Civil Engineering Reference
In-Depth Information
X
2
n
θ
X
1
Figure 3.1
Symbolic representation of an inhomogeneous plane wave with
k
parallel
to the plane x
3
=
0.
components
n
1
,
n
2
and
n
3
of a unit vector
n
such that
n
1
+
n
2
+
n
2
=
1. In the follow-
ing, we will be concerned only with waves with a vector velocity parallel to a coordinate
plane, for instance the plane
x
3
=
0if
k
3
=
0. A complex angle
θ
canbeusedtoindicate
the direction of propagation. Equation (3.4) then reduces to
k
1
=
k(
1
−
sin
2
θ)
1
/
2
,
k
2
=
k
sin
θ,
k
3
=
0
(3.12)
The quantity sin
θ
can be any complex number, associated with a complex angle such
that
(e
jθ
e
−
jθ
)/(
2j
)
sin
θ
=
−
(3.13)
A symbolic representation is given in Figure 3.1.
The components of the unit vector
n
in Equations (3.3) are
(
1
−
sin
2
θ)
1
/
2
n
1
=
k
1
/k
=
=
cos
θ, n
2
=
k
2
/k
=
sin
θ, n
3
=
k
3
/k
=
0
(3.14)
In the previous example
n
1
=
l/k,
n
2
=−
jm/k,
n
3
=
0
The attenuation in the direction
x
2
is not due to the dissipation in the fluid, but to the
fact that
l
is larger than
k
. The equiphase and equiamplitude planes are perpendicular. It is
easy to show that this property is always valid for media with
k
2
n
k
with components
k
1
,
k
2
and
k
3
will be used in the following, even if its components are
complex.
real. The vector
k
=
3.3
Reflection and refraction at oblique incidence
Two fluids with a plane boundary are represented in Figure 3.2.
Let
k
and
k
be the complex wave numbers in fluid 1 and fluid 2. Let
k
i
(k
i
1
,k
i
2
,k
i
3
)
,
k
r
(k
r
1
,k
r
2
,k
r
3
)
and
k
t
(k
t
1
,k
t
2
,k
t
3
)
be the wave number vectors of the incident, reflected
and refracted waves, respectively. The three components of
k
i
are
k(
1
−
sin
2
θ
i
)
1
/
2
k
i
1
=
k
sin
θ
i
,
k
i
2
=
0
,
k
i
3
=
=
k
cos
θ
i
(3.15)
