Introduction to Acoustics

a) In-phase and phase-quadrature string resonance b)
425 450 475
500
Frequency (Hz)
90° phase
400 500 (Hz)
Musical Acoustics 15.2 Stringed Instruments 579
180° phase
400 500 (Hz)
Fig. 15.47a,b Measurements of the in-phase and in-quadrature resonant response of violin strings coupled via the bridge
to a strong body resonance (Gough [15.33]). The shift of the broader resonances relative to the unperturbed narrow
resonance indicates the extent of the perturbative coupling. (a) tuning the A-string resonance through a coupled resonance
at ≈ 460 Hz; (b) the splitting of the string–body normal modes for the more strongly coupled, second partial, of the
heavier G-string
at coincidence there is a slight indication of split modes
somewhat smaller than the widths. The splitting of
modes is clearly seen for the second partial of the much
heavier G-string (Fig. 15.47b), with symmetrically split
broad string/body modes above and below the narrow
uncoupled mode. Not surprisingly, this violin suffered
from a pronounced wolf note when played at 460 Hz in
a high position on the G-string, but not on the lighter
D- or A-string. Such effects tend to be even more pronounced
on cellos due to the very high bridge providing
strong coupling between the vibrating strings and body
of the instrument.
On plucked string instruments the inharmonicity of
the partials of a plucked note induced by coupling at
the bridge to prominent structural resonances causes
beats in the sound of plucked string, which contribute
to the characteristic sound of individual instruments.
Woodhouse [15.81,82] has recently made a detailed theoretical,
computational and experimental study of such
effects for plucked notes on a guitar taking account of
the effect of damping on the coupled string–corpus normal
modes. This is sometimes not taken into proper
account in finite-element software, in which the normal
modes of an interacting system are first calculated ignoring
damping, with the damping of the modes then added.
As is clear from Fig. 15.46, such an approach will always
break down whenever the width of resonances associated
with damping becomes comparable with the splitting of
the normal modes in the absence of damping, as is frequently
the case in mechanical and acoustical systems.
Polarisation
We have already commented on the response of a bridge
mounted centrally on a symmetrically constructed instrument,
with string vibrations perpendicular to the
front plate exciting only symmetric modes of the body
of the instrument, while string vibrations parallel to
the front plate induce a couple on the front plate
exciting only asymmetric modes. The terminating admittance
at the bridge end of the string will therefore
be a strongly frequency dependent function of the
polarisation direction of the transverse string modes.
The angular dependence of the terminating admittance
lifts the degeneracy of the string modes resulting
in two independent orthogonal modes of transverse
string vibration, with different perturbed frequencies and
damping, polarised along the frequency-dependent principal
directions of the admittance tensor. If a string is
excited at an arbitrary angle, both modes will be excited,
so that in free decay the directional polarisation
Part E 15.2