Introduction to Acoustics

undercut on their back surface to improve the harmonicity
of the lower frequency modes (Fig. 15.120). Each bar
has an acoustic resonator immediately below it, consisting
of a cylindrical tube, which is closed at the far end.
Any of the flexural and torsional modes can contribute
to the initial sound, when the bar is struck by a hammer;
however, most modes die away rather quickly so that at
longer times the sound is dominated by the resonances
of the coupled pipe resonator.
Figure 15.123 shows the initial part of the waveform
and spectrum of a typical xylophone note ( ),
illustrating the initial large amplitudes and rapid decay of
the higher frequency bar modes excited and the strongly
excited but slowly decaying resonances of the first two
modes of the air resonator. All modes contribute to the
initial sound but the sound at longer times is dominated
by the lowest-frequency bar modes and resonantly tuned
air resonators.
The marimba is closely related to the xylophone, but
differs largely in its playing range of typically two to
four and a half octaves from A2 (110 Hz) to C7 (2093),
though some instruments play down to C2 (65 Hz). In
contrast to xylophone bars, which are undercut near their
centre to raise the frequency of their second partial from
2.71 to 3.0 above the fundamental, marimba bars are
often thinned still further to raise the frequency of the
second partial to four times the fundamental frequency
(Fig. 15.119).
Marimbas produce a rather mellow sound and are
usually played with much softer sticks than traditionally
used for the xylophone. Although the marimba is nowadays
used mostly as a solo percussion instrument, in the
1930s ensembles with as many as 100 marimbas were
played together. In many ways, such ensembles were
the forerunners of today’s Caribbean steelbands, to be
described later in this section.
The vibraphone is similar to the marimba, but uses
longer-sounding aluminium rather than wooden bars and
typically plays over a range of three octaves from F3 to
F6. Like the marimba, the bar thickness is varied to give
a second partial two octaves above the fundamental.
They are usually played with soft yarn-covered mallets,
which produce a soft, mellow tone. In addition, the vibraphone
incorporates electrically driven rotating discs
at the top of each tuned air resonator, which periodically
changes the coupling. This results in a strong amplitudemodulated
vibrato effect. The wave envelope of audio
is shown in Fig. 15.124 for a succession of
notes played on the vibraphone with vibrato, which are
then allowed to decay freely. The vibrato rate can be
adjusted by changing the speed of the electric motor.
Musical Acoustics 15.4 Percussion Instruments 651
Strikes Free decay
10 ms
Fig. 15.124 Envelope of a succession of notes on the vibraphone,
which freely decay with modulated coupling to
tuned air resonators to produce an amplitude modulated
vibrato effect
Note the very long decay of the sound, which can be
controlled by a pedal-operated damper.
Triangle
The triangle is a very ancient musical instrument formed
from a cylindrical metal bar bent into the shape of a triangle,
with typical straight arm lengths of 15–25 cm. They
are usually struck with a similar-diameter metal rod.
Although the instrument is small and therefore a very
inefficient acoustic radiator, it produces a characteristic
high-frequency ping or repetitive high-pitched rattle,
which is easily heard over the sound of a large symphony
orchestra (audio ). The quality of the
sound can be varied by beating at different positions
along the straight arms. The triangle is usually supported
be a thread around the top bend of the hanging
instrument.
The flexural modes of a freely suspended bar of circular
cross section are fn ∼ (a/2) √ E/ρ(2n + 1) 2 π/8L 2 ,
with frequencies in the ratios 3 2 :5 2 :7 2 :9 2 :11 2 :13 2 (see
Sect. 15.2.4). Transverse flexural vibrations can be excited
perpendicular or parallel to the plane of the
instrument. For vibrations perpendicular to the plane, the
bends are only small perturbations. Transverse modes in
this polarisation are therefore almost identical to those of
the straight bar from which the triangle are bent (Rossing
[15.212] Sect. 7.6). However, for flexural vibrations
in the plane, there is a major discontinuity in impedance
at each bend, because the transverse vibrations within
one arm couple strongly to longitudinal vibrations in
Part E 15.4