Introduction to Acoustics

Fig. 15.40. A surprising aspect of these measurements is
the apparent lack of any such feature for modern master
violins, possibly because of a wider variation in bridge
resonances and effective masses of bridge and plate resonances
in the chosen instruments. From measurements
of the radiated sound of over 700 violins, Dünnwald
proposed that the presence of a number of strong acoustic
resonances in the broad frequency band from 1.5 to
4 kHz was one of the distinguishing features of a really
fine instrument. The influence of the bridge in accounting
for such a peak and the reduced response at higher
frequencies is clearly important.
Woodhouse [15.76] has recently revisited the problem
of the coupling between bridge and body of the
instrument and the origin of the BH peak. A simple theoretical
model shows that the peak depends on many
factors, such as the effective masses, Q-values and
resonant frequencies of the major vibrational modes
of the bridge and the multi-resonant properties of the
instrument. To demonstrate the overall effect of the
bridge without having to consider the detailed vibrational
response of a particular instrument, Woodhouse
first considered coupling to a simplified model for the
vibrational modes of the coupled instrument. This assumed
a set of coupled vibrational modes each having
the same effective mass M and Q-value, with a constant
spacing of resonances ω0 = 2π∆ f . Different values
for these parameters would need to be used to model
the independent rotational or bouncing modes, though
Woodhouse concentrates on the influence of the lowest
frequency “rocking” bridge mode. The merit of such
a model is that the multi-resonant response of such a system
varies monotonically with frequency. The features
introduced by the resonant properties of the bridge can
then be easily identified and the input admittance expressed
relative to the admittance AV for a completely
rigid bridge of the same mass, where
AV(ω) = 1
M
�
n
iω
(nω0) 2 − ω 2 + iωnω0/Q .
(15.55)
The corresponding input admittance for the onedegree-of-freedom
model bridge is then given by
AV + iω/mω 2 B
ABB(ω) =
1 − (ω/ωB) 2 , (15.56)
+ iωmAV
where m is the effective mass of the bridge and ωB its
resonant frequency and internal damping of the bridge
has been neglected.
We can also define a nonlocal admittance or mobility
AVB to describe the induced body motion per unit force
|ABB| (dB)
40
20
Musical Acoustics 15.2 Stringed Instruments 573
0
0 1 2 3 4 5 6
|AVB| (dB)
(kHz)
40
20
Fe iwt
iωm
k/iω
Zviolin
0
0 1 2 3 4 5 6
(kHz)
Fig. 15.41 Response curves for a one-degree-of-freedom
bridge coupled to an artificial set of regularly spaced
(200 Hz), constant effective mass (100 g) and constant Q
(50) structural resonances. The upper curves illustrate the
effect of bridge mass on the admittance ABB measured at
the point of excitation at the top of the bridge, while the
lower curves illustrate the corresponding induced body mobility
AVB. Thecoloured response curves are for lossless
bridges with effective masses 1, 1.5 and 3 g (highest to lowest
response), having the same resonant frequency at 3 kH
(after Woodhouse [15.76]) The black curves show the violin
body response AV that would be measured using a massless
rigid bridge
at the foot of the bridge given by
AV
AVB(ω) =
1 − (ω/ωB) 2 . (15.57)
+ iωmAV
The simulations in Fig. 15.41 illustrate the major
effect of the bridge resonance on both the input response
and induced body motion and hence radiated
sound at, around and above the resonant frequency of
the bridge (3 kHz in the above example). For a real
instrument, the spacing and Q-values of the individ-
Part E 15.2