Building Acoustics

248 Building acoustics

6.5.2.1 Formulas for calculation. Examples

One should not be surprised, taken the complexity of the task to even calculate the sound
reduction index of a single homogeneous wall between two rooms, that a number of
formulas are cited in the literature. A number of these are found in EN 12354–1, in
which Equation (6.103) is given for the transmission factor. The dimensions of the wall
are given by the quantities a and b, ηtot is the total loss factor, σ and σf are the radiation
factor for resonant and non-resonant transmission, respectively. The latter is expressed
by the formula by Sewell (Equation (6.49)), whereas the corresponding one for σ is a
little more involved than the one given in section 6.3.4.2.

()

(^222)
0c
f 22
tot
(^22)
0
c
tot
(^22)
0c
c
tot

2,

,

2

.

2

c

Zfab

f f

fm abf

Z

f f

fm

Zf

f f

fm f

σ

τσ

πη

πσ

τ

πη

πσ

τ

πη

⎛⎞⎡⎤+

=+⋅<⎜⎟⎢⎥

⎝⎠⎢⎥+

⎣⎦

⎛⎞⎡⎤

==⎜⎟⎢⎥

⎝⎠⎢⎥⎣⎦

⎛⎞⎡⎤

=>⎜⎟⎢⎥

⎝⎠⎢⎥⎣⎦

(6.103)

For rough estimates one may make some simplifications. For forced transmission (f < fc),
the simple mass law, given in Equation (6.102), is often sufficient. A slightly better
alternative is to neglect the contribution from the resonant transmission but to include a
slightly simplified area effect. Fahy (1987) has suggested that for the range f < fc one
should use the following expression:

2

f0

0c

2

10 lg ln 20 lg 1 ,

ff

RR ab

cf

⎡⎤⎛⎞ ⎛⎞π ⎡⎢ ⎤⎥

=−⋅⎢⎥⎜⎟ ⎜⎟⋅ +⋅ −

⎢⎥⎣⎦⎝⎠ ⎝⎠⎢ ⎥

⎣ ⎦

(6.104)

where index f on the reduction index indicates that we are dealing with forced
transmission. Inserting for R 0 according to Equation (6.82) we get

()

2

f

0c

2

20 lg 10 lg ln 20 lg 1 42 dB.

ff

Rmf ab

cf

⎡⎤⎛⎞ ⎛⎞π ⎡⎤⎢⎥

≈⋅ ⋅−⋅⎢⎥⎜⎟ ⎜⎟⋅ +⋅ − −

⎢⎥⎣⎦⎝⎠ ⎝⎠⎢⎥

⎣⎦

(6.105)

In the frequency range above the critical frequency we may set σ ≈ 1, and when using the
last entry in (6.103), we obtain

(^) tot
c
20 lg( ) 10 lg 2 47 dB c.
f
Rmf ff
f
η

⎡⎤

=⋅ ⋅+⋅⎢⎥− >

⎣⎦

(6.106)

This expression is identical to the one given for a plate of infinite size (see Equation
(6.101)). Below, we shall present several examples where we compare measured and
calculated data. In all cases we shall use the complete set of equations given in Equation
(6.103) and where the radiation factors are taken from Equations (6.48) and (6.49).