Introduction to Acoustics

and Woodhouse [15.86] have published a detailed account
of the theory and derivation of elastic constants
from measurement of plate vibrations in both uniform
and orthotropic thin plates, including the influence of
damping.
For an isotropic rectangular thin plate, hinged along
on all its edges, a simple two-dimensional (2-D) sinewave
solution satisfies both the wave equation (15.68)
and the boundary conditions, with m and n halfwavelengths
along the x-andy-directions, respectively,
giving modal frequencies
�
E
ωmn = h
12ρ(1 − ν2 ��mπ �2 � � �
2
nπ
+ .
) Lx L y
(15.80)
By analogy with our discussion of flexural waves in
one-dimensional bars, we would expect the modal frequencies
of plates with clamped or free edges to be
raised, with the nodes of the sinusoidal components of
the wave solution moved inwards from the edges by
approximately quarter of a wavelength. For the higherorder
modes, the modal frequencies would therefore be
given to a good approximation by
�
E
ωmn = h
12ρ(1 − ν2 0
1
2
3
)
��(m �2 � � �
2
+ 1/2) π (n + 1/2) π
+
.
Lx
L y
(15.81)
As recognised by Rayleigh ([15.3] Vol. 1, Sect. 223), it
is difficult to evaluate the modal shapes and modal frequencies
of plates with free edges. The method used by
Rayleigh was to make an intelligent guess of the wavefunctions
which satisfied the boundary conditions and
to determine the frequencies by equating the resulting
potential and kinetic energies. Leissa [15.88] hasreviewed
various refinements of the original calculations.
For a plate with free edges, the nodal lines are also no
longer necessarily straight, as they were for plates with
freely hinged edges.
Chladni Patterns
The modal shapes of vibrating plates can readily be visualised
using Chladni patterns. These are obtained by
supporting the plate at a node of a chosen mode excited
electromagnetically, acoustically or with a rosined
bow drawn across an edge. A light powder is sprinkled
onto the surface. The plate vibrations cause the powder
to bounce up and down and move towards the nodes
Musical Acoustics 15.2 Stringed Instruments 585
0 1
2
3
4
Fig. 15.52 Chladni pattern with white lines indicating the
nodal lines of the first few modes of a rectangular plate
(after Waller [15.87])
of the excited mode, allowing the nodal line patterns to
be visualised. Figure 15.52 illustrates Chladni patterns
measured by Waller [15.87] for a rectangular plate with
dimensions Lx/L y = 1.5, with the number of nodal lines
between the boundary edges determining the nomenclature
of the modes. Note the curvature of the nodal lines
resulting from the boundary conditions at the free edges.
Figure 15.53 illustrates the nodal line shapes and
relative frequencies of the first 10 modes of a square plate
with free edges, where f11 = hcL/L 2√ 1 − ν/2 (after
Fletcher and Rossing, [1] Fig. 3.13).
Another important consequence of the anticlastic
bending is the splitting in frequencies of combination
modes that would otherwise be degenerate. This is illustrated
in Fig. 15.54 by the combination (2, 0) ± (0, 2)
normal modes of a square plate with free edges. The
(2, 0) ± (0, 2) modes are referred to as the X- and ringmodes
from their characteristic nodal line shapes. The
(2, 0)−(0, 2) X-mode exhibits anticlastic bending in the
same sense as that induced naturally by the Poisson coupling.
It therefore has a lower elastic energy and hence
lower vibrational frequency than the (0, 2)+(2, 0) ring-
(1,1) (2,0)–(0,2) (2,0)+(0,2) (2,1) (1,2)
1.00 1.52 1.94 2.71 2.71
(2,2) (3,0) (0,3) (3,1)–(1,3) (3,1)+(1,3)
4.81 5.10 5.10 5.30 6.00
Fig. 15.53 Schematic representation of the lowest 10 vibrational
modes of a square plate with free edge
Part E 15.2