116 Basic Seismological Theory
Problems
11. For the strain tensor
e =
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
3 0 0
0 1 1
0 1 2
(a) Find the corresponding stress tensor, assuming an isotropic
solid with Lamé constants λ and µ .
(b) Find the stored elastic strain energy, W = σ ij e ij /2.
12. Give a physical interpretation of the fact that Young’s modulus for
rubber is less than that for steel.
13. An alternative to using potentials to find seismic wave solutions to
the equation of motion in terms of displacements is to formulate
wave equations for the dilatation and curl of the displacement
field. To see this:
(a) Take the divergence of Eqn 2.4.12 to obtain a wave equation
for the dilatation θ. At what velocity does θ propagate?
(b) Take the curl of Eqn 2.4.12 to obtain a wave equation for
∇
∇ ∇
∇ ∇ × u. At what velocity does ∇
∇ ∇
∇ ∇ × u propagate?
14. Derive the constitutive law (Eqn 2.3.70) for an isotropic and
linearly elastic material using the c ijkl in Eqn 2.3.69.
15. Derive the ratio of P- and S-wave velocities in a Poisson solid.
16. Use the gradient operator in spherical coordinates (Eqn A.7.14) to
find the displacement field from the spherical wave scalar potential f(t − r/v)/r. How would you approximate the displacements
near the source? How would you approximate the displacements
far from the source?
17. On a seismometer located at an earthquake hypocenter, the phases
reflected from the core, PcP and ScS, arrive at 8 minutes, 31 seconds, and 15 minutes, 36 seconds, respectively after the earthquake. If the earth’s radius is 6371 km, and the core’s radius is
3480 km:
(a) Find the average P- and S-wave velocities in the earth’s mantle.
(b) Use these average velocities to estimate how close the mantle
is to a Poisson solid.
18. Estimate the P- and S-wave velocities in the upper mantle by
assuming that it is a Poisson solid, and that the earthquake for
which seismograms are shown in Fig. 2.4-8 occurred at a depth of
280 km. Compare these velocities to the average mantle values.
Note that the seismograms do not start at the earthquake origin
time.
19. To get a feel for the distance and time scales in seismic wave propagation, consider waves propagating in a material with velocity
8 km/s.
(a) Find the wavelengths of waves with periods of 0.1 s, 1 s,
and 100 s.
(b) Find the periods and frequencies of waves with wavelengths
of 1 m, 1 km, and 100 km.
20. For waves propagating in an arbitrary direction given by the
wavenumber vector k,
(a) Show that the P-wave displacement due to the scalar potential
φ(x, t) = e i(ωt−k·x)
is parallel to the propagation direction.
1. What are the reflection and transmission coefficients for a junction
between two identical strings? Give a physical interpretation of the
result.
2. In Fig. 2.2-6, find the seismic velocities of the two different string
segments by measuring the distance versus time slope of the wave
pulses on the left and right sides of the figure. Are these velocities
the same as the velocities given in the figure caption?
3. For the stress tensor
σ =
− −
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
2
1 3
1 1 2
3 2 5
find the traction on
(a) the x–y plane,
(b) the y–z plane,
(c) the plane with normal (3, 2, −1).
4. To derive the reflection coefficient for the end of a string:
(a) Express the total displacement due to incident and reflected
harmonic waves of unknown amplitudes.
(b) Find the relation between these amplitudes at a fixed string
end, where the displacement is zero, and at a free end, where
the traction is zero.
5. For the stress tensor
σ =
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0 2 0
2 0 0
0 0 0
(a) Find the principal stresses and their associated directions.
(b) Find the surfaces on which the maximum tangential traction
occurs, and the value of this traction.
6. Estimate the pressure expected at a depth of 1000 km in the earth.
7. Given the stress tensor, whose elements are in kbar:
σ =
−
−
− −
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
150
2
1
2 155
3
1
3 145
(a) What physical situation do the large negative values on the
diagonal represent?
(b) What is the mean stress?
(c) What is the deviatoric stress tensor?
(d) At what depth in the earth might this state of stress be
found?
8. Give an example of a strain tensor for which there is
(a) an increase in volume,
(b) a decrease in volume,
(c) shear strain but no volume change.
Which of these strains could result from a P wave, and which could
result from an S wave?
9. Estimate by what fraction the volume of a block of a Poisson solid
with the rigidity of crustal rock will be compressed at a depth of
30 km relative to its volume at the earth’s surface.
10. Determine whether the Lamé constant λ can be negative and, if so,
under what conditions.
