Building Acoustics

Waves in fluid and solid media 67

+-

h

prt(,)

G

r (^1) r
r 2
Figure 3.4 Dipole, two monopoles in opposite phase. An oscillating sphere is an equivalent source.
Two simple monopole sources, coupled in anti-phase, are called a dipole and are
illustrated in Figure 3.4. Assuming that the distance r to the observation point is large
compared with the source dimensions we may write the pressure as
2
(,)^0011 cos ej( ),where ,
4j
prt ckD tkr D Q h
rkr
ρ φ ω
π

= ⎛⎞+⋅⋅− =⋅

⎜⎟

⎝⎠



  (3.41)

where h is the centre distance between the monopoles. Dis the so-called dipole moment.
As seen from the equation the dipole will not radiate evenly in all directions; we get a
directivity factor expressed by the cosine of the angle φ to the point of observation and
thereby a directivity pattern shaped as a figure-of-eight. When the wavelength is large
compared with the source dimensions the sound power will be given by

42

(^00) ,
12
ck D
W
ρ
π

=



(3.42)

and here we may see the dramatic reduction in sound power at low frequencies as
compared with the monopole source. Comparing with Equation (3.40) we get

22

dipole 22

monopole

.

3

W kh

fh

W

⋅

= ∝⋅ (3.43)

An oscillating sphere or ball, i.e. vibrating back and forth, will also act as a dipole and is
equivalent to our two monopoles vibrating in anti-phase. This is in fact a very useful
example to use later when we will treat the concept of radiation factor.
This kind of dipole action does not apply only to bodies of spherical shape; a
vibrating string, a vibrating pipe or beam will also act like a dipole when it comes to
sound radiation. Another example is a loudspeaker with an open back. At low
frequencies both radiating surfaces will act like a monopole and these will be 180° out of
phase with each other. There will be no efficient sound radiation where one does not
mount the loudspeaker in a closed box, alternatively provide for a distance between the
front and back large in comparison with the wavelength. The latter means that in practice
one mounts the loudspeaker in a large baffle. We have tried to illustrate these effects in
Figure 3.5.