Introduction to Acoustics

Grossmann et al. [15.215] andFletcher [15.214]
have considered the vibrations of a spherical-cap shell
of height H and thickness h. Interestingly, the change in
frequency of the asymmetric vibrations about the equilibrium
point depends only on the ratio of the amplitude
to the arching height, a/H, as illustrated in Fig. 15.129.
When the height of the dome is much less than the
thickness, the frequency increases approximately as a 2 ,
as expected from the induced increase in tension with
amplitude. However, when the arching becomes comparable
with and greater than the thickness, the asymmetry
of the potential energy dominates the dynamics and results
in an initial decrease in frequency, which increases
strongly with the ratio of arching height to thickness. At
very large amplitude, a ≫ H, the frequency is dominated
by the increase in tension and therefore again increases
with amplitude like a flat plate. At large amplitudes,
Legge and Fletcher [15.216] have shown that changes
in the curvature of the plate profile result in a large
transfer of energy to the higher-frequency plate modes.
We now show how many of the above properties
relate to the sounds of cymbals, gongs of various types
and steelpans.
Cymbals
Many types of cymbals are used in the classical symphony
orchestra, marching bands and jazz groups. They
are normally made of bronze and have a diameter of
20–70 cm. The low-frequency modes of a cymbal are
very similar to those of a flat circular plate and can
be described using the same (mn) mode nomenclature
ω/ω0
1.4
1.2
1.0
0.8
0.6
0.4
0
0.2
h/H
2.0
1.2
1.0
0.8
0.6
0.2
0.4
1 2
a/H
Fig. 15.129 Vibration amplitude a dependence of frequency
of lowest axisymmetric mode of a spherical cap of dome
height H as a function of thickness h to H ratio
Musical Acoustics 15.4 Percussion Instruments 655
(Fletcher and Rossing [15.5], Fig. 20.2). However, small
changes in curvature across a cymbal will results in
modes that are linear combinations of ideal circular-plate
modes.
Cymbals are usually played by striking with
a wooden stick or soft beater or a pair can be crashed together,
each method producing a distinctive sound. They
can even be played by bowing with a heavy rosined bow
against the outer rim. Rossing and Shepherd showed
that the characteristic 60 dB decay time of the excited
modes of a large cymbal varies approximately inversly
proportional with frequency, with a typical decay time
for the lowest (20) mode as long as 300 s (Fletcher and
Rossing [15.5], Fig. 20.5)
Figure 15.130 shows the waveform envelope and
spectra of the initial sound of a cymbal crash and after
1 s. Audio illustrates a recorded cymbal
crash followed by the same sound played first through a
0–1 kHz and then a 1–10 kHz band-pass filter, illustrating
the decay of the low- and high-frequency wide-band
noise.
When a cymbal is excited with a drumstick, waves
travel out from the excitation point with a dispersive
group velocity proportional to k, inversely proportional
to the dimensions of the initial flexural indention of the
surface made by the drumstick. The dispersive pulse
strikes and is reflected from the edges of the cymbal
(dB)
0
–20
–40
0s
3s 60 ms
1s
–60
0 1 2 3 4 5 6 7 8 9 10
(kHz)
Fig. 15.130 Wave envelope and spectrum at start and after
1 s of a cymbal clash illustrating wide-band noise at all
times
Part E 15.4