Introduction to Acoustics

652 Part E Music, Speech, Electroacoustics
Part E 15.4
(dB)
0
–100
0 10
(kHz)
2.5 s
Fig. 15.125 The decaying waveform and FFT spectra at the
start (upper trace) and after 1 s (lower trace) of a struck
triangle note
the adjacent arms. Hence each arm will support its own
vibrational modes, which will be coupled to form sets
of normal-mode triplets, since each arm is of similar
length.
Figure 15.125 illustrates the envelope and 50 ms
FFTs of the initial waveform and after 1 s, illustrating
the very well-defined and only weakly attenuated highfrequency
modes of the triangle. Note the wide-band
spectrum at the start of the note from the initial impact
with the metal beater.
Chimes and Tubular Bells
We include orchestral chimes and bells in this section
because their acoustically important vibrations
are flexural modes, just like those of a xylophone or
triangle. The radius of gyration of a thin-walled cylindrical
tube of radius a is ≈ a/ √ 2. The frequency
of the lowest flexural modes is then be given by
fn ∼ a √ E/2ρ(2n + 1) 2 π/8L 2 .
Orchestral chimes are generally fabricated from
lengths of 32–38 mm-diameter thin-walled tubing, with
the striking end often plugged by a solid mass of brass
with an overhanging lip, which provides a convenient
striking point.
Fletcher and Rossing ([15.5], Sect. 19.8) note that
the perceived pitch of tubular bells is determined by the
frequencies of the higher modes, excited with frequencies
proportional to 9 2 ,11 2 , and 13 2 , in the approximate
ratios 2:3:4. The pitch should therefore sound an octave
below the lowest of these. Readers can make there
own judgement from audio , which compares
the rather realistic sound of a tubular bell synthesised
from the first six equal amplitude modes of an ideal
bar sounded together, followed by the 9 2 ,11 2 , and 13 2
modes in combination, and then by a pure tone an octave
below the 9 2 partial. Such comparisons highlight
the problem of subjective pitch perception in any sound
involving a combination of inharmonic partials.
15.4.3 Plates
Flexural Vibrations
This section describes the acoustics of plates, cymbals
and gongs, which involve the two-dimensional
flexural vibrations of thin plates described by the
two-dimensional version of (15.149). Unlike stringed
instruments, we can usually assume that the plates of
percussion instruments are isotropic. Well away from
any edges or other perturbing effects such as slots
or added masses, the standing-wave solutions at high
frequencies will be simple sinusoids. However, close
to the free edges, and across the whole plate at low
frequencies, contributions from the exponentially decaying
solutions will be equally important over a distance
∼ (E/12ρ(1 − ν2 )) 1/4 (h/ω) 1/2 . The nodes of the sinusoidal
wave contributions will be displaced a distance
∼ 1/4λ from the edges. Hence, the higher frequency
modes of a freely supported rectangular plate of length
a, width b and thickness h will be given, to a first
approximation, by
�
E
ωmn ∼ h
12ρ � 1 − ν2� �1/4 π 2
��m �2 � � �
2
+ 1/2 n + 1/2
+
. (15.152)
a
b
A musical instrument based on the free vibrations of
a thin rectangular metal plate is the thunder plate, which
when shaken excites a very wide range of closely spaced