Building Acoustics

68 Building acoustics

Figure 3.5 A loudspeaker open at the back is a dipole source. a) No efficient radiation due to short distance
between the front and back. b) A baffle increases the distance front–back.

Evidently, these results may be extended to poles of higher orders. A quadrupole is
made up of four monopoles alternating in anti-phase or it could be two oscillating
spheres in anti-phase. This type of sound source will be even less effective than a dipole
at low frequencies as the radiated power will be proportional to the wave number with an
exponent of six.
A group of sound sources characterised as multipole are vibrating surfaces such as
plates, shells etc. that will be treated in detail later. Another important group is connected
to fluid flow. A turbulent jet flow is a typical quadrupole source. Turbulent flow
interacting with solid surfaces however constitutes sources of dipole character and as
such will be found for example in air-conditioning terminal units as grilles and diffusers.

3.4.2 Rayleigh integral formulation

The idealized models treated above are, however, only useful in a qualitative way when
it comes to calculating radiation from solid bodies such as plates or shells vibrating in
complex patterns, in particular when the wavelength (in air) becomes comparable or less
than the dimensions of the source. However, there are tools available to calculate the
radiated power in the case when the surface velocity is known, either known in detail or
as a space averaged value.
The German physicist von Helmholtz showed well over 100 years ago that the
sound pressure outside a vibrating surface (see Figure 3.6) could be expressed as the sum
of two integrals:

j( ) j( )

(,) d () d,

44

tkr tkr

SSS

ep e

pRt S pS S

rn n r

ωω

ππ

−−⎛⎞∂∂⎛⎞

=−⎜⎟ ⎜⎟⎜⎟

vv∫∫⎝⎠∂∂⎝⎠

(3.44)

where n indicate the normal to the surface. The term∂pn/∂ will then, as in Equation
(3.36), be proportional to the normal surface velocity un on the vibrating surface S, and
can be written

j(). 00 n
S

p

cku S

n

ρ

⎛⎞∂

⎜⎟=

⎝⎠∂

(3.45)

Plate (baffle)

a) b)