Civil Engineering Reference
In-Depth Information
The components
k
1
,
k
2
and
k
3
of the vector
k
are related by
k
1
k
2
k
3
k
P
+
k
P
+
k
N
=
1
(3.54)
and
A
is an arbitrary constant.
Substitution of Equation (3.53) into Equation (3.48) yields the three components of
the velocity
υ
Ak
3
/k
N
(ρ
N
K)
1
/
2
υ
3
(x
1
,x
2
,x
3
,t)
=
exp[
j(
−
k
1
x
1
−
k
2
x
2
−
k
3
x
3
+
ωt)
]
(3.55)
Ak
i
/k
P
(ρ
P
K)
1
/
2
υ
i
(x
1
,x
2
,x
3
,t)
=
exp[
j(
−
k
1
x
1
−
k
2
x
2
−
k
3
x
3
+
ωt)
]
i
=
1
,
2
Planes waves in fluids equivalent to fibrous materials can be described by Equations
(3.53) - (3.55).
3.8
Impedance at oblique incidence at the surface of a fluid
equivalent to an anisotropic porous material
The layer of a fibrous material is fixed to a rigid, impervious wall, on its rear face, and
is in contact with air on its front face, as shown in Figure 3.11.
The acoustic field in the air is homogeneous and the angle of incidence
θ
is real. The
formalism is the same if
θ
is complex. The Snell - Descartes law for refraction may be
written
k
o
sin
θ
=
k
1
(3.56)
where
k
o
is the wave number in air, and
k
1
the component on
x
1
of the wave number
vector
k
in the fibrous material.
The component
k
3
of
k
, obtained from Equation (3.54) is
k
N
1
−
1
/
2
k
1
k
P
k
3
=
(3.57)
air
fibrous
layer
θ
X
3
X
1
d
Figure 3.11
A layer of fibrous material backed by an impervious rigid wall and in
contact with air on its front face.
