Introduction to Acoustics

probably plays a rather similar role to the bridge, as it is
via the vibrations of this central region that the largerarea
radiating surfaces of the front plate are excited.
At low frequencies this will be mainly by the lowestorder
twisting and flexing modes of the central island
region. It therefore seems likely that the dynamics of the
central island region also contributes significantly to the
BH hill feature and the resulting acoustical properties
of the violin in the perceptually important frequency
range of ≈ 2–4 kHz, as recognised by Cremer and his
colleagues [15.11].
Historically, the role of the soundpost and the coupling
of plates through enclosed air resonances were first
considered analytically using relatively simple mass–
spring models with a limited number of degrees of
freedom to mimic the first few resonances of the violin,
as described in some detail by Cremer ([15.29],
Chap. 10). Now that we can obtain detailed information
about not only the frequencies, but also the shapes
of the important structural modes of an instrument from
finite-element calculations, holography and modal analysis,
there is greater emphasis on analytic methods based
on the observed set of coupled modes.
The Complete Instrument
Bowing, plucking or striking a string can excite every
conceivable vibration of the supporting structure including,
where appropriate, the neck, fingerboard, tailpiece
and the partials of all strings both in front of and behind
the bridge. Many of the whole-body lower-frequency
modes can be visualised by considering all the possible
ways in which a piece of soft foam, cut into the
shape of the instrument with an attached foam neck and
fingerboard, can be flexed and bent about its centre of
mass.
Figure 15.59 illustrates the flexing, twisting and
changes in volume of the shell of a freely supported violin
for two prominent structural resonances computed
by Knott [15.96] using finite-element analysis. However,
not all modes involve a significant change in net
volume of the shell, so that many of the lower-frequency
modes are relatively inefficient acoustic radiators. Nevertheless,
since almost all such modes involve significant
bridge motion, they will be strongly excited by the player
and will produce prominent resonant features in the
input admittance at the point of string support on the
bridge. They can therefore significantly perturb the vibrations
of the string destroying the harmonicity of the
string resonances and resulting playability of particular
notes on the instrument, especially for bowed stringed
instrument.
Musical Acoustics 15.2 Stringed Instruments 591
Helmholtz Resonance
Almost all hand-held stringed instruments and many
larger ones such as the concert harp make use of
a Helmholtz air resonance to boost the sound of their
lowest notes, which are often well below the frequencies
of the lowest strongly excited, acoustically efficient,
structural resonances. For example, the lowest acoustically
efficient body resonance on the violin is generally
around 450 Hz, well above the bottom note G3 of
the instrument at ≈ 196 Hz. Similarly, the first strong
structural resonance on the classical acoustic guitar is
≈ 200 Hz, well above the lowest note of ≈ 82 Hz.
To boost the sound in the lower octave, a relatively
large circular rose-hole is cut into the front plate of
the guitar and two symmetrically facing f-holes are cut
into the front plate of instruments of the violin family.
The air inside the enclosed volume of the shell of such
instruments vibrates in and out through these openings
to form a Helmholtz resonator.
The frequency of an ideal Helmholtz cavity resonator
of volume V, with a hole of area S in one of
its rigid walls is given by
ωH =
�
γ P S
ρ L ′ �
S
= c0
V L ′ , (15.88)
V
Mode 10 @ 436 Hz
Mode 15 @ 536 Hz
Fig. 15.59 Representative finite element simulations of the
structural vibrations of a violin, with greatly exaggerated
vibrational amplitudes (after Knott [15.96])
Part E 15.2