118 Basic Seismological Theory
mode is the average mantle shear wave velocity from problem 17 and find the period you would expect. How does this
compare to the actual period?
(b) Find the phase velocity for the mode 0 T 130 and compare it to
that for the Love wave of this period found in the dispersion
calculation (Section 2.7.4).
(c) Find the phase velocity for three modes with similar periods:
4 T 67 , 10 T 40 , and 13 T 7 , and interpret the differences.
(d) Find the phase velocities and wavelengths of waves corresponding to the modes 0 S 3 , 0 S 30 , and 0 S 130 . Interpret the trend
of the velocities. Which of these modes would you expect to
be most affected by lateral heterogeneity in the earth, and
why?
36. (a) Show that the three vector spherical harmonics T l
m
, S l
m
, R l
m are
orthogonal, and explain this result’s physical significance.
(b) Show that there is no volume change associated with torsional
modes, and explain this result’s physical significance.
37. (a) Estimate the magnitude of the splitting of the 0 S 2 multiplet
in Fig. 2.9-16a as the ratio of the separation in frequency
between the m = ± 2 singlets to the frequency of m = 0, which is
essentially that of the unsplit multiplet.
(b) We expect that the splitting would be of the order of the ratio
of the unsplit mode’s period to that of the earth’s rotation.
Compute this ratio and compare the result to the results of (a).
Computer problems
C-1. Write a subroutine to generate the values of the function
cos (ω t − kx). Use it to plot the function as a
(a) function of time from t = 0 to 10, at x = 1, for ω = 1, k = 1.
(b) function of time from t = 0 to 10, at x = 0, for ω = 4, k = 1.
(c) function of position from x = 0 to 10, at t = 0, for ω = 1,
k = 2.
(d) function of position from x = 0 to 10, at t = 0, for ω = 1,
k = 4.
C-2. Write a subroutine that uses the P and S velocities on either side of
a solid–solid interface and the angle of incidence for a wave of
a specific type to find the angles of reflection and transmission for
both P and S waves. The subroutine should calculate and list
any possible critical angles for that incident wave, and indicate
whether any of the reflected or transmitted waves are past the
critical angle.
C-3. Write a program that takes the velocities and densities on either
side of a solid–solid interface and finds the vertical incidence
displacement reflection and transmission coefficients, and energy
flux ratios, for P and S waves incident from either side. Use the
program to estimate these quantities for the core–mantle boundary (although it is a solid–liquid boundary), if the lower mantle
has α = 13.7 km/s, β = 7.2 km/s, ρ = 5.5 g/cm
3 , and the core has
α 2 = 8.0 km/s, β 2 = 0.0 km/s, ρ 2 = 9.9 g/cm
3 .
C-4. Write a program, using the result of C-2, to generate figures like
the ray paths in Fig. 2.6-11 for an interface with given velocities
on either side. Use the program to show the ray paths for the
possible incident wave types on either side of a planar interface
with the properties of the core–mantle boundary (problem C-3).
28. For a P wave incident on a horizontal solid–solid interface
(Fig. 2.6-9):
(a) Write the potentials for the incident P wave and reflected P
and SV waves.
(b) Derive the four continuity equations at the interface in terms
of the potentials.
29. For the Love waves in a layer over a halfspace, use the model in
Fig. 2.7-9 to derive the cutoff frequencies for the first and second
higher modes. Compare these results to the figure.
30. A second way to study the downgoing slab is to use observations
from Japan showing that earthquakes about 1300 km away can
give rise to two P-wave arrivals, a small direct one and a larger
one presumably reflected off the upper surface of the slab (Fig. 2.615). Using the geometry and velocities assumed in problem 27:
(a) Determine the angles of incidence at the surface if the apparent velocities of the direct and reflected arrivals are 8.5 and
16 km/s.
(b) Determine the angle of incidence at the slab top of the reflection. To see if the large amplitude of the reflection might occur because of near-critical incidence, compute this critical
angle and compare the two.
(c) Suppose that a P-to-S wave conversion also occurred at the
slab top. For the converted wave, find the angle of incidence
at the slab and the angle of incidence and apparent velocity
expected at the surface.
31. For Love waves in a layer over a halfspace, derive a vertical wavelength to show how the displacement oscillates with depth in the
layer. Also, derive a vertical decay constant for the halfspace, a distance over which the displacement decays to e
−1 of its value at the
interface. Show how these quantities vary with apparent velocity for a given period. For different modes at a given period, interpret the result in terms of the rate at which the displacement
oscillates in the layer and the depth of penetration in the halfspace.
32. For a dispersive wave, derive the following relations between
group velocity, phase velocity, wavelength, frequency, and period:
(a)
(b)
(c)
,
,
.
U c
dc
d
U c
dT
d
U
df
d
= −
=
= −
λ λ
λ
λ λ
2
2
33. Find the displacements for 0 T
0
3 as functions of θ and φ in the
manner done for 0 T 0
2 in Eqns 2.9.12 and 2.9.13.
34. (a) Show that for m = 0,
Y
l
P
l
l
0
1 2
2 1
4
( , )
(cos ).
/
θ φ
π
θ
=
+
⎛
⎝
⎜
⎞
⎠
⎟
(b) Use (a) to find the spherical harmonic Y 00 associated with
radial modes n S 0 .
(c) Evaluate the vector spherical harmonics associated with the
radial modes and explain what the results imply for these
modes’ displacements.
35. Using the relation between modes and traveling waves and the data
in Table 2.9-1:
(a) Because 0 T 2 samples the mantle fairly uniformly (Fig. 2.99a), assume that the phase velocity appropriate for this
mode is the average mantle shear wave velocity from problem 17 and find the period you would expect. How does this
compare to the actual period?
