Sound transmission. Characterization and properties of single walls and floors - Building Acoustics - page 236

It could be useful to give an example illustrative of the sound power due to this
bending near field. We shall then make a comparison with the radiated sound power
from a given area of an infinite plate, in conformity with defining the radiation factor.
We envisage this area as a piston having radius a. Setting the velocity amplitude equal to
u 0 the radiated sound power will be given by

22

W = ρπ c a u

0 .

(6.63)

piston 0 0

The task is now to find the radiated sound power due to the near field according to
Equation (6.62) having velocity u 0 in the driving point. Second, we shall calculate the
radius a of the piston when W point is equal to W piston . From Equations (6.50) and (6.51)
together with Equation (6.41) , we obtain

3 2

⋅

8 ρ

π

cu

00 0

W = . (6.64)

point

3 2

f

c

Equating this sound power with W piston we obtain

22 c c

0 0

a = ⋅ ≈ 0.29 . (6.65)

2

f f

π

c c

Example The critical frequency f c of a concrete plate, having a thickness of 50 mm, will
be approximately 380 Hz (see Figure 6.11) . The radius a of the equivalent piston source
will thereby be ≈ 26 cm. Using e.g. an applied force of 10 N (RMS-value) we get from
Equation (6.62) a radiated sound power of approximately 4.2⋅10 -6 watts or a sound power
level L W re 10 -12 watts of 66 dB. Assuming a semicircular radiation centred on the driving
point the sound pressure level L p will be 58 dB at a distance of 1 metre.
In several practical cases, it is important to know the radiated power from the
bending near field, not only when a point force drives the plate, but also equally well
when driven along a line. The latter applies to cases where vibrations are transmitted to a
panel or wall by studs or stiffeners. Corresponding expressions to the ones given in
Equations (6.62) and (6.64) are

2 2 2

ρ F 2 ρ

π

c u

0 A 0 0 0

A A

W = ⋅ = ⋅ when f << f . (6.66)

line c

2

f

m ω

c

The quantities F ℓ and ℓ are the force per unit length and the length of the line,
respectively.

6.4.2.2 Total sound power emitted from a plate
We have already presented an expression giving the total acoustical power emitted from
one side of a point-excited plate (see Equation (6.61) ). By inserting the expression for the
power radiated from the near field, Equation (6.62) and using Equation (6.55) , giving the
mean square velocity, we obtain