The direction of propagation for each wave is given by the
components of the wave vector k. For example, the signs of
the k x and k x r α 1 terms show that the incoming P wave with
amplitude A 1 travels in the +x and +z directions as time increases. Similarly, the reflected P wave with amplitude A 2 and
the reflected SV wave with amplitude B 2 travel in the +x and
−z directions.
The downgoing P wave and SV waves in the second
medium are given by the potentials
φ(x, z, t) = transmitted P = A′ exp (i(ωt − k x x − k x r α 2 z))
ψ(x, z, t) = transmitted SV = B′ exp (i(ωt − k x x − k x r β 2 z)). (21)
A′ and B′ are the amplitudes of the transmitted P and SV waves,
which travel in the +x and +z directions. We generally write
the amplitudes of P waves as A and the amplitudes of S waves
as B.
We can find the incidence angles of the transmitted and
reflected waves from the incidence angle of the incoming wave.
The boundary conditions for the solid–solid interface at z = 0
are that the components of the displacement and traction
vectors are continuous (Section 2.3.10). Because all of the
potentials contain the phase factor, exp (i(ωt − k x x)) times a
factor independent of x and t, all of the displacement and traction components have this phase factor. For the displacement
and traction to be continuous at the interface for all x and all t,
(ωt − k x x) must be equal for each of the potentials. Thus the
horizontal wavenumber k x , and hence the apparent velocity
along the interface c x = ω/k x , must be the same for each wave.
As a result, the waves travel along the interface at the same
speed and stay in phase.
This condition and the definition of c x (Eqn 16) give the
familiar form of Snell’s law:
c
i
j
i
j
x =
=
=
=
sin
sin
sin
sin
,
α
β
α
β
1
1
1
1
2
2
2
2
(22)
the ratio of the sine of the angle of incidence for each wave to
the corresponding velocity is constant. Hence the incident and
reflected P waves have the same incidence angle i 1 . The transmitted P and S waves change direction by a factor depending on
the velocities in the two media. A change in direction upon
transmission into a medium with a different velocity is called
refraction, so the waves in the second medium are called
refracted or transmitted waves. Figure 2.5-5 illustrates the ray
paths for the different waves.
The S wave reflected from the boundary satisfies
sin j 1 = sin i 1 (β 1 /α 1 ).
(23)
Because in any medium P waves travel faster than S waves,
Snell’s law requires that j 1 < i 1 . Hence the reflected S ray is
closer to the vertical, or further from the interface, than the P
ray in the same medium. Physically, this is because the S wave
must be closer to the vertical than the P wave to have the same
apparent velocity along the interface.
The angle of incidence for the refracted P wave is related to
that for the incident P wave by
sin i 2 = sin i 1 (α 2 /α 1 ).
(24)
If the second medium has a higher velocity, then i 2 > i 1 , so the
transmitted ray is further from the vertical than the incident
ray. It travels more horizontally, so the apparent velocities
along the interface are equal. On the other hand, if α 1 > α 2 ,
then the refracted P wave would be closer to normal incidence.
(This effect, for light waves, makes a pencil appear to bend at
the surface of a glass of water.)
The transmitted S wave satisfies
sin j 2 = sin i 1 (β 2 /α 1 ).
(25)
Hence for β 2 > β 1 , we get j 2 > j 1 , so the transmitted S wave
is more nearly horizontal than the reflected S wave. Similar
relations apply for an incident SV wave (Fig. 2.5-5). The
reflected P ray is bent further from the normal than the incident or reflected SV rays.
The fact that an incident P wave generates both P and SV
waves, and vice versa, is a consequence of the displacement
and traction boundary conditions at the interface, as we will
see in Section 2.6. Some insight into why this should be can be
obtained by considering Fig. 2.5-6, in which an incident SV
wave disturbs the boundary, which then generates P waves in
addition to the transmitted and reflected SV waves.
2.5.5 Critical angle
When a P wave impinges on a horizontal boundary, Eqn 24
shows that the incidence angle for the transmitted P wave in the
second medium is
i 2 = sin −1 [sin i 1 (α 2 /α 1 )],
(26)
where the notation sin −1 indicates the inverse sine function.
If the second medium has a higher velocity, the transmitted
P ray is further from the vertical than the incident ray. As the
angle of incidence increases, the transmitted ray approaches
the horizontal interface (Fig. 2.5-7, overleaf ). Eventually, the
incidence angle i 1 reaches a value i c where i 2 = 90° and the argument of the sin −1 term becomes 1, so
sin i c (α 2 /α 1 ) = 1 or sin i c = α 1 /α 2 .
(27)
Thus for a wave incident at this critical angle of incidence, the
transmitted wave grazes the interface.
