Building Acoustics

328 Building acoustics

thickness 100 mm is shown in Figure 9.2, illustrating the general frequency dependence
of the transmission, the typical resonance phenomena (can you predict the frequency of
these “dips”?) and the effect of shaping the aperture as a conical horn.

50 100 200 500 1000 2000 5000

Frequency (Hz)

-20

-10

0

10

20

30

Sound Reduction index (dB)

11

20

30

40

10 (W-S)

Figure 9.2 The sound reduction index at normal incidence of a conical aperture in a wall of thickness 100 mm
as compared with a cylindrical aperture. The entrance radius is 10 mm and the exit radius (in mm) is indicated
on the solid curves. Dashed curve gives results for a cylindrical aperture of radius 10 mm calculated from
Wilson and Soroka (1965). After Vigran (2004).

For practical use in sound insulation in buildings, transmission through sealed
apertures is more relevant and we shall therefore revert to the work by Mechel (1986).
This is, however, a purely theoretical work but we shall make comparisons with results
from other sources. The geometry used in calculating the transmission through a
cylindrical aperture with radius a in a wall of thickness d is shown in Figure 9.3. The
aperture is filled with a porous material characterized by propagation coefficient Γ and
complex characteristic impedance Zc. The aperture may also be sealed at one or both
sides by ideal mass layers having a mass per unit area denoted m. The transmission factor
for the aperture at plane wave incidence at an angle φ is given by

() {}() ()

2

00 c

r2 2

c1 2 c 12

1 1 r1 2 2 r2

2

Re

cos cosh sinh

with j and j.

cZ

Z

Z ZZ d Z ZZ d

ZmZ Z mZ

ρ

τφ

φ

ωω

=⋅ ⋅

+Γ++ Γ

=+ =+

(9.4)

The quantity Zr is the radiation impedance, here being the radiation impedance of a
piston in an infinite baffle (see section 3.4.4). Having a diffuse incident field, Mechel
gives a simple relationship between the transmission factor for diffuse sound incidence
and normal incidence, as