Building Acoustics

242 Building acoustics

Having characterized the plate by its mass impedance, also having no internal energy
losses, the transmission factor τ of the plate must be equal to the absorption factor α.
Inserting for Zg after Equation (6.78), we get

(^2)
0

1

,

1

2

m

Z

τ

ω

=

⎛⎞

+⎜⎟

⎝⎠

(6.80)

giving the sound reduction index

2

0

00

1

10 lg 10 lg 1 20 lg.

2

mfm

R

ZZ

ωπ

τ

⎡⎤⎛⎞ ⎛ ⎞

=⋅ =⋅ +⎢⎥⎜⎟≈⋅⎜ ⎟

⎢⎥⎝⎠ ⎝ ⎠

⎣⎦

(6.81)

This is the so-called mass law in its simplest form; the sound reduction index increases
by 6 dB by each doubling of frequency and/or mass per unit area. The approximation
given by the last expression, however presuppose that the mass impedance is much larger
than the characteristic impedance of air. This condition is normally fulfilled for panels
used in buildings. Inserting the characteristic impedance of air at 20°C we get:

Rmf 0 ≈⋅ −20 lg( ) 42.5 (dB). (6.82)

6.5.1.2 Bending wave field on plate. Wall impedance

Taking the bending stiffness into account we have, as mentioned above, to solve a wave
equation where the sound pressure of the incoming wave is the driving force. The wave
equation may be written as

2

22

Bmpxztt 2 (,,),

ξ

ξ

∂

∇∇ + =

∂

(6.83)

where B and m are the bending stiffness per unit length and the mass per unit area,
respectively. The quantity ξ is the particle displacement, the deflection of the plate
surface. Assuming an harmonic time function ejω t and furthermore, using the velocity u
as a variable we get

(^22) B^4
j
uku pxz(, ).
m
ω
∇∇ − = (6.84)
For plane wave incidence we can cast p(x,z) in the form
pxz(,)=⋅⋅p k k eˆ( xz,) jkxx ejkzz, (6.85)
and by rotating the coordinate system one of the partial wave numbers kx and kz may be
set equal to zero. We shall set kz equal to zero and assume that the solution for the
velocity u have the same form as the one for the pressure. Inserting into Equation (6.84)
we obtain the following relation between the amplitudes of the pressure and the velocity