Introduction to Acoustics

574 Part E Music, Speech, Electroacoustics
Part E 15.2
10
dB
2 5
10
(Hz)
Fig. 15.42 Admittances of a violin measured at the top of
the bridge (top trace) and at the left foot of the bridge (lower
trace) illustrating a strong BH peak when measured at the
top of the bridge but a relatively monotonic dependence of
the body of the instrument (after Cremer [15.29], Fig. 12.9).
The added solid line represents the 1/ f reduction in the
predicted BH response above the bridge resonance
ual modes will be very irregular and highly instrument
dependent; nevertheless, the effect of the bridge resonance
on the overall response will be very similar. In
particular, the bridge resonance gives a broad peak in
input admittance followed by a 6 dB/octave decrease
in the admittance above resonance, where the response
is largely dominated by the bridge dynamics rather
than that of the instrument itself, with ABB ∼ 1/imω.
Note that the height and width of the peak is largely
determined by energy lost to the coupled structural vibrations
(including, in practice, additional energy lost
to all the supported strings) rather than from internal
bridge losses, which have been neglected in this
example.
The bridge resonance introduces a somewhat smaller
peak in the induced body mobility and hence radiated
sound. Well above the bridge resonance, the induced
body velocity is given by AVB(ωB/ω) 2 , with an intensity
decreasing by 12 dB/octave. Unlike the input bridge
admittance, the induced body motion and output sound
retains the characteristic resonances of the instrument,
though attenuated.
The predicted difference in admittance at the top
of the bridge A BB and top of the instrument AV is
illustrated in Fig. 15.42, in measurements by Moral
and Jansson [15.77] reproduced by Cremer ([15.29],
Fig. 15.9). Whereas the average admittance of the violin
varies relatively little with frequency, the admittance
at the bridge shows a pronounced BH peak with a relatively
featureless and approximately 1/ f (the added
solid line) variation above the peak, as anticipated from
the above model.
Woodhouse [15.76] has extended this idealised
model to describe the coupling of the bridge to a more
realistic, but still simplified, model for the vibrational
modes of the violin with a soundpost. This changes the
detailed response, but not the overall qualitative features.
Because the response of a violin depends rather
randomly at higher frequencies on the positions and
Q-values of the structural modes, Woodhouse uses a logarithmic
scale to average the peaks and troughs at the
maxima and minima of the admittance (approximately
proportional to Q and 1/Q), to give a skeleton curve
describing the global variation of the violin’s complex
admittance (more details are given in the later
Sect. 15.2.3 on shell modes). This enables Woodhouse to
illustrate the influence of various bridge parameters on
the acoustical properties of the instrument, suggesting
ways in which violin makers could vary bridge properties
to optimise the sound quality of an instrument,
though that will always be a matter of personal taste
rather than being scientifically defined.
The important role of the bridge in controlling the
sound of the violin or cello has often been overlooked,
even by many skilled violin makers. Indeed one of the
reasons why Cremonese violins generally produce such
highly valued sounds is the experience and skill involved
in adjusting the mass, size and fitting of the
bridge (and the position of the soundpost) to optimize
the sound quality, investigated experimentally by
Hacklinger [15.78].
Added Mass and Muting
A familiar demonstration of the importance of the mass
of the bridge on the sound of an instrument is to place
a light mass or mute on the top of the bridge. This
dramatically softens the tone of the instrument by decreasing
the resonant frequency of the bridge and hence
amplitude of the higher-frequency components in the
spectrum of sound. The added mass ∆m lowers the
resonant frequency ωB by a factor [m/(m + ∆m)] 1/2 .