218 Building acoustics
6.3 Sound radiation from building elements
The sound insulation offered by a building element or a complex construction, either for
airborne sound or impacts will depend on two factors: 1) the dynamic response to the
actual excitation, being an acoustic field or a direct mechanical force or moment and 2)
the efficiency as a sound radiator given the actual response pattern. In this section, we
shall deal with the last item, in particular the sound radiation from plane elements when
given a bending wave velocity distribution. We shall give a definition of a quantity that
is used to characterize the efficiency of a surface as a sound radiator, the radiation
factor. In this connection we shall return to the simple and idealized sound sources, the
monopole and dipole, to illustrate the idea. Following this presentation we shall treat the
problems connected to the generation of the bending wave field and further on the
transmission properties.
6.3.1 The radiation factor
A commonly used quantity to characterize the efficiency of a given vibrating surface, as
a sound radiator is the radiation factor σ, also called radiation efficiency or radiation
ratio. By definition
rad 2
00
,
W
cS uσ
ρ=
(6.25)
where Wrad is the radiated power from the actual vibrating surface, having the area S, to
the surrounding medium with characteristic impedance ρ 0 c 0. The quantity u^2 is the
mean square velocity amplitude taken over the surface. The denominator in the
expression is the power radiated from a partial area S of an infinitely large plane surface,
all parts vibrating in phase with a velocity equal to this mean value, i.e. a plane wave
radiation condition. We shall here refer back to the calculation of the radiated power
from a plane circular piston set in a baffle (see section 3.4.4). Here we found the same
expression when the piston dimensions become much larger than the wavelength.
The brackets in the expression signify that we are taking the mean value in the
spatial domain, i.e. of the square RMS-value taken over all points on the surface. The
condition for doing this, in a practical sense, is that the velocity does not vary too much
from point to point, making it sensible to represent the velocity as a mean value. How
large variations should be allowed will obviously depend on the application. In addition
to taking the mean value in the time and spatial domain, a third type of averaging must be
performed in practice; averaging inside frequency bands of width one-third-octave or
octave. We then assume that the applied bandwidth is large enough to contain several
natural frequency modes of the actual structure.
Determination of the radiation factor is often performed by way of measurements,
as a direct prediction is difficult except for idealized cases. However, there are a number
of analytical expressions available, both for plane surfaces and for shell constructions.
We shall limit our discussions to plane surfaces.