286 Building acoustics
We shall treat all these methods of couplings starting with a double construction of
a special type: a heavy wall or ceiling covered by an additional so-called acoustical
lining. This implies a lining having a low bending stiffness and a high critical frequency
as compared with the primary heavy construction. This is a common method of
improving the sound reduction index of a wall or the impact sound insulation of a ceiling
(floor). The reason behind starting out with this example is that we may assume that the
movement of the primary construction is not affected by the lining; i.e. we assume that
there is no “feedback” in the system, which would certainly be the case of lightweight
walls. A laboratory standard for measuring the improvement of such linings has recently
become available (see ISO 140 Part 16).
8.2.2.1 Acoustical lining
The treatment of this case may be found in the book by Cremer et al. (1988), but was
presented by Heckl as early as in 1959. As seen from Figure 8.8, we shall assume that the
basic or primary construction is a heavy, massive wall (or floor) for which the critical
frequency lies below the observed frequency range. Furthermore, we assume that the
radiation factor is approximately equal to 1.0 in this frequency range. The lining, on the
other hand, has such a high critical frequency that the bending wave near field, caused by
vibrations transmitted through the studs or ties, will dominate in the radiated sound. This
is the basic idea behind such additional acoustical linings; even if the lining is firmly
connected to the basic wall, and thus obtains the same velocity as the latter at the
connections, the total radiation will be reduced.
The sound reduction index for the combination (see Figure 8.8 b)), may be
expressed as
ii
22 ,P
10 lg 10 lg ,WW
R
WWW
⎛⎞
=⋅ =⋅⎜⎜
⎝⎠+ 2,B
⎟⎟ (8.11)
where Wi is the incident power on the primary wall. The radiated power from the lining
is divided into two parts: the power W2,B radiated from the bending wave near field and
the power W2,P due to the transmission through the cavity. The latter may, at frequencies
above the double wall resonance (see Equation (8.6)), be written
4
2,P 1 0
ii.
W W f
WWf⎛⎞
=⋅⎜⎟
⎝⎠
(8.12)
The quantity W 1 is the power transmitted through the primary wall. This equation tells us
that the relationship between the mean velocity amplitude of the primary wall and the
lining is proportional to the frequency squared. The cavity acts as a pure spring situated
between the wall and the lining, these being represented by two masses. The radiated
power from one of the structural bridges, which may be a stud (line connection) or a tie
(point connection), may be written
'= ⋅⋅WcuS2,B U0 0 2,B^2 VB. (8.13)
The quantity VB is the radiation factor, however here defined by the velocity of the
bridge, not as earlier by a mean velocity of the plate. Having a number n bridges
distributed over