Introduction to Acoustics

Relative bow force
1
0.1
0.01
0.001
0
0.01
Sul ponticello
0.02 0.04 0.1 0.2
Maximum bow force
Brilliant
Minimum bow force
Normal
Higher modes
Raucous
Sul tasto
0.7 1.4 2.8 7 14
Distance from bridge to bow (cm)
ß
1000
100
10
Bow force (g)
Fig. 15.31 The playing range for a bowed string as a function
of bow force and distance from bridge, with the bottom
and right-hand axis giving values for a cello open A-string
with a constant bow velocity of 20 cms −1 (after Schelling
[15.56])
β is the fractional bowing point along the string. If the
downwards force is larger than Fmax the string remains
stuck to the string instead of springing free into the slipping
regime, while for downward forces less than Fmin
an additional slip occurs leading to a double-slipping
motion.
Figure 15.31 is taken from the article by Schelling
on the bowed string in the Scientific American special
issue on the Physics of Musical Instruments [15.57]. It
shows how the sound produced by a bowed cello string
changes with bow position and downward bow pressure
for a typical bow speed of 20 cm/s. Note the logarithmic
scales on both axes. In practice, a string can be bowed
over a quite a large range of distances from the bridge,
bow speeds and pressures with relatively little change in
the frequency dependence of the spectrum and quality
of the sound of an instrument, apart from regions very
close and very distant from the bridge. Nevertheless, the
ability to adjust the bow pressure, speed and distance
from the bridge, to produce a good-quality steady tone, is
one of the major factors that distinguish an experienced
performer from the beginner.
Slip–Stick Friction
An important advance was the use of a more realistic
frictional force, dependent on the relative velocity
between bow and string, shown schematically for
three downward bow pressures in Fig. 15.32. Such
a dependence was subsequently observed by Schumacher
[15.58] in measurements of steady-state sliding
friction between a string and a uniformly moving bow.
Vp(t)
Increasing bow pressure
Musical Acoustics 15.2 Stringed Instruments 567
Friction
0 v B
String velocity vs(t)
Fig. 15.32 Schematic representation of the dependence of
the frictional force between bow and string on their relative
velocity and downward pressure of the bow on the string.
The straight line with slope 2R0 passes through the velocity
vp of the string determined by its past history and the
intersection with the friction curves determines its current
velocity. The open circle represents the single intersection
in the slipping regime at low bow pressures, while the closed
circles illustrate three intersections at higher pressures
The frictional force is proportional to the downward bow
pressure.
Friedlander [15.59] showed that a simple graphical
construction could be used to compute the instantaneous
velocity v at the bowing point from the velocity vp(t)at
the bowing point induced by the previous action of the
bow. The new velocity is given by the intersection of
a straight line with slope 2R0 drawn through vp with
the friction curve, where R0 is the characteristic string
impedance. This follows because the localised force between
the bow and string generate secondary waves with
velocity F/2Z0 at the bowing point as previously described
(15.34). In the slipping region well away from
capture, there will be just a single point of intersection,
so the problem is well defined. However, close to capture,
as illustrated by the intersections marked by the
black dots with the upper frictional curve, the straight
line can intersect in three points (two in the slipping
regime and one in the sticking regime) as first noted by
Friedlander.
Computational Models
This model has been used in a number of detailed
computational investigations of both the transient
and steady-state dynamics of the bowed string, notably
by the Cambridge group lead by McIntyre
and Woodhouse [15.60–62], their close collaborator
Schumacher [15.58, 63] from Carnegie-Mellon, and
Part E 15.2