Introduction to Acoustics

conditions at each end) are necessary to solve the
boundary-value problem. Following (22.155), four different
situations are compatible at each end:
Supported v = 0
and M(x) = E(x)Iz(x) ∂2 Clamped
v
= 0 ;
∂x2 ∂v
= 0
∂x
and
Free
v = 0 ;
T (x) = ∂
�
E(x)Iz(x)
∂x
∂2v ∂x2 �
= 0
and M(x) = E(x)Iz(x) ∂2 Guided
v
= 0 ;
∂x2 T (x) = ∂
�
E(x)Iz(x)
∂x
∂2v ∂x2 �
= 0
and
∂v
= 0 ,
∂x
where M(x) is the bending moment and T (x) isthe
shear force.
Free–Free Beam with a Constant Section
As a specific example of (22.154), we consider a homogeneous
and isotropic beam of length L, thickness h,
width b, with a constant rectangular section S = bh
whose moment of inertia with respect to the z-axis is
Iz = I = bh 3 /12 (Fig. 22.10).
In this case, (22.154) reduces to
EI ∂4v ∂x4 + ρS ∂2v = 0 . (22.156)
∂t2 Looking for solutions for (22.156) of the form v(x, t) =
Φ(x)cosωt yields the general solution
z
Φ(x) = A cosh ωx
c
y
C cos ωx
c
x
+ B sinh ωx
c +
+ D sin ωx
c
Fig. 22.10 Geometry of a beam with constant section
L
(22.157)
b
h
Structural Acoustics and Vibrations 22.4 Bars, Plates and Shells 921
with
c = √ ωcL 4
�
I
S
and cL = E
. (22.158)
ρ
This expression shows that the phase velocity c of the
flexural waves is proportional to √ ω. The dispersion
relationship is given by
EIk 4 + ρSω 2 = 0 . (22.159)
From (22.159), we can see that the group velocity cg =
dω/dk is also proportional to √ ω. The group velocity
refers to the propagation velocity of wave energy and
envelope of slowly varying amplitude. Flexural waves in
the beam are thus dispersive with the higher frequencies
propagating faster than the lower ones.
Remark. There is a paradox in (22.157) in the sense
that the group velocity tends to infinity as the frequency
tends to infinity. This results from the Euler–Bernoulli
assumptions, where shear and rotation inertia of the
cross sections are neglected. Introducing these two effects
in the model (the Timoshenko model) leads to
a bounded asymptotic value of the group velocity as
frequency increases.
For a beam free at both ends, the boundary conditions
are
∂2 y
∂x2 (0, t) = ∂2 y
(L, t) = 0 (22.160)
∂x2 and
∂3 y
∂x3 (0, t) = ∂3 y
(L, t) = 0 , (22.161)
∂x3 1
0.5
0
–0.5
–1
0
2
4
6
8
10
12
14
16
18
20
X
Fig. 22.11 Graphical resolution of the eigenfrequency equation
for a free–free beam, with X = ωL/c
Part G 22.4