Building Acoustics

90 Building acoustics

constants for these panels. We shall give an example below (see section 3.7.3.3) and the
reason is that further on we shall calculate the sound transmission through such panels.

3.7.3.1 Free vibration of plates. One-dimensional case

The differential equation describing the wave motion is substantially more complicated
than for the ones treated above. The reason is that each element of the plate, as sketched
in Figure 3.18 c), will be influenced by moments as well as shear forces. We shall not
derive the equation, just state that the equation for the particle velocity normal to the
plate surface may be written as

42

42 0,

Bmvvyy

xt

∂∂

+ =

∂∂

(3.101)

where B and m is the plate bending stiffness per unit length and the mass per unit area,
respectively. The same differential equation applies to other quantities such as
displacement, angular velocity, shear force and bending moment but we shall use the
particle velocity as the characterizing quantity. Assuming a solution of the form

vvyy=⋅ˆ ej(ωtkx−B),

we get the following expression for the wave number kB by insertion into Equation

(3.101):

(^) B^4
B

,

m

k

cB

ω

==⋅ω (3.102)

where the phase speed cB is given by

(^) B^4.

B

c

m

=⋅ω (3.103)

As seen from this equation, the medium will be dispersive for bending waves, which
means that the phase speed will be frequency dependent. A broadband-pulsed signal will
therefore change its shape during propagation; the high frequency wave components will
outrun the components having a lower frequency. For a homogeneous plate having a
thickness h we get from Equation (3.103):

(^) BL1.8 L,
3
cchfchf
π
=≈⋅ (3.104)
where f is the frequency in Hz and where the phase speed cL for longitudinal waves in the
medium is given by Equation (3.96). We arrive at this expression by substituting for the
quantities m and B, respectively, using the following formulae
3
and 112212.
EEh
mhρ B I
υυ

=⋅ =⋅=⋅

−−

(3.105)