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Theoretical Seismology 2 Wave Propagation
Rays Snells Law Structure of the
Earth Seismic Waves Near-Field Terms
(Static Displacements) Far-Field Terms (P, S,
Surface waves) Normal modes Free
oscillations of the Earth
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Faulting
Seismic waves
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Homogeneous Earth
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Structure in the Earth
Crust-Mantle Core-Mantle 440 km 660 km
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Snells Law Fermats Principle
Rays
q1
Air
Water
q2
sin q1 / sin q2 n21
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Ray Paths in a Layered Medium
a1
Faster
q1
Slower
Slower
q2
Faster
a2
a1 lt a2
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Ray Paths in a Layered Medium
a1
a2
a3
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Moho
Andrija Mohorovicic (1857-1936) Found seismic
discontinuity at 30 km depth in the Kupa Valley
(Croatia). Mohorovicic discontinuity or
Moho Boundary between crust and mantle
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Backward Branch
Forward Branch
PKP
Forward Branch
PcP
Shadow Zone
P
Forward Branch
Backward Branch
Forward Branch
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PcP
Core Reflections
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P Mantle P
S Mantle S
K Outer core P
I Inner core P
c Reflection from the outer core
i Reflection from the inner core
diff Diffracted arrival
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Seismic Waves
Aspects of Waves not Explained by Ray Theory
Different types of waves (P, S) Surface
Waves Static Displacements Frequency
content

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Wave Equation
1-D wave equation c propagation speed
Slinky constant velocity wave
propagation, no mass transfer, different from
circulation eq.
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1-D Wave Equation
Solution
T wave period w angular frequency
LW 3.2.1
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Wave Period and Wavelength
Velocity 6 km/s
Space
x
wavelength 300 km
wavelength
Time
t
period 50 s frequency 1/period 0.02 hz
period
Velocity Wavelength / Period
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Period
Wavelength
Body waves (PS) 0.01 to 50 sec 50 m to 500 km
Surface waves 10 to 350 sec 30 to 1000 km
Free Oscillations 350 to 3600 sec 1000 to 10000 km
Static Displacements -
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3-D Wave Equation with Source
source
spatial 2nd derivative
Near-field Terms (Static Displacements)
Solution
Far-field Terms (P, S Waves)
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Near-field terms

• Static displacements
• Only significant close to the fault
• Source of tsunamis

t ?
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Static Displacements
Bei-Fung Bridge near Fung-Yan city, 1999 Chi-Chi,
Taiwan earthquake
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Static displacements
Co-seismic deformation of 2003 Tokachi-oki Earthqu
ake (M8.0)
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Generation of Tsunami from Near-field Term
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Far-field Terms

• Propagating Waves
• No net displacement
• P waves
• S waves

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surface waves
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Surface Waves
Group Velocity (km/sec)
Love
Rayleigh
Period (sec)
S
Shearer, Fig. 8.1
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January 26, 2001 Gujarat, India Earthquake (Mw7.7)
vertical
Rayleigh Waves
radial
transverse
Love Waves
Recorded in Japan at a distance of 57o (6300 km)
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Amplitude and Intensity
Seismic waves loose amplitude with distance
traveled - attenuation
So the amplitude of the waves depends on distance
from the earthquake. Therefore unlike magnitude
intensity is not a single number.
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Modified Mercalli Intensity
I Barely felt II Felt by only few people
III Felt noticeably, standing autos rock
slightly IV Felt by many, windows and walls creak
V Felt by nearly everyone, some dished and
windows broken VI Felt by all, damaged plaster
and chimneys VII Damage to poorly constructed
buildings VIII Collapse of poorly constructed
buildings, slight damage to well built
structures IX Considerable damage to well
constructed buildings, buildings
shifted off foundations X Damage to well built
wooden structures, some masonary
buildings destroyed, train rails bent, landslides
XI Few masonary structure remain standing,
bridges destroyed, ground
fissures XII Damage total
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Normal Modes
(Stein and Gellar 1978)
Free Oscillations of the Earth 1960 Chile
Earthquake
(Daishinji, Fukui Prefecture)
Useful for studies of Interior of the Earth
Largest earthquakes
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Toroidal and Spheroidal Modes
Toroidal
Spheroidal
Dahlen and Tromp Fig. 8.5, 8.17
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Natural Vibrations of the Earth
Shearer Ch.8.6 Lay and Wallace, Ch. 4.6
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Free Oscillations l1 m1
Houseman http//earth.leeds.ac.uk/greg/?Sphar/ind
ex.html
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Free Oscillations l1 m2
Houseman http//earth.leeds.ac.uk/greg/?Sphar/ind
ex.html
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Free Oscillations l1 m3
Houseman http//earth.leeds.ac.uk/greg/?Sphar/ind
ex.html
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Summary
Rays Earth structure causes complicated ray
paths through the Earth (P, PKP,
PcP) Wave theory explains P and S waves
Static displacements Surface waves
Normal Modes The Earth rings like a bell
at long periods