Introduction to Acoustics

566 Part E Music, Speech, Electroacoustics
Part E 15.2
coupling between motions in orthogonal transverse directions,
causing the orbits to precess (Elliot [15.53],
Gough [15.49], Villagier [15.47]), as illustrated by computational
simulations and measurements in Fig. 15.30.
The precessional frequency Ω is given by Ω ab
ω = L∆L ,
where a and b are the major and minor semi-axes of the
orbital motion and ∆L is the amount by which the string
is stretched to bring it to tension [15.49].
Such precession can lead to the rattling of the string
against the fingerboard on a strongly plucked instrument,
as the major axis of the orbiting string precesses
towards the fingerboard. The nonlinear origin of such effects
can easily by distinguished from other linear effects
causing degeneracy of the string modes and hence beats
in the measured waveform, by the very strong dependence
of the precession rate on amplitude, as illustrated
in Fig. 15.30b.
15.2.3 The Bowed String
Realistic Models
Although the main features of the bowed string can be
described by a simple Helmholtz wave, it is important
to consider how such waves are excited and maintained
by the frictional forces between the bow and string.
The simple Helmholtz solution is clearly incomplete
for a number of reasons including:
1. the unphysical nature of infinitely sharp kinks,
2. the insensitivity of the Helmholtz bowed waveform
to the position and pressure of the bow on the string.
In particular, the simple Helmholtz waveform involves
partials with amplitudes proportional to 1/n,
whereas such partials must be absent if the string
is bowed at any integer multiple of the fraction 1/n
along its length, since energy cannot be transferred
from the bow to the string at a nodal position of
a partial,
3. the neglect of frictional forces in the slipping regime,
4. the neglect of losses and reaction from mechanical
coupling to structural modes at the supporting
bridge,
5. the excitation of the string via its surface, which must
involve the excitation of additional torsional modes.
Understanding the detailed mechanics of the strongly
nonlinear coupling between the bow and string has been
a very active area of research over the last few decades,
with major advances in our understanding made possible
by the advent of the computer and the ability to
simulate the problem using fast computational methods.
Cremer ([15.29], Sects. 3–8) provides a detailed account
of many of the important ideas and techniques used to investigate
the dynamics of the bowed string. In addition,
Hutchins and Benade ([15.27], Vol. 1), includes a useful
introduction to both historical and recent research prefacing
20 reprinted research papers on the bowed string.
Woodhouse and Galluzzo [15.54] have recently reviewed
present understanding of the bowed string.
Pressure, Speed and Position Dependence
In the early part of the 20th century, Raman [15.35], later
to be awarded the Nobel prize for his research on optoacoustic
spectroscopy, confirmed and extended many
of Helmholtz’s earlier measurements and theoretical
models of the bowed string. Raman used an automated
bowing machine to investigate systematically the effect
of bow speed, position and pressure on bowed string
waveforms. He also considered the attenuation of waves
on the string and dissipation at the bridge. From both
measurements and theoretical models, he showed that
a minimum downward force was required to maintain
the Helmholtz bowed waveforms on the string, which
was proportional to bow speed and the square of bow
distance from the bridge. He also measured and was
able to explain the wolf-note phenomenon, which occurs
when the pitch of a bowed note coincides with an
over-strongly coupled mechanical resonance of the supporting
structure. At such a coincidence, it is almost
impossible for the player to maintain a steady bowed
note, which tends to stutter and jump in a quasi-periodic
way to the note an octave above, illustrated previously
for a cello with a bad wolf note, audio .
Saunders [15.55], well known for his work in
atomic spectroscopy (Russel–Saunders spin-orbit coupling)
was a keen violinist and a cofounder of the
Catgut Acoustical Society. He showed that, for any given
distance of the bow from the bridge, there was both
a minimum and a maximum bow pressure required for
the Helmholtz kink to trigger a clean transition from
the sticking to slipping regimes and vice versa. Subsequently,
Schelling [15.56] derived explicit formulae for
these pressures in terms of the downward bow force F
as a function of bow speed vB, assuming a simple model
for friction between bow hair and string in the slipping
region of µdF and a maximum sticking force of µsF,
R
Fmin =
2 0vB 2Rβ2 and
(µs − µd)
Fmax = 2R0vB R
= 4β Fmin , (15.51)
β (µs − µd) R0
where R0 is the characteristic impedance of the string
terminated by a purely resistive load R at the bridge, and