Civil Engineering Reference
In-Depth Information
prt
(,)
G
r
r
r
+
-
h
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
⎛
⎞
ρ
ckD
1
j(
ω
tkr
−
)
00
prt
(,)
=
1
+
⋅
cos
ϕ
⋅
e
,where
D
= ⋅
Q h
,
(3.41)
⎜
⎟
4
π
r
j
kr
⎝
⎠
where
h
is the centre distance between the monopoles.
D
is 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
ρ
ck D
00
W
=
,
(3.42)
12
π
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
W
22
kh
⋅
dipole
22
=
∝⋅
f
h
.
(3.43)
W
3
monopole
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
