Building Acoustics

80 Building acoustics

j( cos sin )

ii

i j( cos sin )

i,

00

(, , ) ˆ e and

ˆ

(, , ) cos e.

kx y

kx y

x

pxy p

p

vxy

c

φφ

φ φ

φ

φφ

ρ

−+

−+

=⋅

=⋅

(3.75)

As seen, we tacitly infer the time dependence ejωt. In a similar manner we get for the
reflected wave

j( cos sin )

rr

i j( cos sin )

r,

00

(, , ) ˆ e and

ˆ

(, , ) cos e.

kx y

p kx y

x

pxy p

Rp

vxy

c

φφ

φ φ

φ

φφ

ρ

−

−

=⋅

=− ⋅

(3.76)

In analogy with the use of the Equations (3.66) to (3.69) we now get

g0
g0

cos

.

p cos

Z Z

R

Z Z

φ

φ

−

=

+

(3.77)

Equation (3.73), giving the total sound pressure in front of the surface, will be modified
to read

1

ˆi (^22)
(, ) 1 2 cos(2 cos ).
2
pp
p
pxy=++⎡⎤⎢⎥R R kx⋅+φδ
⎣⎦

 (3.78)

Figure 3.13 Sound incidence at an angle φ. Locally reacting boundary of impedance Zg.

According to our assumption on local surface reaction, which implies that the
impedance Zg is independent of the angle φ, we may then calculate the statistical
absorption factor αstat. This is an average value for α over all angles of incidence using
the expression

pi

pr

Zg

x = 0

y

x

φ

φ