Civil Engineering Reference
In-Depth Information
In the receiving domain, the acoustic pressure reads:
p(M)
=
p
ray
(M)
(12.2)
Since the panel is baffled, the radiated pressure is given by
∂ p(M
0
)
∂n
p
ray
(M)
=−
G(M,M
0
)
d
S(M
0
)
(12.3)
S
(x
jk
0
R
]
(
2
πR)
,
R
y
0
)
2
and
n
denotes
the outward normal to the radiating surface
S
pointing into the receiving domain.
Let
υ
n
denote the normal surface velocity, the radiated (transmitted) power is given by
where
G(M,M
0
)
=
exp[
−
=
−
x
0
)
2
+
(y
−
2
Re
p
ray
(M)υ
n
(M)
d
S(M)
1
t
=
S
(12.4)
2
Re
∂n
(M
0
)G(M,M
0
)
∂ p
∗
j
ωρ
0
∂ p
1
=
∂n
(M)
d
S(M
0
)dS(M)
S
As
in
the
classical
implementation
of
the
TMM,
we
assume
that
the
surface
impedance,
Z
m
,
seen
from
the
radiating
surface
is
governed
by
the
plane
wave
assumption and thus is given by
ρ
0
c/
cos
θ
. Using the relationship
∂ p/∂n
+
jkβ p
=
0
with
β
=
ρ
0
c/Z
m
=
cos
θ
the transmitted power reads
Re
jk
0
S
p(M
0
)G(M,M
0
) p
∗
(M)
d
S(M
0
)
d
S(M)
S
cos
2
θ
ρ
0
c
1
2
t
=
(12.5)
S
The parietal pressure, on the receiver side, is related to the incident pressure by the
pressure transmission coefficient:
p
T
∞
p
i
.Here
T
∞
is calculated using the TMM and
in consequence is constant over the surface. The subscript
∞
is used to denote that the
panel is assumed of infinite extent and its radiation impedance given by the plane wave
approximation. Defining the energy transmission coefficient
τ
∞
=|
=
2
, the transmitted
T
∞
|
power becomes
1
2
cos
θ
Re
jk
0
S
p
i
(M
0
)G(M,M
0
)p
i
(M)
d
S(M
0
)
d
S(M)
(12.6)
S
cos
θτ
∞
ρ
0
c
t
=
S
The first term of Equation (12.6) represents the classical expression of the transmitted
power, assuming an infinite structure. The second term represents a geometrical correction
accounting for the finite size effect. It is given by the ratio of the radiation efficiency
σ
R
of the incident plane wave forced finite structure radiator to the radiation efficiency
σ
∞
of the infinite structure:
σ
R
σ
∞
cos
θ
Re
jk
0
S
p
i
(M
0
)G(M,M
0
)p
i
(M)
d
S(M
0
)
d
S(M)
(12.7)
=
σ
R
cos
θ
=
S
Indeed, consider a flat baffled panel forced to vibrate by the incident plane wave
υ(x,y)
=|
υ
|
exp[
−
jk
0
(
cos
φ
sin
θx
+
sin
φ
sin
θy)
]
(12.8)
