Wave front
x
z
c x ∆t
i
v∆t
Surface
i
k z
k
k x
Wave
vector
Surface
Wave front
k = k z (k x = 0)
c x = ∞
Surface
Wave front
k = k x
(k z = 0)
c x = v
Fig. 2.5-3 The wave vector, k, is normal to the wave front and points in
the direction of propagation. Top: For a plane wave traveling in the
x–z plane, the propagation direction is given by the wave vector (k x , k z )
or the incidence angle, i, between the wave vector and the vertical. In a
time increment ∆t the wave front moves a distance v∆t, where v is the
medium velocity, and sweeps out a distance along the surface c x ∆t, where
c x is the apparent velocity along the surface. Middle: For a plane wave
traveling vertically, the incidence angle i = 0°, k equals k z , and c x is infinite.
Bottom: For a plane wave propagating horizontally, i = 90°, k equals
k x , and c x equals the medium velocity.
1 Because seismological observations are made at the earth’s surface, the apparent
velocity along the earth’s surface is sometimes written as c rather than c x , and k is
sometimes used to denote k x .
incidence angle i in a medium with velocity v moves forward a
distance v∆t and moves across the horizontal surface a distance
c x ∆t. Thus the horizontal apparent velocity is
c x = v/sin i.
(16)
The apparent velocity is always greater than or equal to the
medium velocity, α for P waves and β for S waves. A horizontally propagating wave, with i = 90°, has an apparent velocity
equal to the medium velocity. A vertically incident plane wave
arrives everywhere on the surface at the same time, so it has an
infinite apparent velocity.
The horizontal apparent velocity
1 can be written in terms of
the horizontal component of the wave vector using Eqns 15
and 16:
2.5 Snell’s law 65
so that
∇
=
2
2
2
2
1
u x z t
u x z t
t
y
y
( , , )
( , , )
.
β
∂
∂
(12)
Thus the SH-wave displacement satisfies a scalar wave equation, and can be found without using the SH potential.
2.5.3 Angle of incidence and apparent velocity
We now consider P–SV waves propagating in the x–z plane
that are described by harmonic plane wave solutions of the
scalar wave equations (10),
(P)
φ(x, z, t) = A exp (i(ωt − k x x ± k z α z))
(13)
(SV) ψ(x, z, t) = B exp (i(ωt − k x x ± k z β z)).
The direction of wave propagation is described by the wave
vector, which is the normal to the wave fronts. For propagation in the x–z plane, the direction is given by k x and k z
because k y is zero. Thus Eqn 13 represents waves propagating
in the +x direction (because of the negative sign in −k x x), and
in both the +z and −z directions.
Subscripts on k and k z are needed because the magnitude of
the wave vector differs for P and SV waves. We will see shortly
that in this geometry k x is the same for the P and the SV waves.
The components of the wave vectors satisfy
| k α | 2 = k x
2 + k 2
z α
= ω 2 /α 2 | k β | 2 = k x
2 + k 2
z β
= ω 2 /β 2 .
(14)
Because k y = 0, k x is the horizontal component of the wave
vector.
The direction of propagation can also be expressed by the
angle of incidence that the wave vector makes with the vertical
(Fig. 2.5-3). Because the wave vectors, and therefore incidence
angles, differ for P and S waves, we adopt the convention that
i refers to P-wave incidence angles and j to S-wave incidence
angles. Thus
sin
(
)
, sin
(
)
.
/
/
i
k
k
k
k
j
k
k
k
k
x
x
z
x
x
x
z
x
=
+
=
=
+
=
2
2 12
2
2 12
α
β
α
β
|
|
| |
k
k
(15)
We will see shortly that plane waves change direction when
they cross an interface into a material with different seismic
velocity (Fig. 2.5-4), so the orientation of the wave vector and
the angle of incidence change. Hence the propagation of a
plane wave is characterized by the changing orientations of the
wave vector. We thus speak of a seismic ray that follows this
ray path. Figures like Fig. 2.5-4 are often drawn showing only
the ray paths and omitting the wave fronts that are normal to
the ray.
