Civil Engineering Reference
In-Depth Information
Using Equation (12.28)
2
S
ρ
0
c
1
2
|
p
i
|
4Re
Z
A
abs,f
(θ,φ)
=
(12.31)
|
Z
A
+
Z
R
|
2
On the other hand, the incident power is given by
2
S
ρ
0
c
cos
θ
1
2
|
p
i
|
inc
(θ,φ)
=
(12.32)
In consequence, the absorption coefficient for a given incidence
(θ,ϕ)
is:
abs,f
inc
1
cos
θ
4Re
Z
A
α
f
(θ,φ)
=
=
(12.33)
|
Z
A
+
Z
R
|
2
1
cos
θ
and the classical incidence absorption
For an infinite extent material
Z
R
(θ,φ)
=
formula is recovered
4Re
Z
A
cos
θ
α
∞
(θ)
=
(12.34)
|
Z
A
cos
θ
+
1
|
2
The diffuse field incident and absorbed powers are given by
π/
2
2
π
inc
=
inc
(θ)
sin
θ
d
θ
d
φ
(12.35)
0
0
π/
2
2
π
abs,f
=
abs,f
(θ,ϕ)
sin
θ
d
θ
d
φ
(12.36)
0
0
The corresponding energy absorption coefficient is
π/
2
2
π
4Re
Z
A
sin
θ
d
θ
d
φ
abs,f
inc
=
2
|
Z
A
+
Z
R
|
0
0
α
f,st
=
(12.37)
π/
2
2
π
cos
θ
sin
θ
d
θ
d
φ
0
0
A practical approximation to ease the cost of the numerical evaluation of this equation
is to replace the radiation impedance
Z
R
(θ,φ)
for a given
θ
by its average over heading
angle
φ
2
π
1
2
π
Z
R,avg
(θ)
=
Z
R
(θ,φ)
d
φ
(12.38)
0
The corresponding absorption coefficient becomes
π/
2
4Re
Z
A
sin
θ
d
θ
|
Z
A
+
Z
R,avg
|
2
0
α
f,st,avg
=
(12.39)
π/
2
cos
θ
sin
θ
d
θ
0
