Building Acoustics

220 Building acoustics

()

2

0

10 lg 10 lg 1 0.1 2 ,

c

fd

σ

⎡ ⎤

⋅=−⋅+⋅⎢ ⎥

⎢⎣ ⎥⎦

(6.28)

where f is the frequency and d is a typical dimension, which for a monopole is the

diameter of the sphere. This implies that dS= /π or dV=^32 ⋅ , where S and V are

equal to the source radiating area and volume, respectively. It is easy to see that the two

expressions are identical.
Figure 6.8 also shows the corresponding radiation factor for a dipole source
exampled by an oscillating sphere having the same radius. As pointed out earlier on, a
dipole is a much less effective source than a monopole at low frequencies. A practical
example is the radiation from a loudspeaker mounted in a large baffle or in a closed box
as compared to being freely suspended in the air. For illustration see section 3.4.1.
Calculating the radiation factor for an oscillating sphere is however a little more
involved than for a pulsating one. We shall therefore just give the result, which is

(^) dipole 2
22

1

22

1j

ka ka

σ =

−−⋅

(6.29)

and where | | indicates the modulus of the expression. The expression is furthermore
based on setting the mean squared velocity of the oscillating sphere equal to one-third of
the same for the pulsating sphere. Hence

2

2 puls.sphere

osc.sphere 3

u

u =



.

This may also be formulated by saying that the mean particle velocity on the surface of
the oscillating sphere is (1/3)1/2 of the maximum velocity.

6.3.2 Sound radiation from an infinite large plate

We shall use an idealized example to show which parameters are important in sound
radiation from plates, namely radiation from an infinitely large plate where a simple
plane bending wave is propagating (see Figure 6.9). We shall calculate the sound
pressure p in a point with coordinates (x,y) above the plate and further on, the radiation
factor when the velocity is given by

uueB=⋅ˆ j(ωtkx−B), (6.30)

where kB is the wave number for the bending wave that is propagating in the x-direction.
We now assume that the sound pressure above the plate can be expressed as

ˆ j( )

(, ) xy,

tkxky

pxy p e

ω−−

=⋅ (6.31)