Download In this exercise we will consider plane, harmonic waves

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UNIVERSITY OF OSLO
Faculty of Mathematics and Natural Sciences
Exam in: GEO4620 - Seismic waves and seismology
Day of exam: December 12th 2007
Exam hours: 2.30pm – 5.30pm
This examination paper consists of 2 pages.
Appendices: None
Permitted materials: All printed and written, calculator.
Make sure that your copy of this examination paper
is complete before answering.
NB! Exercise 1 counts 40% and exercise 2 and 3 count 30% each.
Exercise 1.
In this exercise we will consider a SKS phase when it is incident on the Moho from below.
The P and S velocities are α1 = 6.8 and 1 = 3.9 km/s, respectively, in the crust and α2 =8.1 and
2 = 4.5 km/s, respectively, in the uppermost mantle.
In the computations below you can assume that the waves propagate in the (x,z)-plane, the xaxis being aligned with the radial direction in the horizontal plane and the z-axis being
positive downward, and that the Moho is located at z = 0.
a) Which waves can be generated in the crust and uppermost mantle when a SKS phase is
incident on the Moho from below?
Assume that the incident SKS phase and the waves that are generated all are plane, harmonic
waves, and write the potentials for these waves.
b) What are the boundary conditions at the Moho?
c) Express the boundary conditions in terms of the potentials.
d) If the angle of incidence for the SKS phase at the Moho is 12º, are inhomogeneous waves
generated above or below the Moho then? State the reason for your answer.
e) Does it happen that the SKS phase has displacement on the transverse component? State the
reason for your answer.
Exercise 2.
a) What does it mean that surface waves are dispersive?
b) Assume that a homogeneous layer of thickness 35 km and with S velocity 1 = 3.7 km/s
overlies a homogeneous halfspace with S velocity 2 = 4.4 km/s.
At which period will the 3rd Love wave overtone arise in this structure? And what is its phase
velocity at that period?
c) Show that the Love wave displacement in the halfspace decays with depth z as
exp{–2π(z/T)[(1/cx)2 – (1/β2)2]½}
where T is the period and cx is the phase velocity.
At 20 s period the phase velocity of the fundamental mode of the Love wave in the structure
given in b) is 3.97 km/s. What is the ratio of its displacement at 50 km depth to that at 80 km
depth then?
Exercise 3.
a) What does the quality factor Q represent physically?
b) For a Poisson medium, determine Qα and Qβ when Qµ = 312.
c) The P phase from an earthquake is recorded at an epicentral distance of 66º. The travel time
is 10 min 48.25 s. Assume that the Q-values found in b) apply to the whole propagation path,
and compute the percentage amplitude reduction due to anelasticity at 1.0 Hz and 0.02 Hz.
Give a physical explanation of the difference in amplitude reduction between the two
frequencies.
d) Why is it necessary to introduce dispersion in connection with anelasticity?
Assume that between the earth’s surface and the core-mantle boundary the average S velocity
is 6.176 km/s at 1 Hz and the average Q-values are those found in b).
For a ScS phase which is radiated from and recorded at the same position at the earth’s
surface, how much longer will the travel time be for the 0.02 Hz component than for the 1.0
Hz component? The depth to the core-mantle boundary is 2889 km.