Civil Engineering Reference
In-Depth Information
B
mn
are modal amplitudes obtained from the modes orthogonality properties
(p
a
p
rad
ϕ
mn
(x
1
,x
2
)
d
S
x
=
−
p
b
)ϕ
mn
(x
1
,x
2
)
d
S
x
(13.35)
p
1
p
1
which leads to
1
N
mn
(p
a
B
mn
=
−
p
b
)ϕ
mn
(x
1
,x
2
)
d
S
x
(13.36)
p
1
where
2
d
S
x
N
mn
=
1
|
ϕ
mn
(x
1
,x
2
)
|
(13.37)
p
is the norm of mode
(m,n)
.
The calculation for the second term of
I
2
is carried out in the same way, taking into
account the fact that Equation (13.31) reduces to
p
a
=
p
rad
. Finally,
1
jω
I
i
=−
δ(pu
n
)
d
S
−
A(
x,y
)p(
y
)δp(
x
)
d
S
x
d
S
y
p
i
p
i
p
i
(13.38)
jω
+
A(
x,y
)p
b
(
y
)δp(
x
)
d
S
x
d
S
y
i
=
1
,
2
i
i
where
ε
=
1if
i
=
1and0if
i
=
2. In Eq (13.38),
x
=
(x
1
,x
2
)
and
y
=
(
y
1
,
y
2
)
are two
points on the interface.
A(
x
,
y
)
is an admittance operator given by
k
mn
ρ
0
ωN
mn
ϕ
mn
(x
1
,x
2
)ϕ
mn
(y
1
,y
2
)
A(
x, y
)
=
(13.39)
mn
Equation (13.38) has the advantage of depicting the coupling with the waveguide
in terms of radiation admittance in the emitter and receiver media, together with a
blocked-pressure loading. To see the radiation effect, note for example that the power
radiated (transmitted) into the receiving part of the waveguide is given by
p
a
u
a
n
d
S
jω
1
2
Re
2
=
−
(13.40)
p
2
This can be written
2
Re
p
a
(
x
)A
∗
(
x
,
y
)p
a
∗
(
y
)
d
S
x
d
S
y
1
2
=
(13.41)
p
2
p
2
Note that at low frequencies (below the cutoff frequency of the waveguide), higher
modes lead to a purely imaginary admittance of an inertance type. These modes are
evanescent and do not radiate in the tube.
For completion note that, if the porous medium is replaced by an elastic medium (e.g.
the case of a porous medium sandwiched between two plates; a medium made up from
