Waves in fluid and solid media - Building Acoustics - page 80

− j( cos kx ϕ + y sin ) ϕ

ˆ

pxy (, , ) ϕ =⋅ p

p

e and

i i

(3.75)

ˆ

− j( cos kx ϕ + y sin ) ϕ

i

vxy (, , ) ϕ = cos ϕ ⋅ e .

i, x

ρ c

00

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

j( cos kx ϕ − y sin ) ϕ

ˆ

pxy (, , ) ϕ =⋅ p

Rp

e and

r r

ˆ (3.76)

p i

j( cos kx ϕ − y sin ) ϕ

vxy (, , ) ϕ =− cos ϕ ⋅ e .

r, x

ρ c

00

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

Z cos ϕ

ϕ

− Z

g 0

R = . (3.77)

p

Z cos + Z

g 0

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

1

ˆ

p

2

⎡ ⎤

2

i

pxy (, ) = 1 + R + 2 R cos(2 kx ⋅ cos ϕδ + ) . (3.78)

⎢

p p

⎥

⎣ ⎦

2

p r

ϕ

ϕ Z g

p i

y

x

x = 0

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

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