Introduction to Acoustics

578 Part E Music, Speech, Electroacoustics
Part E 15.2
(Ω� –ωB)/ωB
0.08
Weak coupling K 1
ωS – ωB
ωB
–0.04 0 0.04 ωS – ωB
–0.08 0.08
Fig. 15.46 Normal modes of coupled oscillators illustrating
the profound effect of damping on the behaviour in the
cross-over region illustrated for K-values of 0.75 and √ 5
for an undamped string resonance coupled to a body resonance
with a typical Q = 25. The solid line shows the
shifted frequencies of the normal modes as the string frequency
is scanned through the body resonance, while the
dashed lines show the 3 dB points on their damped resonant
response (Gough [15.80])
In the strong-coupling limit, K > 1, the normal
modes are split at the crossover point. The losses
are also shared equally between the split modes. As
the string frequency is varied across the body resonance,
one mode changes smoothly from a normal
mode with a predominantly string-like character, to
a mixed mode at cross over, and to a body-like mode at
ωB
higher frequencies, and vice versa for the other normal
mode.
Our earlier discussion of the perturbation of string
resonances by the terminating admittance is therefore
justified in the weak-coupling regime (K ≪ 1), which
is the usual situation for most string resonances on musical
instruments. However, if the fundamental mode of
a string is over-strongly coupled at the bridge to a rather
light, weakly damped body resonance, such that K > 1,
the normal-mode resonant frequency of the vibrating
string, when coincident in frequency with the coupled
body mode, will be significantly shifted away from its
position as the fundamental member of the harmonic set
of partials. It is then impossible to maintain a steady
Helmholtz bowed waveform on the string at the pitch of
the now perturbed fundamental, which is the origin of
the wolf-note problem frequently encountered on otherwise
often very fine-stringed instruments, and cellos in
particular.
To overcome such problems, it is sometimes possible
to reduce K by using a lighter string, but more commonly
the effective Q-value is reduced by extracting energy
from the coupled system by fitting a resonating mass
on one of the strings between the bridge and tailpiece.
A lossy material can be placed between the added mass
and the string to extract energy from the system, which
might otherwise simply move the wolf note to a nearby
frequency.
String Resonances
Figure 15.47 illustrates: (a) the frequency dependence of
the in-phase and phase-quadrature resonant response of
an A-string as its tension increased, so that its frequency
passes through a relatively strongly coupled body resonance
at ≈460 Hz; (b) the splitting in frequency of the
normal modes of the second partial of the heavier Gstring
frequency tuned to coincide with the frequency
of the coupled body resonance. Superimposed on these
relatively broad resonances is a very sharp resonance
arising from transverse string vibrations perpendicular
to the strong coupling direction, to be explained in the
next section. This very weakly perturbed string resonance
provides a marker, which enables us to quantify
the shifts and additional damping of string vibrations in
the strong coupling direction.
When the frequency of the lighter A-string is tuned
below that of the strongly coupled body resonance, the
coupling lowers the frequency of the coupled string
mode, as anticipated from our earlier discussion. In
contrast, when tuned above the coupled resonance the
frequency of the coupled string mode is increased, while