Building Acoustics

Multilayer elements 293

The radiation factor VB of each of the n bridges is given by Equation (8.18). As
distinct from the case of the acoustical lining treated in section 8.2.2.1, where we
assumed that the primary wall was not influenced by the lining, we have now got an
added term depending on the mass and input impedance of the leaves. From these
impedances we shall understand the input impedance of the leaves seen from the bridges.
For point connections we may use the expression valid for the point impedance of an
infinitely large plate. As explained earlier (see Chapter 6, section 6.4.1), this expression
also gives the space averaged mean value for a plate of finite dimensions. We shall write

2

0

point

c

4

8

cm

ZBm,

πf

=⋅= (8.21)

where the critical frequency is introduced in the last expression. Inserting this expression
into the term giving 'R together with the expression for VB,point in Equation (8.18), we get

point ^ g,point`

1g,2 2g,1

g,point

12

20 lg 45 dB,

where.

Raf

mf m f

f

mm

'≈⋅⋅ −

+

=

+

(8.22)

Here we have assumed that the point connections are arranged in a square pattern, the
quantity a being the centre-to-centre distance between points.
In a similar way we shall make use of the expression for the point impedance of an
infinitely long beam to calculate the input impedance of a plate driven along a line. The
point impedance of an infinite beam having a mass mA per unit length is, according to
Cremer et al. (1988):

Zbeam=21 j( +⋅)mcA B, (8.23)

where cB is the bending wave speed. This impedance is, as opposed to Zpoint in Equation
(8.21), a complex quantity. Driving the plate along a line by a force F distributed over a
length Ly (see Figure 8.15), we may envisage that the plate is built up from a set of beams
having cross-section 'Ly⋅ h. The impedance of each of these beams is

(^) ()line 21 j()y B.

F

Z Lh c

u

' U

'

==+'⋅⋅ (8.24)

'Ly

'F

h

x

y

Figure 8.15 A plate driven along a line.