Introduction to Acoustics

ural waves is much smaller than the overall curvature,
the vibrational modes will be closely related to the flexural
waves of a circular disc, with frequencies satisfying
Chladni’s generalised empirical law, fmn ∼ c(m + 2n) p ,
as confirmed by Perrin et al. [15.218].
The flexural modes involve radial displacements
proportional to cos(mφ). Continuity requires that there
must also be a tangential displacement, such that u +
∂v/∂φ = 0, where u and v are the radial and tangential
velocities respectively. Coupling to the tangential
motion explains why it is possible to feed energy into
a vibrating wine glass or the individual glass resonators
of a glass harmonica (Rossing [15.201] Chap. 14), by
rubbing a wetted finger around the rim. The excitation is
very similar to the slip–stick mechanism used to excite
the bowed string (Sect. 15.2.4).
Figure 15.136 illustrates a set of holographic measurements
by Rossing ([15.201], Fig. 12.4), which is
typical of most bell shapes. The (m, 1) and (m, 2)
modes can immediatly be related to the (m, n) modes
of a cupped disc. However, there is a distinct change
in character for n > 0, with an additional node appearing
close to the rim – referred to as a (m, 1#)
mode. Some insight is provided by the finite element
solutions for a typical large English church bell
illustrated in Fig. 15.137 [15.221]. The three modes
illustrated are very similar to the (3, 0), (3, 1) and
(3, 2) modes expected from the simple cupped disc
model, except for the lowest frequency (3, 0) mode,
2,0 3,0 4,1# 5,1# 6,1#
2,1 3,1 4,1 5,1 6,1
2,2 3,2 4,2 5,2 6,2
Fig. 15.136 Holographic interferograms and nomenclature
for vibrational modes of a hand bell (after Rossing [15.201])
Musical Acoustics 15.4 Percussion Instruments 659
(3,1) (3,1#) (3,2)
Fig. 15.137 Finite-element solutions for the lowest-order m
= 3 modes of an English church bell (after Perrin et al.
[15.219]) illustrating the (3,1), (3,1#) and (3,2) modes
in which the top surfaces of the bell move in antiphase
with the rim, to give a nodal line about
half-way along the length. Similarly, the anticipated
(4, 0), (5, 0) and (6, 0) modes of the handbell investigated
by Rossing [15.201] acquire an additional
nodal line close to the end rim denoted as (4, 1#),
(5, 1#) and (6, #) modes. Such features can generally
only be accounted for by detailed computational
analysis.
Bell Tuning
Figure 15.138 shows the frequencies and associated
modes of a well-tuned traditional church bell (Rossing
and Perrin [15.220]), ordered into groups based on mode
Group 0
Group I
Group II
Group III
End view
(2,0) 0.5
0.30
Prime
1.0
0.54
0.19
(2,2) 2.7
0.54
Tierce Nominal (Twelfth) (Upper octave)
(3,1) 1.2 (4,1) 2.0 (5,1) 3.0 (6,1) 4.2
Quint (Major third)
0.19
(3,1#) 1.5 (4,1#) 2.5 (5,1#) 3.7 (6,1#) 5.0
(3,2) 2.6 (4,2) 3.3 (5,2) 4.5 (6,2) 5.9
Fig. 15.138 Measured frequencies for a typical D5 church bell,
indicating the relative frequencies of the observed mode, with the traditional
names associated with such modes indicated (after Rossing
and Perrin [15.220])
Part E 15.4