Civil Engineering Reference
In-Depth Information
where we assume that
k
B
<
k
. For increasing
k
, i.e. when the wavelength λ
B
in the plate
increases in relation to the wavelength λ in air, the radiation factor approaches 1 (one).
This applies, as already demonstrated, for the idealized source types but also for plates of
finite dimensions.
6.3.3
Critical frequency (coincidence frequency)
In the above we introduced the notion of
trace matching
. The term was first introduced
in German literature (see e.g. Cremer et al. (1988)), describing the condition of the trace
wavelength in an incident wave equal to the wavelength of the plate, i.e. a reversed
situation of the one being described in the last section. In either case, there will be a
limiting or
critical frequency
where this coincidence phenomenon may occur, also called
the
coincidence frequency.
We shall use the former notion. At this frequency
f
c
the
wavelength λ
B
is equal to the wavelength λ in the surrounding medium. In other words:
the phase speed
c
B
in the solid medium is equal to the phase speed
c
0
in the surrounding
medium (air).
10000
5000
Chipboard
Plasterboard
2000
Steel, Al.
Glas
1000
500
Concrete
Lightw. concrete
200
100
1
2
5
10
20
50
100
200
Plate thickness (mm)
Figure 6.11
Critical frequency of homogeneous plates as a function of thickness.
For thin plates, i.e. when the wavelength is larger than approximately six times the
plate thickness, we have shown that the phase speed is expressed as
B
c
=⋅
,
(6.40)
4
B
m
where
B
is the bending stiffness per unit length and
m
is the mass per unit area. By
putting
c
B
equal to
c
0
and solving with respect to frequency, we get
