Civil Engineering Reference
In-Depth Information
6.5.1
Sound transmitted through an infinitely large plate
We shall assume that the plate lies in the xz-plane and that it is driven by an external
force per unit area, a sound pressure
p
(
x, z, t
), due to the incident plane wave (see
Figure
6.21)
.
p
i
p
r
ϕ
y
= 0
p
t
Figure 6.21
Plane wave incidence on a thin, infinitely large plate.
The first task will be, analogous to the case of point force excitation, to find an
expression for the plate velocity as a function of the sound pressure driving the plate. To
solve this we have to use a differential equation, a wave equation, where the driving
pressure is represented on the right side of the equation. Hence, to calculate the sound
reduction index we shall have to find the sound pressure in the transmitted wave. We
shall use this procedure but as an introduction we shall treat a special case neglecting the
bending stiffness of the plate, i.e. characterizing the plate by its mass impedance only.
6.5.1.1
Sound reduction index of a plate characterized by its mass impedance
We may visualize such a wall or plate as a membrane (without tensional forces) or a
collection of loosely connected point masses. A plastic curtain or something comparable
will in practice behave, acoustically speaking, in such a way. For simplicity, we shall
also assume normal sound incidence. The resulting input impedance
Z
g
in this case (see
section 3.5) will be
(6.78)
Z
=
ρ
c
+
j
ω
mZ
=
+
j
ω
m
.
g
0
0
0
This is a series connection of the mass impedance of the plate and the characteristic
impedance of the air behind the plate. Seen from the side of the incident wave the plate
will represent a boundary surface giving an absorption factor α that we may calculate
using the following equation, derived in section 3.5.1
⎧
⎪
⎨
⎪
⎩⎭
Z
Z
g
4Re
0
α
=
.
(6.79)
2
Z
⎧
⎪
⎪
Z
g
g
+
2Re
+
1
⎨
⎪
⎩⎭
Z
Z
0
0