(b) Find the phase velocity for the mode 0 T 130 and compare it to
that for the Love wave of this period found in the dispersion
calculation (Section 2.7.4).
(c) Find the phase velocity for three modes with similar periods:
4 T 67 , 10 T 40 , and 13 T 7 , and interpret the differences.
(d) Find the phase velocities and wavelengths of waves corresponding to the modes 0 S 3 , 0 S 30 , and 0 S 130 . Interpret the trend
of the velocities. Which of these modes would you expect to
be most affected by lateral heterogeneity in the earth, and
why?
36. (a) Show that the three vector spherical harmonics T l
m
, S l
m
, R l
m are
orthogonal, and explain this result’s physical significance.
(b) Show that there is no volume change associated with torsional
modes, and explain this result’s physical significance.
37. (a) Estimate the magnitude of the splitting of the 0 S 2 multiplet
in Fig. 2.9-16a as the ratio of the separation in frequency
between the m = ± 2 singlets to the frequency of m = 0, which is
essentially that of the unsplit multiplet.
(b) We expect that the splitting would be of the order of the ratio
of the unsplit mode’s period to that of the earth’s rotation.
Compute this ratio and compare the result to the results of (a).
Computer problems
C-1. Write a subroutine to generate the values of the function
cos (ω t − kx). Use it to plot the function as a
(a) function of time from t = 0 to 10, at x = 1, for ω = 1, k = 1.
(b) function of time from t = 0 to 10, at x = 0, for ω = 4, k = 1.
(c) function of position from x = 0 to 10, at t = 0, for ω = 1,
k = 2.
(d) function of position from x = 0 to 10, at t = 0, for ω = 1,
k = 4.
C-2. Write a subroutine that uses the P and S velocities on either side of
a solid–solid interface and the angle of incidence for a wave of
a specific type to find the angles of reflection and transmission for
both P and S waves. The subroutine should calculate and list
any possible critical angles for that incident wave, and indicate
whether any of the reflected or transmitted waves are past the
critical angle.
C-3. Write a program that takes the velocities and densities on either
side of a solid–solid interface and finds the vertical incidence
displacement reflection and transmission coefficients, and energy
flux ratios, for P and S waves incident from either side. Use the
program to estimate these quantities for the core–mantle boundary (although it is a solid–liquid boundary), if the lower mantle
has α = 13.7 km/s, β = 7.2 km/s, ρ = 5.5 g/cm
3 , and the core has
α 2 = 8.0 km/s, β 2 = 0.0 km/s, ρ 2 = 9.9 g/cm
3 .
C-4. Write a program, using the result of C-2, to generate figures like
the ray paths in Fig. 2.6-11 for an interface with given velocities
on either side. Use the program to show the ray paths for the
possible incident wave types on either side of a planar interface
with the properties of the core–mantle boundary (problem C-3).
28. For a P wave incident on a horizontal solid–solid interface
(Fig. 2.6-9):
(a) Write the potentials for the incident P wave and reflected P
and SV waves.
(b) Derive the four continuity equations at the interface in terms
of the potentials.
29. For the Love waves in a layer over a halfspace, use the model in
Fig. 2.7-9 to derive the cutoff frequencies for the first and second
higher modes. Compare these results to the figure.
30. A second way to study the downgoing slab is to use observations
from Japan showing that earthquakes about 1300 km away can
give rise to two P-wave arrivals, a small direct one and a larger
one presumably reflected off the upper surface of the slab (Fig. 2.615). Using the geometry and velocities assumed in problem 27:
(a) Determine the angles of incidence at the surface if the apparent velocities of the direct and reflected arrivals are 8.5 and
16 km/s.
(b) Determine the angle of incidence at the slab top of the reflection. To see if the large amplitude of the reflection might occur because of near-critical incidence, compute this critical
angle and compare the two.
(c) Suppose that a P-to-S wave conversion also occurred at the
slab top. For the converted wave, find the angle of incidence
at the slab and the angle of incidence and apparent velocity
expected at the surface.
31. For Love waves in a layer over a halfspace, derive a vertical wavelength to show how the displacement oscillates with depth in the
layer. Also, derive a vertical decay constant for the halfspace, a distance over which the displacement decays to e
−1 of its value at the
interface. Show how these quantities vary with apparent velocity for a given period. For different modes at a given period, interpret the result in terms of the rate at which the displacement
oscillates in the layer and the depth of penetration in the halfspace.
32. For a dispersive wave, derive the following relations between
group velocity, phase velocity, wavelength, frequency, and period:
(a)
(b)
(c)
,
,
.
U c
dc
d
U c
dT
d
U
df
d
= −
=
= −
λ λ
λ
λ λ
2
2
33. Find the displacements for 0 T
0
3 as functions of θ and φ in the
manner done for 0 T 0
2 in Eqns 2.9.12 and 2.9.13.
34. (a) Show that for m = 0,
Y
l
P
l
l
0
1 2
2 1
4
( , )
(cos ).
/
θ φ
π
θ
=
+
⎛
⎝
⎜
⎞
⎠
⎟
(b) Use (a) to find the spherical harmonic Y 00 associated with
radial modes n S 0 .
(c) Evaluate the vector spherical harmonics associated with the
radial modes and explain what the results imply for these
modes’ displacements.
35. Using the relation between modes and traveling waves and the data
in Table 2.9-1:
(a) Because 0 T 2 samples the mantle fairly uniformly (Fig. 2.99a), assume that the phase velocity appropriate for this