Civil Engineering Reference
In-Depth Information
From Equation (3.22) it follows that
k
i
1
=
k
t
1
(3.26)
and
(k
2
k
2
sin
2
θ
i
)
1
/
2
k
t
3
=
−
(3.27)
The appropriate choice of the square root to avoid the amplitude of the refracted wave
tending to infinity with
x
3
is the one with Im
(k
t
3
)<
0.
In terms of the wave speeds
c
and
c
in the two fluids, if the wave numbers are real,
Equation (3.26) reads
sin
θ
i
c
=
sin
θ
t
c
(3.28)
This is the classical form of Snell's law of refraction.
3.4
Impedance at oblique incidence in isotropic fluids
3.4.1 Impedance variation along a direction perpendicular
to an impedance plane
An ingoing and an outgoing plane wave are represented in Figure 3.3. Let
n
and
n
be
the unit vectors associated with the incoming and outgoing wave, respectively. Both are
parallel to the plane
x
2
=
0. The 0
x
2
axis is perpendicular to the plane of the figure. Let
θ
be the angle between the
x
3
axis and
n
, and between the
x
3
axis and
n
. This angle
can be real or complex. First, it will be shown that with this geometry, the ratio of the
pressure
p
to
υ
3
(the component
υ
3
of velocity on the
x
3
axis) is constant on planes
parallel to
x
3
=
0. These planes are impedance planes on which the impedance
Z
in the
normal direction
Z
=
p/υ
3
(3.29)
is constant, this impedance depending only on
x
3
.
By employing Equations (3.1) and (3.4), the pressure and the velocity component on
the
x
3
axis can be written for the two waves respectively
p(x
1
,x
3
)
=
A
exp
(j(
−
k
1
x
1
−
k
3
x
3
+
ωt))
(3.30)
A
Z
c
k
3
k
exp
(j(
−
k
1
x
−
k
3
x
3
+
ωt))
υ
3
(x
1
,x
3
)
=
(3.31)
p
(x
1
,x
3
)
A
exp
(j(
=
−
k
1
x
1
+
k
3
x
3
+
ωt))
(3.32)
A
Z
c
k
3
k
exp
(j(
υ
3
(x
1
,x
3
)
=−
−
k
1
x
1
+
k
3
x
3
+
ωt))
(3.33)
On the plane
x
3
=
x
3
(M
1
)
, the impedance in the direction
x
3
is equal to
p
(x
1
,x
3
)
υ
3
(x
1
,x
3
)
+
υ
3
(x
1
,x
3
)
p(x
1
,x
3
)
+
Z(M
1
)
=
(3.34)
