Building Acoustics

Sound transmission 237

22 2 2

000 0 c

ac 222 2

00

1.

28 2 4

W FcFkB F f

cm m cm f

ρρ ρ π

σ σ

πηω π η

⎡ ⎤

=+ = +⋅⎢ ⎥

⎣ ⎦

 

(6.67)

Concerning the force F it is still presupposed that it has a given bandwidth although we
allowed ourselves to leave out this index. Also, as pointed out above, the first term is
only applicable at frequencies below the critical frequency fc. Another presumption is
that the total energy loss of the plate is dominated by the inner material losses and
boundary losses; i.e. the radiated acoustical power makes up a minor part of the total
power dissipated. The latter condition is normally fulfilled when the surrounding
medium is air (refer to the introduction to section 6.4.2).
From Equation (6.67) we may arrive at a couple of conclusions of great practical
interest in general noise abatement. The radiation factor will, as seen from Figure 6.15,
strongly increase when the frequency approaches the critical frequency, which implies
that above a given frequency the reverberation field will dominate the radiated power.
This again implies that the loss factor η will be of great importance. By artificially
increasing this factor, adding e.g. viscoelastic layers to thin plates is therefore favourable.
Conversely, in the lower frequency range, the contribution from the near field
could be dominant given that the loss factor is not too small. We may prove this by using
Equation (6.48) to find a low frequency approximation for the radiation factor, this being

(^230) c
c

2

when.

cU f

f f

Sf

σ

π

≈⋅ << (6.68)

Thus, the second term in Equation (6.67) will be proportional to 1/(η ⋅ f^ ½) in the low
frequency range, whereas the first term, representing the radiation from the near field,
will be constant and frequency independent; thereby determining the radiated power
above a certain frequency. Increasing the loss factor will in this case have no effect on
the radiated power. Denoting the crossover frequency by fk, i.e. the frequency where
Wpoint is equal to Wreverberant, we get
2
0
kkc
c

,where.

2

cU

f ff

πηSf

⎛⎞

=<⎜⎟ <

⎜⎟

⎝⎠

(6.69)

If the task is to reduce the radiated power for excitation frequencies f in the range given
by fk < f << fc, no further increase in η will accomplish this.

Example We shall assume that a steel panel, of 1 mm thickness, has a loss factor of 0.05,
partly due to material losses and partly to boundary losses. For simplicity, we will further
assume that the panel is square with dimensions 2.5 meters. The crossover frequency fk
will then be approximately 250 Hz. A loss factor η > 0.05 will therefore not reduce the
radiated power in the frequency range from 250 Hz and upwards to several kHz (the
critical frequency for 1 mm steel is 12.5 kHz). It must, however, be noted that by
increasing the loss factor with a viscoelastic layer, glued or sprayed on to the panel, there
will be a reduction in the radiated power due to the added mass.