Civil Engineering Reference
In-Depth Information
Figure 6.5
The third 'gedanken experiment' with a glass wool. Due to the stiffness of
the glass, the frame is not affected by the increase of pressure.
The third 'gedanken experiment' is represented in Figure 6.5. Due to the stiffness
of the glass, at a first approximation, one assumes that the frame is not affected by the
increase of pressure.
Equation (6.12) can be rewritten
R
=
φK
f
(6.26)
By using Equations (6.24) - (6.26),
Q
and
P
can be written
Q
=
K
f
(
1
−
φ)
(6.27)
(
1
−
φ)
2
φ
4
P
=
3
N
+
K
b
+
K
f
(6.28)
These expressions for
P
,
Q
and
R
can be obtained directly from Equations
(6.18) - (6.20) for
K
s
infinite (the material the frame is made of is not compressible),
and can be used for most of the sound-absorbing porous materials. It may be pointed
out that the description of the third 'gedanken experiment' is valid only under the
hypothesis that the frame is homogeneous. A more complicated formulation has been
developed by Brown and Korringa (1975) and Korringa (1981) for the case of a frame
made from different elastic materials.
6.2.4 Determination of
P
,
Q
and
R
The complex dynamic coefficients of elasticity must be used at acoustical frequencies.
The rigidity of an isotropic elastic solid is commonly characterized by a shear modulus
and a Poisson coefficient. The bulk modulus
K
b
in Equation (6.28) can be evaluated by
the following equation:
2
N(ν
+
1
)
3
(
1
−
2
ν)
K
b
=
(6.29)
ν
being the Poisson coefficient of the frame. We have used
N
and
ν
instead of
N
and
K
b
to specify the dynamic rigidity of the frame. For the case of porous materials
saturated with air, it is necessary to insert the frequency-dependent parameter
K
f
defined
previously (denoted by
K
in Chapters 4 and 5), in order to obtain from the Biot theory
a correct model of sound propagation similar to the model developed in Chapter 5 for
materials having a rigid frame. The bulk modulus
K
f
can be obtained from Equations
(5.34), and (5.35) or (5.38).
