Building Acoustics

244 Building acoustics

The equation clearly shows the presumption for the derivation in the preceding section;
the wall impedance will become a pure mass impedance at frequencies far below the
critical frequency.

6.5.1.3 Sound reduction index of an infinitely large plate. Incidence angle dependence

The transmission factorτ and the sound reduction index R are calculated from the ratio of
the sound pressure amplitudes in the transmitted and incident wave. By definition:

2

tt

ii

ˆ

.

ˆ

Wp

Wp

τ== (6.92)

We may by using Equation (6.89) express the velocity as

jsin ( irt) jsin

w

ˆˆˆ

ˆeekx kx.

ppp

uu

Z

== ⋅−−φ +− φ (6.93)

The normal component of the acoustic particle velocity v on both sides of the plate must
be equal to the plate velocity u. Hence, the following relationship must apply,

vv uvˆˆ ˆˆir+ ==t. (6.94)

The relationship between these velocity amplitudes and the corresponding pressure
amplitudes is easily found by applying the force equation (Euler equation),

(^0)
(^00)

1

.

j

y

y

p

v

ωρ y

=

=

⎛⎞∂

=− ⎜⎟

⎝⎠∂

(6.95)

Applying this to Equations (6.87), we get

(^) irirtt
00 0

ˆˆ ˆ

ˆˆcos , cos and ˆcos.

pp p

vv v

ZZ Z

==φ − =φφ (6.96)

The Equations (6.93), (6.94) and (6.96) give us the relationship between the pressure
amplitudes we are looking for as we find

i t

(^00)
w
2 ˆ ˆcos
ˆ.
2
cos
p p
u
Z Z
Z
φ
φ

==

+

(6.97)

The transmission factor and the reduction index will then be given by