Building Acoustics

222 Building acoustics

This result shows that the important factor for the sound radiation is the ratio of the wave
numbers in the plate and the surrounding medium. When kB > k, i.e. the wavelength λB in
plate is smaller than the wavelength λ in the air, the sound pressure will decrease
exponentially with the distance y. We only get an exponentially diminishing near field, as
the exponent containing y becomes a real number.

Figure 6.10 Sound radiation from a plate. The wavelength of the bending wave in the plate is larger than the
wavelength in the surrounding medium.

If, on the other hand, kB < k (or λB > λ) we have an ordinary propagating plane wave
where the sound pressure increases with increasing ratio k/kB. This may be expressed by
the angle φ of the radiated wave (see Figure 6.10). We get

(^) B
B

1

or.

sin sin

k

k

λ

λ

φ φ

== (6.37)

The condition having λB > λ is sometimes called trace matching; the wavelength of the
radiated wave is equal to the plate bending wavelength projected in the direction of the
wave. In this case we may calculate the radiation factor by finding the radiated power
from a partial surface S. This power may be expressed as

(^) {}
2 2
(^0000)
22
BB
22

1 ˆ

Re.

2

21 1

cu S cu S

WpvS

kk

kk

∗ ρ ρ

=⋅⋅= =

−−



(6.38)

Hence, the radiation factor is given by

2

B

2

1

,

1

k

k

σ=

−

(6.39)

y

x

λΒ

λ

φ