to Romantic” 2009.
