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12.510 Introduction to Seismology
Spring 2008
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Problem Set 2 (Due on Wed., April 9, 2008)
Please don’t just copy the results from books: show full derivations!
1. For (i) SH wave incidence on the solid-solid interface and (ii) P wave incidence
on the free surface:
(a) Give the potentials, boundary conditions, Zoeppritz’ equations and all the
elements of the scattering matrix. (20 points)
(b) In some cases an angle exists where the reflection coefficient is zero; the
angle for which that happens is called the intramission angle: compute –
for the wavespeeds given – the intramission angle. (10 points)
(c) Make plots of reflection and transmission coefficients as function of
incidence angle (from i = 0 o through i = ic to i = 90 o ). (10 points)
(d) Explain the plots. (10 points)
SH incidence ( ρ1 = 2.5 g ⋅ cm −3 ρ 2 = 3g ⋅ cm −3 β1 = 3km ⋅ s −1 β 2 = 4km ⋅ s −1 )
P incidence ( α = 5.5km ⋅ s −1 β = 3.2km ⋅ s −1 )
\
/
u1
u 1
i
ρ1 , β1, µ1
Free surface
i
ρ 2 , β 2, µ 2
j
z
SH incidence
\
u2
x
α ,β
x
j
i
\
φ1
/
φ1
\
z
P incidence
ψ1
2. Give the Zoeppritz’ equations of the P-SV system (P incidence) across a
solid-solid interface. (20 points)
/
ψ1
\
φ 1
/
φ1
ρ 1 , α 1 , β 1 , µ1
x
ρ2, α 2 , β 2 , µ 2
\
φ2
\
z
ψ2
P-SV system (P incidence)
3. Suppose we have an incidence plane wave S H as depicted in Figure below:
prove that the total response has the same amplitude as that of the incidence
wave. (30 points)
Free surface
ρ1 , β1, µ1
ρ 2 , β 2, µ 2
SH