Building Acoustics

260 Building acoustics

(^) ()

( )

2

2 00 00

22

4 2

''.

c ckSF

pW

AAm

ρ ρ

πω

=⋅=



 (6.119)

The sound pressure p ́ will then drive the plate, resulting in a velocity uP that in the
linear case must be proportional to the applied pressure. Hence, we may write

(^) ()
2 2
ubpP=⋅', (6.120)
where b is a second factor of proportionality. Using Equation (6.119) we get

()

2 2

P 00

222

2

u ckS

b

FAm

ρ

πω

=⋅



 (6.121)

Looking at Equations (6.117) and (6.121) we observe that we have just interchanged the
source and receiving point. From the principle of reciprocity, these equations shall then
be identical resulting in

()

(^2222)
(^0000)
22 2
00

2

84

aAckS mck

bAm cS

ρ ω ρ

πω ρ π

=⋅= (6.122)

From this follows that we may generally find the velocity of a structure placed in a
diffuse field (see Equation (6.120)), if we know the power radiated from the structure by
a point force excitation given by Equation (6.114). It should be noted that this
specifically applies to a given point-to-point relationship. The proportionality factors a
and b will generally be space dependent.

6.6.2 Sound reduction index and impact sound pressure level: a relationship

Finally, we shall follow Cremer et al. (1988), giving an example on how to use Equation
(6.122) to derive a simple functional relationship between the impact sound pressure
level of a massive floor construction and its sound reduction index. We shall assume that
the frequency is above the critical frequency, i.e. setting the radiation factor σ ≈ 1 is
applicable both for airborne and impact sound. We shall cast the impact sound pressure
level Ln, given by Equation (6.21), into the form

2

n 2

00

10 lg ,

pA

L

pA

⎧ ⎫

=⋅ ⋅⎨ ⎬

⎩⎭



(6.123)

where p 0 and A 0 are the reference values 2⋅ 10 -5 Pa and 10 m^2 , respectively. This is
equivalent to a radiated power Wn to the receiving room

2

2

n

00

.

4

p

WAaF

ρc

= ⋅=⋅

 

(6.124)

The last expression is introduced from Equation (6.114). We may then write for Ln