Sound transmission. Characterization and properties of single walls and floors - Building Acoustics - page 244

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

Wp
Wp

ˆ

t t

τ== . (6.92)

ˆ

i i

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

( )

ppp ˆ +− ˆ ˆ

− j n kx ϕ i r t − j n kx ϕ

ˆ e

uu = = ⋅ e . (6.93)

Z

w

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,

ˆˆ ˆˆ .

vv uv + == (6.94)

i r t

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

1 ⎛⎞ ∂ p

v =− . (6.95)

⎜⎟
∂ ⎝⎠

y = 0

j ωρ y

0

y = 0

Applying this to Equations (6.87) , we get

p ˆ p ˆ p ˆ

ˆ

i

ˆ

r

ˆ

t

v = cos ϕ , v = − cos ϕ and v = cos ϕ . (6.96)

i r t

Z Z Z

0 0 0

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

2 p ˆ p ˆ cos ϕ

ˆ

i t

u = = . (6.97)

2
cos

Z

Z

0

Z +

0

w

ϕ

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