Problems
11. For the strain tensor
e =
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
3 0 0
0 1 1
0 1 2
(a) Find the corresponding stress tensor, assuming an isotropic
solid with Lamé constants λ and µ .
(b) Find the stored elastic strain energy, W = σ ij e ij /2.
12. Give a physical interpretation of the fact that Young’s modulus for
rubber is less than that for steel.
13. An alternative to using potentials to find seismic wave solutions to
the equation of motion in terms of displacements is to formulate
wave equations for the dilatation and curl of the displacement
field. To see this:
(a) Take the divergence of Eqn 2.4.12 to obtain a wave equation
for the dilatation θ. At what velocity does θ propagate?
(b) Take the curl of Eqn 2.4.12 to obtain a wave equation for
∇
∇ ∇
∇ ∇ × u. At what velocity does ∇
∇ ∇
∇ ∇ × u propagate?
14. Derive the constitutive law (Eqn 2.3.70) for an isotropic and
linearly elastic material using the c ijkl in Eqn 2.3.69.
15. Derive the ratio of P- and S-wave velocities in a Poisson solid.
16. Use the gradient operator in spherical coordinates (Eqn A.7.14) to
find the displacement field from the spherical wave scalar potential f(t − r/v)/r. How would you approximate the displacements
near the source? How would you approximate the displacements
far from the source?
17. On a seismometer located at an earthquake hypocenter, the phases
reflected from the core, PcP and ScS, arrive at 8 minutes, 31 seconds, and 15 minutes, 36 seconds, respectively after the earthquake. If the earth’s radius is 6371 km, and the core’s radius is
3480 km:
(a) Find the average P- and S-wave velocities in the earth’s mantle.
(b) Use these average velocities to estimate how close the mantle
is to a Poisson solid.
18. Estimate the P- and S-wave velocities in the upper mantle by
assuming that it is a Poisson solid, and that the earthquake for
which seismograms are shown in Fig. 2.4-8 occurred at a depth of
280 km. Compare these velocities to the average mantle values.
Note that the seismograms do not start at the earthquake origin
time.
19. To get a feel for the distance and time scales in seismic wave propagation, consider waves propagating in a material with velocity
8 km/s.
(a) Find the wavelengths of waves with periods of 0.1 s, 1 s,
and 100 s.
(b) Find the periods and frequencies of waves with wavelengths
of 1 m, 1 km, and 100 km.
20. For waves propagating in an arbitrary direction given by the
wavenumber vector k,
(a) Show that the P-wave displacement due to the scalar potential
φ(x, t) = e i(ωt−k·x)
is parallel to the propagation direction.
1. What are the reflection and transmission coefficients for a junction
between two identical strings? Give a physical interpretation of the
result.
2. In Fig. 2.2-6, find the seismic velocities of the two different string
segments by measuring the distance versus time slope of the wave
pulses on the left and right sides of the figure. Are these velocities
the same as the velocities given in the figure caption?
3. For the stress tensor
σ =
− −
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
2
1 3
1 1 2
3 2 5
find the traction on
(a) the x–y plane,
(b) the y–z plane,
(c) the plane with normal (3, 2, −1).
4. To derive the reflection coefficient for the end of a string:
(a) Express the total displacement due to incident and reflected
harmonic waves of unknown amplitudes.
(b) Find the relation between these amplitudes at a fixed string
end, where the displacement is zero, and at a free end, where
the traction is zero.
5. For the stress tensor
σ =
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0 2 0
2 0 0
0 0 0
(a) Find the principal stresses and their associated directions.
(b) Find the surfaces on which the maximum tangential traction
occurs, and the value of this traction.
6. Estimate the pressure expected at a depth of 1000 km in the earth.
7. Given the stress tensor, whose elements are in kbar:
σ =
−
−
− −
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
150
2
1
2 155
3
1
3 145
(a) What physical situation do the large negative values on the
diagonal represent?
(b) What is the mean stress?
(c) What is the deviatoric stress tensor?
(d) At what depth in the earth might this state of stress be
found?
8. Give an example of a strain tensor for which there is
(a) an increase in volume,
(b) a decrease in volume,
(c) shear strain but no volume change.
Which of these strains could result from a P wave, and which could
result from an S wave?
9. Estimate by what fraction the volume of a block of a Poisson solid
with the rigidity of crustal rock will be compressed at a depth of
30 km relative to its volume at the earth’s surface.
10. Determine whether the Lamé constant λ can be negative and, if so,
under what conditions.