Once the incidence angle exceeds the critical angle, which is a
situation called postcritical incidence, no transmitted plane
wave exists in the second medium. This phenomenon is sometimes called total internal reflection. In this case, as we will see
2.5 Snell’s law 67
components of the wave vector k. For example, the signs of
the k x and k x r α 1 terms show that the incoming P wave with
amplitude A 1 travels in the +x and +z directions as time increases. Similarly, the reflected P wave with amplitude A 2 and
the reflected SV wave with amplitude B 2 travel in the +x and
−z directions.
The downgoing P wave and SV waves in the second
medium are given by the potentials
φ(x, z, t) = transmitted P = A′ exp (i(ωt − k x x − k x r α 2 z))
ψ(x, z, t) = transmitted SV = B′ exp (i(ωt − k x x − k x r β 2 z)). (21)
A′ and B′ are the amplitudes of the transmitted P and SV waves,
which travel in the +x and +z directions. We generally write
the amplitudes of P waves as A and the amplitudes of S waves
as B.
We can find the incidence angles of the transmitted and
reflected waves from the incidence angle of the incoming wave.
The boundary conditions for the solid–solid interface at z = 0
are that the components of the displacement and traction
vectors are continuous (Section 2.3.10). Because all of the
potentials contain the phase factor, exp (i(ωt − k x x)) times a
factor independent of x and t, all of the displacement and traction components have this phase factor. For the displacement
and traction to be continuous at the interface for all x and all t,
(ωt − k x x) must be equal for each of the potentials. Thus the
horizontal wavenumber k x , and hence the apparent velocity
along the interface c x = ω/k x , must be the same for each wave.
As a result, the waves travel along the interface at the same
speed and stay in phase.
This condition and the definition of c x (Eqn 16) give the
familiar form of Snell’s law:
c
i
j
i
j
x =
=
=
=
sin
sin
sin
sin
,
α
β
α
β
1
1
1
1
2
2
2
2
(22)
the ratio of the sine of the angle of incidence for each wave to
the corresponding velocity is constant. Hence the incident and
reflected P waves have the same incidence angle i 1 . The transmitted P and S waves change direction by a factor depending on
the velocities in the two media. A change in direction upon
transmission into a medium with a different velocity is called
refraction, so the waves in the second medium are called
refracted or transmitted waves. Figure 2.5-5 illustrates the ray
paths for the different waves.
The S wave reflected from the boundary satisfies
sin j 1 = sin i 1 (β 1 /α 1 ).
(23)
Because in any medium P waves travel faster than S waves,
Snell’s law requires that j 1 < i 1 . Hence the reflected S ray is
closer to the vertical, or further from the interface, than the P
ray in the same medium. Physically, this is because the S wave
must be closer to the vertical than the P wave to have the same
apparent velocity along the interface.
The angle of incidence for the refracted P wave is related to
that for the incident P wave by
sin i 2 = sin i 1 (α 2 /α 1 ).
(24)
If the second medium has a higher velocity, then i 2 > i 1 , so the
transmitted ray is further from the vertical than the incident
ray. It travels more horizontally, so the apparent velocities
along the interface are equal. On the other hand, if α 1 > α 2 ,
then the refracted P wave would be closer to normal incidence.
(This effect, for light waves, makes a pencil appear to bend at
the surface of a glass of water.)
The transmitted S wave satisfies
sin j 2 = sin i 1 (β 2 /α 1 ).
(25)
Hence for β 2 > β 1 , we get j 2 > j 1 , so the transmitted S wave
is more nearly horizontal than the reflected S wave. Similar
relations apply for an incident SV wave (Fig. 2.5-5). The
reflected P ray is bent further from the normal than the incident or reflected SV rays.
The fact that an incident P wave generates both P and SV
waves, and vice versa, is a consequence of the displacement
and traction boundary conditions at the interface, as we will
see in Section 2.6. Some insight into why this should be can be
obtained by considering Fig. 2.5-6, in which an incident SV
wave disturbs the boundary, which then generates P waves in
addition to the transmitted and reflected SV waves.
2.5.5 Critical angle
When a P wave impinges on a horizontal boundary, Eqn 24
shows that the incidence angle for the transmitted P wave in the
second medium is
i 2 = sin −1 [sin i 1 (α 2 /α 1 )],
(26)
where the notation sin −1 indicates the inverse sine function.
If the second medium has a higher velocity, the transmitted
P ray is further from the vertical than the incident ray. As the
angle of incidence increases, the transmitted ray approaches
the horizontal interface (Fig. 2.5-7, overleaf ). Eventually, the
incidence angle i 1 reaches a value i c where i 2 = 90° and the argument of the sin −1 term becomes 1, so
sin i c (α 2 /α 1 ) = 1 or sin i c = α 1 /α 2 .
(27)
Thus for a wave incident at this critical angle of incidence, the
transmitted wave grazes the interface.
Once the incidence angle exceeds the critical angle, which is a
situation called postcritical incidence, no transmitted plane
wave exists in the second medium. This phenomenon is sometimes called total internal reflection. In this case, as we will see
2.5 Snell’s law 67