Civil Engineering Reference
In-Depth Information
2
⎡
⎤
⎛
⎞
f
f
(
)
Z
=
j
ω
m
f
⎢ −
1
cos
2
θ
+
sin
2
θ
1
+
j
η
sin
4
ϕ
⎥
.
(6.108)
⎜
⎟
w
f
⎢
⎥
⎝
⎠
⎣
⎦
c1
c2
The expression, however without the inclusion of the loss factor, was given back in 1960
by Heckl. The transmission factor for a given angle of incidence, according to Equation
(6.98),
hence becomes
−
2
Z
cos
ϕ
( )
τϕθ
,
=+
1
w
.
(6.109)
2
Z
0
Setting out to calculate the transmission factor for diffuse field incidence we have to
integrate this expression over all angles of incidence (see Equation
(6.99)
). Hence
π
⎡
π
⎤
2
2
2
( )
∫ ∫
⎢
⎥
τ
=
2
τϕθ ϕ ϕϕ θ
,
cos
sin
d
d
.
(6.110)
d
π
⎢
⎥
0
⎣
0
⎦
Inserting
τ
from Equation
(6.109)
, we may write
(
)
π
dsin
2
ϕ θ
d
1
2
2
∫∫
τ
=
.
(6.111)
d
2
π
Z
Z
00
1
+
w
cos
ϕ
2
0
The evaluation of this expression must be performed numerically. However, Heckl
(1960) also gives some approximate expressions (for η equal zero). In the frequency
range below the lowest critical frequency, we may use ordinary mass law. In two other
frequency ranges, being the range between the critical frequencies and above the highest
one, respectively, Heckl gives the following expressions:
2
Zf
⎛
4
f
⎞
τ
≈
0
⋅
c1
ln
for
f
< <
f
f
⎜
⎟
d
c
1
c
2
2
2
2
π
mf
f
⎝
⎠
c1
(6.112)
ff
Z
c1
c2
and
τ
≈⋅
0
for
f
>
f
.
d
c2
2
2
mf
It should be noted that the expressions above only apply to an infinitely large plate.
Taking the finite dimensions into account, we may as before (see section
6.5.2.1)
introduce the correction factor after Sewell (1970). Hansen (1993) introduces a
variable limit
c
f
(
)
2
sin
ϕ
=−
1
0
,
(6.113)
2
π
S
upper limit
where
S
is the area of the panel and where we recognise the last term from the
expressions given in section
6.5.2.1.
In the examples we shall use, we have for simplicity
