Civil Engineering Reference
In-Depth Information
material area leads to
p(M)δ p
∗
(M)
d
S(M)
=
2
p
i
(M)δ p
∗
(M)
d
S(M)
S
S
jkβ
(12.24)
p(M
0
)G(M,M
0
)δ p
∗
(M)
d
S(M
0
)
d
S(M)
−
S
S
that is
B p
i
(M) p
i
(M)δ B
∗
d
S(M)
=
2
p
i
(M) p
i
(M)δ B
∗
d
S(M)
S
S
−
jkβ
(12.25)
B p
i
(M
0
)G(M,M
0
) p
i
(M)δ B
∗
d
S(M
0
)
d
S(M)
S
S
δ B
∗
being arbitrary, the above equation leads to
2
p
i
(M) p
i
(M)
d
S(M)
B
S
jkβ
=
p
i
(M) p
i
(M)
d
S(M)
p
i
(M
0
)G(M,M
0
) p
i
(M)
d
S(M
0
)
d
S(M)
(12.26)
+
S
S
S
Using the fact that
|
p
i
|=
1 (arbitrarily normalized)
2
S
B
=
(12.27)
S
+
Z
R
βS
where
S
denotes the area of the material and
Z
R
is the normalized radiation impedance
obtained from Equation (12.11) wherein symbol
Z
R
denotes a non-normalized impedance.
Finally, using
B p
i
=
(
1
+
V
f
) p
i
, the surface pressure is related to the incident
p
=
pressure by
2
p
i
(x,y,
0
)
1
+
2
p
i
(x,y,
0
)Z
A
Z
A
+
p(x,y,
0
)
=
=
(12.28)
Z
R
β
Z
R
with
Z
A
=
1
β
the material normalized surface impedance of the material.
In the case where the material is of infinite extent, the normalized radiation impedance
Z
R
=
1
cos
θ
and the classical formula for the parietal pressure is recovered
2
p
i
(x,y,
0
)Z
A
p(x,y,
0
)
=
(12.29)
1
cos
θ
Z
A
+
12.3.2 Absorption coefficient
The power absorbed at an incidence (
θ
,
φ)
reads
2
Re
p υ
n
dS
2
ρ
0
c
Re
Z
A
d
S
p
p
∗
1
1
abs,f
(θ,φ)
=
=
(12.30)
S
S
