Sound transmission through multilayer elements - Building Acoustics - page 294

As we are driving all these “beams” in parallel, we get

∑

Δ F

F

Z == = + ⋅ ⋅ 2(1 j) Lmc

line y B

u u

(8.25)

f

or Z =+ ⋅ ⋅ 2 (1 j) L c m .

line y 0

f

c

The critical frequency is introduced here in the last expression. Inserting this expression
into Equation (8.20), which gives Δ R , together with σ B,line from Equation (8.18) , we
arrive at the approximate equation

Δ≈ ⋅ R 10 lg ( b f ) − 23 dB,

line c,line

2

⎡ ⎤

(8.26)

mf + m f

1 c,2 2 c,1

where f = ⎢ ⎥ .

c,line

mm +

⎢ ⎥

⎣

1 2

⎦

The quantity b is the centre-to-centre distance between the line connectors (studs). It is
apparent that this expression is identical to the one depicted in Figure 8.10. Here,
however, the modified critical frequency f c,line takes the position of the critical frequency
of the lining.
Before showing examples based on this expression, we shall also refer to a work by
Davy (1991), having extended the work by Sharp by taking into account the elasticity of
the connections. He uses the case of stud connections and introduces the compliance
(inverse stiffness) C M of these connectors. The expressions, which are also cited in Bies
and Hansen (1996), become relatively complicated and may be a little difficult to follow.
However, we shall repeat them here and also make a comparison with the equations
given above.
In the same manner as above, the energy transmission is divided into two parts, one
part transmitted by way of the cavity, the other by way of the connections. In the
frequency range between the double wall resonance f 0 and f c,1 , where the latter is the
lowest of the critical frequencies of the two leaves, the transmission factor for the part
caused by the connections is

23
00

64 ρ c

(8.27)

τ = ⎧ ,

B, line

( )

2

⎫ 3/2 2

2

( ) ( )

g + 42 c π f mCg ⋅ − β b 2 π f

⎨ ⎬

0 1 2 M

⎩ ⎭

where

g = mf 2 π + m f 2 π

1 c,2 2 c ,1

and

⎡

2

⎤⎡

2

⎤
⎥
⎦

⎛ ⎞ ⎛ ⎞

f f

⎢ ⎥⎢

β =− 1 1 − .

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

f f

⎢ ⎥⎢

⎝ ⎠ ⎝ ⎠

c,1 c,2

⎣ ⎦⎣

The quantity b denotes, as before, the centre-to-centre distance between the studs. For
“commonly used” steel studs, a compliance C M equal to 10 -6 m 2 ⋅N -1 is indicated and for