Theoretical Seismology
Spring Semester, 2005
Due 16 Feb. 2005
Homework #1--Refraction/Reflection of Plane Waves
1. For body waves in a uniform medium, the displacement potential
, is associated with longitudinal waves,
, is associated with shear waves. Assume
a plane wave propagating in the x1-direction. For the expressions relating
displacement to potential:
a) For P-waves show that all components of
must be zero
(or at least contribute nothing to the displacement).
b) For the P and SV SV waves, what is the advantage of solving the wave equation in terms of the displacement potentials, rather than the displacement directly?
c) For the SH waves, what is the advantage of solving the wave equation in terms of the displacement, rather than the displacement potential?
2. For the case of a vertically incident SH-waves using the expressions for reflection and refraction across a solid/solid boundary:
a) Show that the conservation of energy holds across a boundary for the incident, reflected and transmitted SH waves at an interface.
b) Comment on the significance of a body-wave amplification for body waves incident into a low-velocity layer from a source "at infinity".
3. For a P wave incident on a horizontal solid-solid interface:
a) Write the potentials for the incident P wave and reflected P and SV wave.
b) Derive the four continuity equations at the interface in terms of potentials.
c) Comment on how the potentials are solved for displacement.
4. The 1-D Zoeppritz equation program 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.
a) Use it to estimate these quantities for the core-mantle boundary, for the lower
mantle with 
= 13.7
km/s, 
= 7.2 km/s,
= 5.5 g/cm3 and
the core with = 8.0 km/s,
= 0.0 km/s,
= 9.9 g/cm3.
Discuss your results and how they relate to our interpretation of whole earth
structure. Don't forget this is a spherical earth problem.
b) Use Zoeppritz equations to show the conservation of energy for SH waves as in Problem 2.
5. A P wave incident from the air (density = 1.3 x 10 g/cm3, velocity = 333 m/s) onto water (density = 1.0 g/cm3, velocity = 1500 m/s) demonstrates the effect of an inhomogeneous wave produced by incidence from a low velocity medium onto a high velocity medium.
a) Compute the critical angle of incidence, ic.
b) For an angle of incidence of 30° and a frequency of 500 Hz, compute the phase shift in the reflected wave and the penetration depth of the refracted wave into the water (the penetration depth refers to the depth where the amplitude has dropped to 1/e of its value at the surface).
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