Geoscience Reference
In-Depth Information
Figure 4.39.
Reflected
and transmitted energy
(energy/incident energy)
for the situation illustrated
in Fig. 4.34:aP-wave
incident on an interface
between two solid media.
For this example,
1.0
0.8
Reflected P
Transmitted P
0.6
α
1
/
α
2
=
0.4
0.5,
ρ
1
/
ρ
2
=
2.0,
α
1
/
β
1
=
1.87 and
1.73.
Solid lines, reflected and
transmitted P-wave;
short-dashed line,
reflected S-wave; dashed
and dotted line,
transmitted S-wave; and
long-dashed lines,
reflected and transmitted
P-wave when
ρ
1
/
ρ
2
= 1.0.
(After Tooley
et al.
(1965).)
α
2
/
β
2
=
Transmitted S
0.2
Reflected P
Reflected S
0
0
90
30
60
Angle of incidence
where
Z
1
=
ρ
1
α
1
,
Z
2
=
ρ
2
α
2
W
1
=
ρ
1
β
1
,
W
2
=
ρ
2
β
2
β
1
α
1
,
β
2
α
2
γ
1
=
2
=
Equations (4.56) and (4.57) come from requiring continuity of the vertical and
horizontal displacements, and Eqs. (4.58) and (4.59) are required by continuity
of the vertical and horizontal stresses on the boundary.
Z
and
W
, the product of
the density and seismic velocity, are termed the impedance.
The ratio of the reflected or transmitted amplitude to the incident ampli-
tude is called the
reflection
or
transmission coefficient
. (Unfortunately, and
confusingly, the fractions of the incident energy which are reflected or trans-
mitted are also sometimes referred to as the reflection and transmission coef-
ficients.) Calculation of reflection and transmission coefficients for anything
other than normal incidence is lengthy because the equations have to be solved
numerically.
Figure 4.39 illustrates the reflection and transmission of energy for a P-wave
incident on a solid-solid boundary. For this example there are two critical angles,
one for P-waves and the other for S-waves, at 30
◦
and 60
◦
.Beyond the first
critical angle, that for P-waves (sin 30
◦
=
α
1
/
0.5), there are no transmitted
P-waves. The reflected P-wave energy increases greatly as the angle of incidence
increases towards the first critical angle; these are the
wide-angle reflections
which
are used extensively in seismic-refraction work to determine critical distances.
Similarly, beyond the critical angle (sin
−
1
α
2
=
α
1
/
β
2
)) for S-waves at 60
◦
there are
(
no transmitted S-waves.
Reflection and transmission coefficients take a very much simpler form for
the case of normal incidence on the boundary (
i
=
0). In this case,
B
1
=
B
2
=
0
