Civil Engineering Reference
In-Depth Information
In Equation (11.12)
δ
1
and
δ
3
are the squares of the wave numbers of the longitudinal
and shear waves in the elastic solid layer, respectively. They are given by:
$
ω
2
ρ
λ
+
2
µ
δ
1
=
(11.13)
ω
2
ρ
µ
%
δ
3
=
where
ρ
indicates the density for the elastic solid, and
λ
and
µ
are the first and second
Lame coefficients (these quantities are complex in general), respectively. The constants
A
1
,A
2
,A
3
and
A
4
represent the amplitudes of the four waves (incident and reflected)
which can propagate in the layer. The acoustic field in the elastic solid layer can be
predicted if these four amplitudes are known. Instead of these parameters, four mechanical
variables may be chosen to express the sound propagation everywhere in the medium.
However, different sets of four independent quantities may be chosen. Following Folds
and Loggins (1977), the four chosen quantities are
v
1
,v
3
,σ
33
and
σ
13
.Let
V
s
be the
vector
=
v
1
(M) v
3
(M)
σ
33
(M) σ
13
(M)
T
V
s
(M)
(11.14)
In these equation,
v
1
and
v
3
are the
x
1
and
x
3
components of the velocity at point
M
, respectively.
σ
33
and
σ
13
are the normal and tangential stresses at point
M
.These
velocities and stresses are written as
#
$
v
1
=
jω
∂ϕ
∂ψ
∂x
3
∂x
1
−
(11.15)
jω
∂ϕ
%
∂ψ
∂x
1
v
3
=
∂x
3
+
σ
33
=
λ
∂
2
ϕ
+
2
µ
∂
2
ϕ
$
∂
2
ϕ
∂x
3
∂
2
ψ
∂x
1
∂x
3
∂x
1
+
∂x
3
+
µ
2
∂
2
ϕ
(11.16)
%
∂
2
ψ
∂x
1
−
∂
2
ψ
∂x
3
σ
13
=
∂x
1
∂x
3
+
4 transfer matrix [
T
] of the elastic solid layer, the vector
V
s
(M)
is
first connected to vector
A
To obtain the 4
×
A
4
)
] by a matrix
[
(x
3
)
] such that
V
s
(M)
=
[
(x
3
)
]
A
. Equations (11.15) and (11.16) can be used to eval-
uate [
(x
3
)
]
=
[
(A
1
+
A
2
),(A
1
−
A
2
),(A
3
+
A
4
),(A
3
−
[
(x
3
]
ωk
1
cos
(k
13
x
3
)
−
jωk
1
sin
(k
13
x
3
)jω
33
sin
(k
33
x
3
)
−
ωk
33
cos
(k
33
x
3
)
−
jωk
13
sin
(k
13
x
3
)ωk
13
cos
(k
13
x
3
)
k
1
cos
(k
33
x
3
)
−
jωk
1
sin
(k
33
x
3
)
=
−
D
1
cos
(k
13
x
3
) j
1
sin
(k
13
x
3
)
j
2
k
33
sin
(k
33
x
3
)
−
D
2
k
33
cos
(k
33
x
3
)
jD
2
k
13
sin
(k
13
x
3
)
−
D
2
k
13
cos
(k
13
x
3
)D
1
cos
(k
33
x
3
)
−
jD
1
sin
(k
33
x
3
)
(11.17)
