Civil Engineering Reference
In-Depth Information
Adding this new equation to the system of Equations (11.85), a new system is formed
with a (
N
+
2
)
×
(N
+
2) square matrix
T
0
V
=
0
···
−
(
1
+
R)
0
(11.93)
[
D
]
The determinant of this matrix is equal to zero, so
T
is calculated by
R
)
det[D
N
+
1
]
det[D
1
]
T
=−
(
1
+
(11.94)
where det[
D
N
+
1
] is the determinant of the matrix obtained when the (
N
+
1)th column
has been removed from matrix [
D
].
For a plane wave of incidence
θ
, the transmission loss is defined by
TL
=−
10 log
τ(θ)
(11.95)
T
2
(θ)
where
τ(θ)
|
is the transmission coefficient for the angle of incidence
θ
. In case
of a diffuse field excitation, the transmission loss is defined as:
=|
θ
max
2
cos
θ
sin
θdθ
|
τ(θ)
|
θ
min
TL
d
=−
10 log
(11.96)
θ
max
cos
θ
sin
θdθ
θ
min
where
τ(θ)
is the transmission coefficient for a given angle of incidence
θ
, varying from
θ
min
to
θ
max
.
In the numerical implementation, the diffuse field integration is done numerically.
For example, 21-point Gauss - Kronrod rules can be used. Moreover, in the case where
the transfer matrix of a layer depends on the wave heading in the
(x
1
,x
3
)
plane (e.g.
the case of an orthotropic plate), integration over heading needs to be accounted for. An
example is given in Chapter 12 where the size effects are investigated.
The solution procedure described in the previous section relies on the numerical eval-
uation of three determinants at each frequency and/or incidence angle. For a small number
of layers, the computation time is not an issue unless a diffuse field indicator is needed
where a numerical integration is performed. This may be expensive, especially at high fre-
quencies. Moreover, the evaluation of a determinant may be sensitive to ill-conditioning
of the matrix. A variant solution procedure is based on fixing arbitrarily the amplitude
of the incident pressure to 1. Consider the system given by either Equation (11.87) or
(11.93). In both cases, let the (
N
1) square matrix obtained by eliminating
the first column of matrix [
D
] be denoted by [
D
1
]. Let
F
represent the vector obtained
by multiplying the eliminated first column by
−
1. Finally, let
V
1
represent the variables
vector less its first entry. The solution of the (
N
+
1)
×
(
N
+
1) linear system, [
D
1
]
V
1
=
F
allows for the calculation of the acoustic indicators (depending on the problem at hand):
+
1)
×
(
N
+
1
V
1
(
1
)
Z
s
=
(11.97)
T
=
(
1
+
R)V
1
(N)
(11.98)
