Introduction to Acoustics

584 Part E Music, Speech, Electroacoustics
Part E 15.2
Another case of practical importance in musical
acoustics is a bar clamped at one end and free at the
other. This would, for example, describe the bars of
a tuning fork or could be used to model the vibrations of
the neck or finger board on a stringed instrument. In this
case, there is an addition m = 0 vibrational mode, with
exponential decay length comparable with the length of
the bar. The modal frequencies are then given [Fletcher
and Rossing [15.5], (2.64)] by
�
ωm = h
�
π
�2 E
4 L 12ρ(1 − ν2 )
�
1.194 2 , 2.988 2 , 5 2 ,... ,(2m + 1) 2�
.
(15.77)
In the above discussion, we have described the modes in
terms of the number m of half-wavelengths of the sinusoidal
component of the wave solutions within the length
of the bar. A different nomenclature is frequently used
in the musical acoustics literature, with the mode number
classified by the number of nodal lines (or points in
one dimension) m in a given direction not including the
boundaries rather than the number of half-wavelengths
m between the boundaries, as in Fig. 15.51.
Twisting or Torsional Modes
In addition to flexural or bending modes, bars can
also support twisting (torsional) modes, as illustrated
in Fig. 15.51 for the z = xy (1,1) mode.
The frequencies of the twisting modes are determined
by the cross section and shear modulus G, equal
to E/2(1 + ν) for most materials (Fletcher and Rossing
[15.5], Sect. 2.20). The wave velocity of torsional
waves is dispersionless (independent of frequency) with
ωn = ncTk, where
cT = ω
k =
�
GKT
= α
ρI
�
E
, (15.78)
2ρ(1 + ν)
where GKT is the torsional stiffness given by the couple,
C = GKT∂θ/∂x, required to maintain a twist of the bar
through an angle θ and I = � ρr 2 dS is the moment of
inertia per unit length along the bar. For a bar of circular
cross section α = 1, for square cross section α = 0.92,
and for a thin plate with width w>6h, α = (2h/w). For
a bar that is fixed at both ends, fn = ncT/2L, while for
a bar that is fixed at one end and free at the other, fn =
(2n + 1)cT/4L,wheren is an integer including zero.
Thin bars also support longitudinal vibrational
modes, but since they do not involve any motion perpendicular
to the surface they are generally of little acoustic
Fig. 15.51 Schematic illustration of the lowest-frequency
twisting (1,1) and bending (2,0) modes of a thin bar with
free ends
importance, other than possibly for the lowest-frequency
soundpost modes for the larger instruments of the violin
family.
Two-Dimensional Bending Modes
Solutions of the thin-plate bending wave solutions in two
dimensions are generally less straightforward, largely
because of the more-complicated boundary conditions,
which couple the bending in the x-andy-directions. For
a free edge parallel to the y-axis, the boundary conditions
are (Rayleigh [15.3] Vol. 1, Sect. 216)
∂2z ∂x2 + ν ∂2z = 0
∂y2 and
�
∂ ∂2z ∂x ∂x2 + (2 − ν) ∂2z ∂y2 �
= 0 . (15.79)
Thus, when a rectangular plate is bent downwards along
its length, it automatically bends upwards along its width
and vice versa. This arises because downward bending
causes the top surface of the plate to stretch and the
bottom surface to contract along its length. But by Poisson
coupling, this causes the top surface to contract
and lower surface to stretch in the orthogonal direction,
causing the plate to bend in the opposite direction
across its width. This is referred to as anticlastic bending.
The Poisson ratio ν can be determined from the ratio
of the curvatures along the bending and perpendicular
directions.
In addition, for orthotropic materials like wood, from
which soundboards and the front plates of most stringed
instruments are traditionally made, the elastic constants
are very different parallel and perpendicular to the grain
structure associated with the growth rings. McIntyre