Introduction to Acoustics

at 300 Hz, corresponding to the two plates vibrating in
the same phase, with no volume change and hence no
coupling to the air resonance. However, the coupling
via the Helmholtz resonance splits the degenerate plate
modes, to give a normal mode at a raised frequency, with
the plates vibrating in opposite directions in a breathing
mode. The coupling also decreases the frequency of
the Helmholtz cavity resonance. The unperturbed mode
may dominate the measured admittance and affect the
playability of the instrument via its perturbation of string
resonances. But, because there is no associated volume
change, it will be an inefficient acoustic radiator. One
should note that, because of the changes in phase of the
air resonance on passing through resonance, it appears as
a dispersive curve superimposed on the low-frequency
wings of the stronger higher-frequency body resonances.
The frequency of the excited normal mode is not the
peak in the admittance curve (i. e. its modulus), as often
assumed but is more nearly mid-way between the
maximum and minimum, where its phase lags 90 ◦ relative
to the phase of the higher frequency normal modes.
Similarly, the upper body mode results in a dispersive
feature in the opposite sense, as its phase changes from
almost 180 ◦ to 0 ◦ relative to the unshifted normal mode.
Very similar, but narrower, dispersive features are also
observed in admittance-curve measurements from string
resonances, unless they are purposely damped.
Cavity Modes
In addition to the Helmholtz air resonance, there will
be many other cavity resonances of the air enclosed
within the shell of stringed instruments, all of which can
in principle radiate sound through the f- or rose-holes.
Alternatively, the internal air resonances can radiate
sound via the vibrations they induce in the shell of
the instrument, as discussed in some detail by Cremer
([15.29], Sect. 11.4). Because of the relatively
small height of the violin ribs, below around 4 kHz
the cavity air modes are effectively two dimensional
in character. Simple statistical arguments based on the
overall volume of the violin cavity show that there are
typically ≈ 28 resonances below this frequency, as observed
in measurements by Jansson [15.101]. Whether
or not such modes play a significant role in determining
the tonal quality of an instrument remains a somewhat
contentious issue. However, at a given frequency, the
wavelengths of the flexural modes of the individual
plates and the internal sound modes will not, in general,
coincide. The mutual coupling and consequent
perturbation of modes will therefore tend to be rather
weak. Even if such coupling were to be significant, it is
Musical Acoustics 15.2 Stringed Instruments 593
likely to be far smaller than the major changes in modal
frequencies and shapes introduced by the f-holes and
soundpost.
Finite-Element Analysis
To progress further in our understanding of the complex
vibrations of instruments like the violin and guitar, it
is necessary to include the coupled motions of every
single part of the instrument and to consider the higherorder
front and back plate modes, which will be strongly
modified by their mutual coupling via the connecting
ribs and, for the violin, the soundpost as well.
Such a task can be performed by numerical simulations
of the vibrations of the complete structure using
finite-element analysis (FEA). This involves modelling
any plate or shell structure in terms of a large number
of interconnected smaller elements of known shape
and elastic properties. This division into smaller segments
is known as tessellation. Provided the scale of
the tessellation is much smaller than the acoustic wavelengths
at the frequencies being considered, the motion
of the structure as a whole can be described by the threedimensional
translations and rotations of the tessellated
elements. The motion of each element can be related to
the forces and couples acting on the adjoining faces of
each three-dimensional (3-D) element. The problem is
then reduced to the solution of N simultaneous equations
proportional to the number of tessellated elements.
Deriving the frequencies and mode shapes of the resulting
normal modes of the system involves the inversion
of a N × N matrix. Such calculations can be performed
very efficiently on modern computer systems, though the
computation time, proportional to N 2 , can still be considerable
for complex structures, particularly if a fine
tessellation is used to evaluate the higher-frequency,
shorter-wavelength, modes.
Figure 15.59 has already illustrated the potential
complexity of the vibrational modes of a violin. The
displacements have been greatly exaggerated for graphical
illustration. In practice, the displacements of the
plates are typically only a few microns, but can easily
be sensed by placing the pad of a finger lightly on
the vibrating surfaces. The first example shows a typical
low-frequency mode involving the flexing and bending
of every part of the violin, but with little change in its
volume, so that it will radiate very little sound. The second
example illustrates a mode involving a very strong
asymmetrical vibration of the front plate, excited by the
rocking action of the bridge with the soundpost inhibiting
motion on the treble side of the instrument. Such
a mode involves an appreciable change in volume of the
Part E 15.2