Civil Engineering Reference
In-Depth Information
are described in Chapter 4. Viscosity and heat conduction in tubes lead to dissipative
processes, and in a macroscopic description of sound propagation, the density
ρ
and
the bulk modulus
K
must be replaced by complex quantities. The wave number
k
and
the characteristic impedance
Z
c
given by Equations (2.2) and (2.7) respectively, become
complex:
k
=
Re
(k)
+
j
Im
(k)
(2.8)
Z
c
=
Re
(Z
c
)
+
j
Im
(Z
c
)
If the amplitude of the waves decreases in the direction of propagation, the quantity
Im
(k)/
Re
(k)
must be negative if the time dependence is chosen as exp
(jωt)
.Inthe
alternative convention, exp
(
−
jωt)
,Im
(k)/
Re
(k)
must be positive (see Section 2.7).
2.2.4 Superposition of two waves propagating in opposite directions
The subscript
x
is removed for clarity. The pressure and the velocity, for a wave propa-
gating toward the negative abcissa are, respectively,
p
(x,t)
A
exp[
j(kx
=
+
ωt)
]
(2.9)
A
Z
c
exp[
j(kx
υ
(x,t)
=−
+
ωt)
]
(2.10)
If the acoustic field is a superposition of the two waves described by Equations (2.3)
and (2.5) and by Equations (2.9), (2.10), the total pressure
p
T
and the total velocity
v
T
are
A
exp[
j(kx
p
T
(x,t)
=
A
exp[
j(
−
kx
+
ωt)
]
+
+
ωt)
]
(2.11)
A
Z
c
exp[
j(kx
A
Z
c
exp[
j(
υ
T
(x,t)
=
−
kx
+
ωt)
]
−
+
ωt)
]
(2.12)
A superposition of several waves of the same
ω
and
k
propagating in a given direction
is equivalent to one resulting wave propagating in the same direction. The acoustic field
described by Equations (2.11) and (2.12) is the most general unidimensional monochro-
matic field. The ratio
p
T
(x,t)/v
T
(x,t)
is called the impedance at
x
. The main properties
of the impedance are studied in the following sections.
2.3
Main properties of impedance at normal incidence
2.3.1 Impedance variation along a direction of propagation
In Figure 2.1, two waves propagate in opposite directions parallel to the
x
axis. The
impedance
Z(M
1
)
at
M
1
is known. By employing Equations (2.11) and (2.12) for the
pressure and the velocity, the impedance
Z(M
1
)
can be written
A
exp[
jkx(M
1
)
]
A
exp[
−
jkx(M
1
)
]
−
A
exp[
jkx(M
1
)
]
p
T
(M
1
)
v
T
(M
1
)
=
Z
c
A
exp[
−
jkx(M
1
)
]
+
Z(M
1
)
=
(2.13)
