Introduction to Acoustics

ically in Fig. 15.33. The frictional force fn entering
(15.53) is evaluated from the pressure- and velocity-dependent
frictional force using the Friedlander
construction with the computed string velocity under
the bow given by un + u ′ n .
Pressure Broadening and Flattening
As an example, Fig. 15.34 illustrates the computed
velocity of the string under the bow as a function of
increasing bow pressure (McIntyre et al. [15.60]). In contrast
to the rectangular waveform predicted by the simple
Raman model, the waveform is considerably rounded,
especially at low bow pressures. This results in a less strident,
less intense sound, with the higher partials strongly
attenuated. At higher pressures, but at the same position
and with the same bow velocity, the rounding is less
pronounced, so that higher partials become increasingly
important. The increased intensity of the higher partials
leads to an increased perceived intensity with bow
pressure, in contrast to the Raman model, in which the
waveform and hence intensity remains independent of
bow pressure. This is referred to as the pressure effect.
At even higher pressures, the ambiguity in intersections
a)
b)
c)
vs
Fig. 15.34 Computed velocity of string at bowing point
for increasing bow pressures in the ratios 0.4 : 3 : 5(after
McIntyre and Woodhouse [15.60]) illustrating both the
broadened waveform and pitch dependence on bow pressure
compared with the idealised rectangular Helmholtz
bowed waveform
t
t
t
Musical Acoustics 15.2 Stringed Instruments 569
noted by Friedlander leads to a pronounced increase in
the capture period and hence the pitch of the bowed
note, known as the flattening effect. These features are
discussed in considerable detail along with his own important
research and that of his collaborators on such
effects by Cremer ([15.29], Chaps. 7 and 8).
Initial Transients
Computational models can also describe the initial
transients of the bowed string before the steady-state
Helmholtz wave is established. Figure 15.35 compares
the computed and measured initial transients of the string
velocity under the bow for a string played with a sharp attack
(a martelé stroke) (McIntyre, Woodhouse [15.60]).
These computations also include the additional excitation
of torsional waves, which are excited because the
bowing force acts on the outer diameter of the wire, exerting
a couple in addition to a transverse force. The
excitation and loss of energy to the torsional waves appears
to encourage the rapid stabilisation of the bowed
Helmholtz waveform.
For low-pitched stringed instruments such as the
double bass, it is very important that the Raman bowed
waveform is established very quickly, otherwise there
will be a significant delay in establishing the required
pitch. Remarkably, Guettler [15.64] has shown that, by
simultaneously controlling both bow speed and downward
pressure, the player can establish a regular Raman
waveform in a single period. The speed with which
a steady-state bowed note can be established can be
represented on a Guettler diagram, where the number
of slips before a steady-state Helmholtz motion is established
can be illustrated as a two-dimensional histogram
as a function of bowing force and acceleration of the
bow speed from zero.
a)
b)
Fig. 15.35 (a) Computed transient string velocity at the
bowing point for a strongly bowed string including coupling
to both transverse and torsional modes and (b) the
measured string velocity for a strongly played martelé bow
stroke (after McIntyre and Woodhouse [15.60])
Part E 15.2