282 Building acoustics
In our example (see Figure 8.3), we get f 0 ≈ 140 Hz when using this expression. It
should be noted that we presuppose that there is no elastic coupling of the leaves due to a
possible elasticity in the porous material.
In the frequency range above the resonance frequency the sound reduction index
will increase by 18 dB per octave. It is not evident from the figure that above a given
frequency fd ≈ 55/d (see Equation (8.7), the increase is less strong and according to
empirical data estimated as 12 dB per octave. In our example fd will be approximately
1100 Hz, thus the effect will be partly masked by the “dip” due to coincidence.
These phenomena are all to be found in results from laboratory measurements. An
example is given in Figure 8.4, presenting the sound reduction index of a double wall,
two leaves of 13 mm plasterboard mounted on separate studs. The cavity depth is 150
mm, and measurements were performed both leaving the cavity empty and also being
completely filled up by rock wool of density 20 kg/m^3. One cannot detect the double wall
resonance as the measurements were limited downwards to 100 Hz, whereas the
resonance should be around 65 Hz. The frequency fd, marking the transition from a 18
dB per octave to a 12 dB per octave should be approximately equal to 370 Hz.
The predicted results are based on an empirical model by Sharp (1978). Using
classical expressions and a large measurement database, he presented the following
simple set of equations to predict the sound reduction index for double walls without
structural connections, however having the cavity filled with a porous absorber:
() (8.7)
0
12 0 d
12 d,
20 lg 29 dB ,
6dB ,RfM
RRR fd fff
RR f f⎧ <
⎪
=++⋅ ⋅−⎨ <<
⎪ ++!
⎩
fwhere fd, as given above, is equal to 55/d. The index M indicate that the reduction index
is to be calculated from the total mass of the leaves, M = m 1 +m 2. The predicted results
shown in Figure 8.4 for the frequency range f < fd does not fit too well to the measured
ones but due to lack of accurate specifications only the simple mass law is applied for
calculating R 1 and R 2 , i.e. the one given in Chapter 6 (section 6.5.2):
Rfm=⋅20 lg( )−47 dB. (8.8)
It may seem odd that no specifications as to the porous material, filling the cavity, enter
into Equations (8.7). The attenuation caused by this material certainly depends on
parameters such as flow resistance etc. Brekke (1979) suggested the following
expression to be used for the frequency range above fd :
12 i
ref
20 lg ,Z
RR R ATT
Z
=++ −⋅ (8.9)
where Zi and ATT is the input impedance of the absorber (as seen from the first leaf) and
the attenuation offered by it. The reference impedance Zref is equal to the impedance of a
porous material having a flow resistivity of 7 kPa⋅s/m^2.
In spite of the influence of the cavity material there are several reasons for not
gaining much by using more complicated expressions for the high frequency range than
the one given in Equation (8.7). In practice, the reduction indices are normally much
larger than the ones in the lower frequency range, even when taking the weighting curve
into account (see Figure 6.4). The calculation accuracy which is certainly interesting