Sound transmission 247
Being pragmatic, we could investigate the applicability of the infinite plate model
to real situations and in the preceding section we gave an expression for the sound
reduction index for diffuse field incidence at low frequencies (see Equation (6.100)). In
practice this equation will give values a little too low, a slightly better model for the field
situation is
RRd0=− ≈⋅ ⋅−5dB 20 lg(fm) 47dB, (6.102)
a result which will come close to the one obtained by performing the integration of
Equation (6.99), using an upper limit of approximately 78°. This is explained by the fact
that wave components near to grazing incidence will be of less importance for finite size
partitions. It does not explain, however, why such a simple expression gives quite a good
prediction at low frequencies, i.e. for frequencies below the critical frequency.
This is connected to the phenomena outlined in the introduction to this section. The
wave field in a finite plate will have the following two components: 1) A forced field set
up by the sound field in the same way as in the “infinite” case and 2) a free field
originating at the boundaries due to the impact of the forced field. The forced field
cannot by itself satisfy the boundary conditions. The point now is, as we demonstrated in
section 6.3.4, that the radiation from the free field or resonant modes is very inefficient at
frequencies below the critical frequency. The forced field due to its longer wavelengths
therefore mainly determines the transmission. This is the reason behind the fact that
results calculated for an infinitely large wall with considerable success are transferable to
one having a finite size.
The simple expression given above needs, however, some modification. The finite
sized area influences the forced transmission. Sewell (1970) has calculated this effect
based on calculating the transmission by diffuse field incidence of a plate surrounded by
an infinite baffle. In the expressions given below for the transmission factor (see
Equation (6.103)), the mentioned effect shows up in the radiation factor for forced
transmission.
The importance of the loss factor should also be noted. The resonant modes will
certainly be reduced in amplitude by increasing the loss factor. However, these modes
are of minor importance in the acoustic radiation below the critical frequency. The effect
of increasing the loss factor will therefore be very small. However, in the frequency
range around the critical frequency and upwards, where the resonant transmission is
dominant, any increase in the loss factor will be beneficial.
It is also worth noting that below a certain frequency, below the fundamental
natural frequency, a plate will pass from the mass-controlled area to the stiffness-
controlled one. Ideally, the sound reduction index will then increase with decreasing
frequency. This effect is normally not observed in measurement data for walls in
buildings. The reason is partly that this frequency range in normally below the one used
for measurement, partly that the coupling to the resonant room modes makes the sound
reduction index vary in an irregular and not very transparent manner.
This, however, should not make us believe that low frequency and stiffness-
controlled transmission cannot be important in design of sound insulating devices.
Enclosures designed for noise control of various machines and equipment will often
include small size panels. The offending noise will often contain frequencies below the
fundamental frequency of these panels, maybe even below the fundamental frequency of
the air cavity of the enclosure. The stiffness of the panels, not their mass, is therefore of
vital importance. This case is not covered by the formulae given below.