Civil Engineering Reference
In-Depth Information
This occurs if
|
Z(M
3
)
|
is infinite or equal to zero. If
|
Z(M
3
)
|
is finite, a more
general condition is
Z
∗
(M
3
)Z
c
+
Z(M
3
)Z
c
=
0. If
Z
c
is real, this occurs if
Z(M
3
)
is
|
is greater than 1, the amplitude of the outgoing wave is larger than
the amplitude of the ingoing wave. If
Z
c
is real, this occurs if the real part of
Z(M
3
)
is
negative. More generally, the coefficient
R
can be defined everywhere in a fluid where an
ingoing and an outgoing wave propagate in opposite directions. For instance, it has been
shown previously that the ratio
p/v
for a travelling wave propagating in the positive
x
direction is
Z
c
. As indicated by Equation (2.20) there exists only an ingoing wave at
M
inside a fluid if the impedance at
M
is the characteristic impedance. The behaviour
of the reflection coefficient as a function of
x
is much simpler than the behaviour of
the impedance. Returning to Figure 2.1, Equations (2.3) and (2.9) provide the following
relation between
R(M
2
)
and
R(M
1
)
:
imaginary. If
|
Z(M
3
)
R(M
2
)
=
R(M
1
)
exp
(
−
2
jkd)
(2.22)
where
d
x(M
2
)
. Hence, the reflection coefficient describes a circle in the
complex plane if
k
is real. If
k
is complex, the reflection coefficient describes a spiral.
It should be noted that the propagation of electromagnetic plane waves in a waveguide
can be described with impedances and reflection coefficients in a similar way to the use
of those concepts in describing sound propagation. The Smith chart, which provides a
graphical representation for the propagation of electromagnetic waves, can also be used
to describe the acoustic plane-wave propagation.
=
x(M
1
)
−
2.4.2 Absorption coefficient
The absorption coefficient
α(M)
is related to the reflection coefficient
R(M)
as follows
2
α(M)
=
1
−|
R(M)
|
(2.23)
The phase of
R(M)
is removed, and the absorption coefficient does not carry as much
information as the impedance or the reflection coefficient. The absorption coefficient is
often used in architectural acoustics, where this simplification can be advantageous. It
can be rewritten as
E
(M)
E(M)
α(M)
=
−
1
(2.24)
where
E(M)
and
E
(M)
are the average energy flux through the plane
x
=
x(M)
of the
incident and the reflected waves, respectively.
2.5
Fluids equivalent to porous materials: the laws
of Delany and Bazley
2.5.1 Porosity and flow resistivity in porous materials
Porosity
Materials such as fibreglass and plastic foam with open bubbles consist of an elastic
frame which is surrounded by air. The porosity
φ
is the ratio of the air volume
V
a
to the
