Civil Engineering Reference
In-Depth Information
( )
( )
−
1
⎛
⎞
J
s
−
j
2
1
⎜
⎟
⎟
ρρ
=
1
−
.
(5.30)
0
⎜
s
−
j
J
s
−
j
0
⎝
⎠
Assuming that the length
d
of the tube in question is much shorter than the wavelength
we may express the specific acoustic impedance as
−
1
( )
( )
⎛
⎞
J
s
−
j
Δ
p
2
1
⎜
⎟
⎟
Z
=
=
j
ωρ
d
1
−
.
(5.31)
s
0
⎜
v
s
−
j
J
s
−
j
x
0
⎝
⎠
At low frequencies, setting
s
in Equation
(5.28)
less than ≈ 2.0, a very good
approximation for the term enclosed in the parenthesis is 4/3 - j⋅8/
s
2
, which gives
8
μ
d
j
3
Z
≈
+
ωρ
d
,
(5.32)
s
0
2
a
or expressed by the acoustic impedance
8
μ
π
d
j
3
ωρ
d
0
Z
≈
+
.
(5.33)
a
4
a
2
a
π
The viscosity then gives us two effects. First, we get a resistive part being analogous to
the mechanical damping coefficient, and we shall note the relationship with a quantity to
be introduced later, the
airflow resistance.
This is a very important material parameter
for all porous media. Using the definitions found in the international measurement
the flow resistivity of the tube, having symbol
r
and dimension Pa⋅s/m
2
. Second, the
viscosity also affects the mass term in the expression for the impedance. We get an
increase of one-third compared with our earlier calculations (see Equation
(5.22)
).
For a panel, either a slatted one, i.e. an assemblage of parallel beams, or perforated
by thin slits, we shall need an expression for the equivalent viscous losses in a single
long slit (see
Figure 5.10)
. We shall assume that the input pressure is the same along the
whole length of the slit, the length being long in comparison with the wavelength.
Furthermore, the pressure
p
in the slit varies only in the x-direction, and the velocity in
this direction is only dependent on the z-coordinate. It may be shown (see e.g. Vigran
and Pettersen (2005)), that the effective density of the air in the slit of width
b
may be
written
−
1
kb
′
⎛
⎞
tan(
)
⎜
⎟
ωρ
2
′
0
ρρ
=
1
−
,
where
k
=
.
(5.34)
⎜
⎟
0
kb
′
j
μ
⎜
⎟
⎝
2
⎠
For an (infinitely) long slit in a plate of thickness
d
the specific impedance will then be
