Introduction to Acoustics

586 Part E Music, Speech, Electroacoustics
Part E 15.2
(2,0)
(0,2)
(2,0)+(0,2)
(0,2)–(2,0)
Fig. 15.54 Formation of the ring- and X-modes by the
superposition of the (2, 0) and (0, 2) bending modes
mode, with curvatures in the same sense in both the xand
y-directions. The ring- and X-modes will therefore
be split in frequency above and below the otherwise degenerate
mode, as illustrated in Fletcher and Rossing
([15.5], Fig. 13.11).
Plate Tuning
The modal shapes and frequencies of the lowest-order
(1,1) twisting mode and the X-and ring-modes are
widely used for the scientific tuning of the front and back
plates of violins following plate-tuning guidelines developed
by Hutchins [15.89, 91]. These are referred to as
violin plate modes 1, 2 and 5, as illustrated in Fig. 15.55
by Chladni patterns for a “well-tuned” back plate. The
violin maker aims to adjust the thinning of the plates
286 533 628 672 731 Hz
873 980 1010 1124 1194 Hz
Fig. 15.56 Typical modal shapes for a number of low-frequency
modes of a guitar top plate from time-averaged holographic measurements
by Richardson and Roberts [15.90]
Mode 1 Mode 2 Mode 5
Fig. 15.55 Chladni patterns for the first twisting-(#1),
X-(#2) and ring-(#5) modes of a viola back plate (after
Hutchins [15.89])
across the area of the plate to achieve these symmetrical
nodal line shapes at specified modal frequencies.
The use of such methods undoubtedly results in
a high degree of quality control and reproducibility of
the acoustic properties of the individual plates before
assembly and, presumably, of the assembled instrument
also, especially for the lower-frequency structural resonances.
Unfortunately, they do not necessarily result
in instruments comparable with the finest Italian instruments,
which were made without recourse to such
sophisticated scientific methods. Traditional violin makers
instinctively assess the elastic properties of the plates
by their feel as they are twisted and bent, and also by
listening to the sound of the plates as they are tapped
or even rubbed around their edges, rather like a bowed
plate. From our earlier discussion, it is clear that the
mass of the plates is also important in governing the
acoustical properties.
Geometrical Shape Dependence
The above examples demonstrate that the lowerfrequency
vibrational modes of quite complicated
shaped plates can often be readily identified with those
of simple rectangular plates, though the frequencies of
such modes will clearly depend on the exact geometry
involved. This is further illustrated in Fig. 15.56 by
the modal shapes of a guitar front plate obtained from
time-averaged holography measurements by Richardson
and Roberts [15.90], where the contours indicate
lines of constant displacement perpendicular to the surface.
For the guitar, the edges of the top plate are rather
good nodes, because of the rather heavy supporting ribs
and general construction of the instrument. The boundary
conditions along the edges of the plate are probably
intermediate between hinged and clamped. The modes
can be denoted by the number of half-wavelengths along
the length and width of the instrument. Note that cir-