Building Acoustics

66 Building acoustics

if the wavelength is large in relation to the dimensions of the source (ka << 1) but for
large values of ka, it will go to zero. The first term will, when multiplied by the velocity
of the sphere, give us the “active” intensity, as opposed to second term which only gives
a “reactive” intensity resulting in an exponentially decreasing near field.
This situation is not unique for this idealized type of source but generally applies to
all acoustic sources. This implies that for broadband vibrating sources, at distances from
the source less than 1–2 wavelengths, one will experience variations in the spectral
content of the sound pressure. In practical measurements, using sound pressure
measurements to determine the sound power of sources, one is therefore advised to
perform the measurements at distances from the source greater than its largest dimension,
at the same keeping the distance from the surface larger than 1–2 wavelengths. These
specific requirements do not apply when it comes to direct measurements of intensity but
certain recommendations, as to the measurement distances, do apply in this case.
The radiated real power from the monopole source will then be given by

22 22

222

(^00220022)
(^1) ˆ
4,
2 aa 11
ka ka
Wcu acu S
ka ka
==ρπρ
++

 (3.39)

where S is the area of the spherical surface and the symbol ∼ indicate RMS-value. Later
we shall show that the frequency-dependent factor k^2 a^2 /(1 + k^2 a^2 ) represents the
radiation factor of the monopole. This quantity is very important in building acoustics
and we shall later give a general definition. In the literature one also comes across the
notion of source strength, which is the effective volume velocity QSu= ⋅  of the
source. The parentheses indicate, as in Equation (3.34), a space averaged value. For the
monopole we then get

22 22

1

00 22 00.

4(1 ) 4

WckQ ka ckQ

ka

ρρ

π π

=⎯⎯⎯→<<

+



(3.40)

From these expressions we can see that the monopole is not an efficient radiator at low
frequencies. Maintaining the sound power when lowering the frequency implies that the
surface velocity must be increased in inverse ratio to the frequency. This again means
that the displacement amplitude must increase in inverse ratio to the frequency squared.
It goes without saying that this will, in the end, be impossible. Sound sources radiating
bass sounds efficiently will therefore never be of small dimensions.

3.4.1.2 Multipole sources

When combining several simple monopole sources, assuming that the surface velocity is
fixed and equal on all of them, we may show that the combination may radiate more or
less power than each of them alone. The simplest case will be to combine two
monopoles, vibrating either in phase or in anti-phase. The sound pressure on the surface
of each monopole will be equal to the pressure produced by it plus the pressure caused
by the vibration of the other. If the distance between them is small (compared to the
wavelength) and they are working in phase the pressure may be nearly doubled and the
sound power radiated will correspondingly be increased. However, when working in
anti-phase the pressure may be small and the sound power may be drastically reduced.
This is easily demonstrated by putting two loudspeakers in a stereo system close together
and listening to the amount of bass being produced when playing music, coupling the
loudspeakers either in phase or anti-phase.