Civil Engineering Reference
In-Depth Information
k
i
k
r
θ
i
θ
r
Fluid 1
X
1
Fluid 2
X
2
θ
t
X
3
k
t
Figure 3.2
Plane wave reflection and refraction at a plane interface between two fluids.
The angle of incidence
θ
i
can be either real or complex. The incident wave is repre-
sented by the following equations which are similar to Equations (3.1) and (3.5):
p
i
=
A
i
exp[
j(
−
k(
sin
θ
i
x
1
+
cos
θ
i
x
3
)
+
ωt)
]
(3.16)
A
i
Z
c
sin
θ
i
exp[
j(
υ
i
1
=
−
k(
sin
θ
i
x
1
+
cos
θ
i
x
3
)
+
ωt)
]
(3.17)
A
i
Z
c
sin
θ
i
exp[
j(
υ
i
3
=
−
k(
sin
θ
i
x
1
+
cos
θ
i
x
3
)
+
ωt)
]
(3.18)
υ
i
2
=
0
(3.19)
Two similar sets of equations describe the reflected and the transmitted waves. At the
boundary, velocities and pressures are the same in both media. From these conditions, it
can be shown that reflection and refraction are described by the laws of Snell - Descartes.
The vectors
k
i
,
k
r
and
k
t
are in the same plane. In the given example, the
x
2
com-
ponent of the three wave number vectors vanishes:
k
r
2
=
k
t
2
=
k
i
2
=
0
(3.20)
Continuity of the velocity at the interface shows that the angles of incidence and
reflection,
θ
i
and
θ
r
, are equal:
θ
i
=
θ
r
(3.21)
and that the angles of incidence
θ
i
and of transmission
θ
t
are related by the equation
k
sin
θ
t
k
sin
θ
i
=
(3.22)
It follows that
k
i
1
=
k
sin
θ
i
=
k
sin
θ
r
=
k
r
1
(3.23)
k
i
3
=
k
cos
θ
i
=
k
cos
θ
r
=−
k
r
3
(3.24)
The component
k
t
3
is given by
k
cos
θ
t
k
(
1
−
sin
θ
t
)
1
/
2
k
t
3
=
=
(3.25)
