Civil Engineering Reference
In-Depth Information
The Schroeder cut-off frequency represents an important division in the prediction
of the variance. A satisfactory theory does not exist which covers the frequency range
below this cut-off frequency. However, we shall present an estimate also for this range, a
range where investigations are best conducted by FEM modelling. As for the frequency
range above
f
S
, statistical models will have limited validity if the absorption becomes so
large that the direct field is significant, which may happen at sufficiently high
frequencies.
Lubman (1974) presented the following expressions for the relative variance:
For the range given by 0.2⋅
f
S
≤
f
≤ 0.5⋅
f
S
he got
1
( )
22
r
σ
p
=
,
(4.52)
Δ
N
1
+
π
where Δ
N
is the number of natural modes inside the frequency band Δ
f
(see Equation
1
( )
22
r
σ
p
=
,
(4.53)
Δ ⋅
fT
1
+
6.9
where
T
is the usual reverberation time. It should be noted that both expressions
presuppose that the product Δ
f
⋅
T
is numerically equal or larger than 20.
Normally, one is looking for the corresponding standard deviation
s
(
L
p
) in the
sound pressure level. However, to calculate this one needs to know the probability
distribution of
p
2
. If the relative variance is less than approximately 0.5 we may make an
estimate based on transforming the sound pressure level in the following manner:
2
⎛⎞
=⋅
⎜
⎜
⎝⎠
= ⋅
p
L
10 lg
p
2
0
p
(4.54)
()
()
2
2
2
2
into
L
10 lg(e) ln(
⋅
p
)
− ⋅
10 lg
p
≈
4.34 ln(
⋅
p
)
− ⋅
10 lg
p
.
p
0
0
Differentiating the last expression with regards to
p
2
, we get
( )
( )
2
d
L
sp
1
()
()
p
22
=⋅
4.34
or
s L
=⋅
4.34
=⋅
4.34
σ
p
.
(4.55)
p
r
2
2
2
d(
p
)
p
p
Up until now we have concentrated on the spatial variance. In measurements on
stochastic signals there will also be a corresponding relative time variance given by
1
( )
22
t
(4.56)
σ
p
=
Δ ⋅
,
f
T
i
where
T
i
is the measuring or integration time used to determine
p
2
in a given microphone
position. Certainly, we are able to make this time variance arbitrarily small by extending
the measuring time but there is, of course, a trade-off here. In practice, one normally
