Civil Engineering Reference
In-Depth Information
being independent of the angle of incidence. We also pointed out that this assumption is
reasonably correct for many porous materials. However, one will often encounter cases
where one cannot use this assumption but we shall postpone the treatment of calculation
models for such cases until later.
Simple models for a homogeneous and isotropic porous material consider it to
behave as a fluid; we shall use the term
equivalent fluid.
Such a fluid may be
characterized by its propagation coefficient Γ (or by the complex wave number
k
′ = -
j⋅Γ) and its complex characteristic impedance
Z
c
. Alternative descriptions use the
bulk
modulus K
and the
equivalent
(or
effective
)
density
ρ
.
The relations between these
quantities are given by the following expressions
ρ
−
p
ZK
=
ρ
and
Γ = ⋅
j
ω
with
K
V
=
Δ
.
(3.82)
c
K
V
The last equation defines the bulk modulus; the ratio between the pressure and the
relative change in volume.
As an introduction to these simple models, which we shall treat in more detail in
Chapter 5, we shall assume we have an infinitely thick wall of a porous material with a
given characteristic impedance
Z
2
= ρ
2
c
2
and a wave number
k
2
. The medium of the
incident wave is characterized using an index 1 as shown in
Figure 3.15
. We shall, as
before, calculate the reflection coefficient and further examine the conditions necessary
for the porous material to behave as locally reacting. The sound pressure for the three
partial waves is given by
ˆ
−
j(
kx
cos
ϕ
+
ky
sin
ϕ
)
pp
pp
pp
=⋅
=⋅
=⋅
e
,
1
1
i
i
ˆ
j(
kx
cos
ϕ
−
ky
sin
ϕ
)
e
,
(3.83)
1
1
r
r
ˆ
−
j(
kx
cos
ψ
+
ky
sin
ψ
)
e
.
2
2
t
t
We have as before omitted the time dependence e
jω
t
. Furthermore, the reflection is
specular; a condition that immediately will follow from the boundary conditions without
being shown in detail here.
Another important law will, however, follow from these equations. The pressure
must be equal on both sides of the boundary, i.e. for
x
= 0 we get
p
i
+
p
r
=
p
t
. Applying
this to Equations
(3.83)
we obtain
Snell's law
:
k
sin
ϕ
=
k
sin
ψ
,
(3.84)
1
2
which, by using the sound speeds, may be written
2
sin
ϕ
sin
ψ
⎛⎞
c
2
ψ
2
ϕ
(3.85)
=
or
cos
=
1
−
⋅
sin
.
⎜⎟
⎝⎠
c
c
c
1
2
1
As the two media are in contact with each other, this implies that the normal component
of the particle velocity on both sides will be the same. This will give another boundary
condition stating that