It is useful to define the apparent velocity, c x , the velocity
at which a plane wave appears to travel along a horizontal
surface. Figure 2.5-3 shows that in a time ∆t a plane wave with
x
z
c x ∆t
i
v∆t
Surface
i
k z
k
k x
Wave
vector
Surface
Wave front
k = k z (k x = 0)
c x = ∞
Surface
Wave front
k = k x
(k z = 0)
c x = v
Fig. 2.5-3 The wave vector, k, is normal to the wave front and points in
the direction of propagation. Top: For a plane wave traveling in the
x–z plane, the propagation direction is given by the wave vector (k x , k z )
or the incidence angle, i, between the wave vector and the vertical. In a
time increment ∆t the wave front moves a distance v∆t, where v is the
medium velocity, and sweeps out a distance along the surface c x ∆t, where
c x is the apparent velocity along the surface. Middle: For a plane wave
traveling vertically, the incidence angle i = 0°, k equals k z , and c x is infinite.
Bottom: For a plane wave propagating horizontally, i = 90°, k equals
k x , and c x equals the medium velocity.
1 Because seismological observations are made at the earth’s surface, the apparent
velocity along the earth’s surface is sometimes written as c rather than c x , and k is
sometimes used to denote k x .
incidence angle i in a medium with velocity v moves forward a
distance v∆t and moves across the horizontal surface a distance
c x ∆t. Thus the horizontal apparent velocity is
c x = v/sin i.
(16)
The apparent velocity is always greater than or equal to the
medium velocity, α for P waves and β for S waves. A horizontally propagating wave, with i = 90°, has an apparent velocity
equal to the medium velocity. A vertically incident plane wave
arrives everywhere on the surface at the same time, so it has an
infinite apparent velocity.
The horizontal apparent velocity
1 can be written in terms of
the horizontal component of the wave vector using Eqns 15
and 16:
2.5 Snell’s law 65
so that
∇
=
2
2
2
2
1
u x z t
u x z t
t
y
y
( , , )
( , , )
.
β
∂
∂
(12)
Thus the SH-wave displacement satisfies a scalar wave equation, and can be found without using the SH potential.
2.5.3 Angle of incidence and apparent velocity
We now consider P–SV waves propagating in the x–z plane
that are described by harmonic plane wave solutions of the
scalar wave equations (10),
(P)
φ(x, z, t) = A exp (i(ωt − k x x ± k z α z))
(13)
(SV) ψ(x, z, t) = B exp (i(ωt − k x x ± k z β z)).
The direction of wave propagation is described by the wave
vector, which is the normal to the wave fronts. For propagation in the x–z plane, the direction is given by k x and k z
because k y is zero. Thus Eqn 13 represents waves propagating
in the +x direction (because of the negative sign in −k x x), and
in both the +z and −z directions.
Subscripts on k and k z are needed because the magnitude of
the wave vector differs for P and SV waves. We will see shortly
that in this geometry k x is the same for the P and the SV waves.
The components of the wave vectors satisfy
| k α | 2 = k x
2 + k 2
z α
= ω 2 /α 2 | k β | 2 = k x
2 + k 2
z β
= ω 2 /β 2 .
(14)
Because k y = 0, k x is the horizontal component of the wave
vector.
The direction of propagation can also be expressed by the
angle of incidence that the wave vector makes with the vertical
(Fig. 2.5-3). Because the wave vectors, and therefore incidence
angles, differ for P and S waves, we adopt the convention that
i refers to P-wave incidence angles and j to S-wave incidence
angles. Thus
sin
(
)
, sin
(
)
.
/
/
i
k
k
k
k
j
k
k
k
k
x
x
z
x
x
x
z
x
=
+
=
=
+
=
2
2 12
2
2 12
α
β
α
β
|
|
| |
k
k
(15)
We will see shortly that plane waves change direction when
they cross an interface into a material with different seismic
velocity (Fig. 2.5-4), so the orientation of the wave vector and
the angle of incidence change. Hence the propagation of a
plane wave is characterized by the changing orientations of the
wave vector. We thus speak of a seismic ray that follows this
ray path. Figures like Fig. 2.5-4 are often drawn showing only
the ray paths and omitting the wave fronts that are normal to
the ray.
It is useful to define the apparent velocity, c x , the velocity
at which a plane wave appears to travel along a horizontal
surface. Figure 2.5-3 shows that in a time ∆t a plane wave with