Civil Engineering Reference
In-Depth Information
where
p
and
v
3
are the pressure and the
x
3
component of the fluid velocity, respectively.
The superscript T indicates a transposition,
V
f
(M)
being a column vector.
Let
ρ
and
k
be the density and the wave number of the fluid medium, respec-
tively. Let
k
3
be the
x
3
component of the wave number vector in the fluid, equal to
k
2
k
2
sin
2
θ
1
/
2
. The dependence in
x
1
and time being removed,
p
and
v
3
in the fluid
can be written as
−
p(x
3
)
=
A
1
exp
(
−
jk
3
x
3
)
+
A
2
exp
(jk
3
x
3
)
(11.4)
k
3
ωρ
[
A
1
exp
(
v
3
(x
3
)
=
−
jk
3
x
3
)
−
A
2
exp
(jk
3
x
3
)
]
(11.5)
By arbitrarily setting the coordinate
x
3
equal to zero at
M
, Equations (11.4) and
(11.5) can be rewritten as
p(M
)
=
A
1
+
A
2
(11.6)
k
3
ωρ
(A
1
−
v
3
(M
)
=
A
2
)
(11.7)
Denoting by
h
the thickness of the layer, Equations (11.4) and (11.5) at
x
3
=−
h
can
be rewritten as
V
f
(M)
[
T
]
V
f
(M
)
=
(11.8)
where the 2
×
2 transfer matrix [
T
] is given by:
j
ωρ
k
3
cos
(k
3
h)
sin
(k
3
h)
[
T
]
=
(11.9)
j
k
3
ωρ
sin
(k
3
h)
cos
(k
3
h)
11.3.2 Solid layer
In a layer consisting of an elastic solid, an incident and a reflected longitudinal wave, and
an incident and a reflected shear wave can propagate. The acoustic field in the material
can be completely described using the four amplitudes of these waves. The associated
displacement potentials can be written as, respectively
ϕ
=
exp
(jωt
−
jk
1
x
1
)
[
A
1
exp
(
−
jk
13
x
3
)
+
A
2
exp
(jk
13
x
3
)
]
(11.10)
ψ
=
exp
(jωt
−
jk
1
x
1
)
[
A
3
exp
(
−
jk
33
x
3
)
+
A
4
exp
(jk
33
x
3
)
]
(11.11)
where the
x
3
components
k
13
and
k
33
of the wave number vectors are
k
13
(δ
1
−
k
t
)
1
/
2
=
(11.12)
k
t
)
1
/
2
(δ
3
−
k
33
=
