・ Rays Snell ’ s Law Structure of the Earth ・ Seismic Waves
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Transcript of ・ Rays Snell ’ s Law Structure of the Earth ・ Seismic Waves
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・ Rays Snell’s 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
Theoretical Seismology 2: Wave Propagation
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Seismic waves
Faulting
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Homogeneous Earth
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Structure in the Earth
Crust-MantleCore-Mantle
440 km660 km
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Snell’s LawFermat’s Principle
1
2 sin 1 / sin 2 = n21
Air
Water
Rays
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1
2
1 < 2
Ray Paths in a Layered Medium
1
2
1 > 2
S lower
Faster
Faster
Slower
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1
2
3
Ray Paths in a Layered Medium
1/1
1/2
1/3
Distance
Time
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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
Moho
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Forward Branch
Backward Branch
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Forward Branch
Backward Branch
Forward Branch
Shadow Zone
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Forward Branch
Backward Branch
Forward Branch
Shadow Zone
PcP
・ 1912 Gutenberg observed shadow zone 105o to 143o
・ 1939 Jeffreys fixed depth of core at 2898 km (using PcP)
ForwardBranch
BackwardBranch
ForwardBranch
PPcP
PKP
Shadow Zone
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PcP
Core Reflections
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P Mantle PS Mantle SK Outer core
PI Inner core
Pc Reflection
from the outer core
i Reflection from the inner core
diff Diffracted arrival
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Aspects of Waves not Explained by Ray Theory
・ Different types of waves (P, S) ・ Surface Waves ・ Static Displacements ・ Frequency content
Seismic Waves
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Wave Equation
21
2
21
12 1
tu
cxu
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 21
2
21
12 1
tu
cxu
LW 3.2.1
2
T T = wave period = angular frequency
)]/(sin[),( cxtAtxu
Solution
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Wave Period and Wavelength
wavelength 300 km
Velocity 6 km/sx
t
wavelength
period
Space
Time
period 50 sfrequency = 1/period= 0.02 hz
Velocity = Wavelength / Period
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Body waves ( P ・ S ) 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 -
Period Wavelength
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)()()2(2
2
uuftu
3-D Wave Equation with Source source spatial 2nd derivative
Solution
dtM
rAtxu
r
r
N )(14
1),(/
/ 04 )(14
1)(14
1022022
rtMr
ArtMr
A ISIP
)(14
1)(14
10303
rtMr
ArtMr
A FSFP
Near-field Terms (Static Displacements)
Far-field Terms (P, S Waves)
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Near-field terms
・ Static displacements
・ Only significant close to the fault
・ Source of tsunamis
r/ r/
t →r/ r/
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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 deformationof 2003 Tokachi-okiEarthquake (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
)(14
1)(14
10303
rtMr
ArtMr
A FSFP
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surface waves
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Surface Waves
S
Shearer, Fig. 8.1
Period (sec)
Love
RayleighGr
oup
Velo
city
(km
/sec
)
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January 26, 2001 Gujarat, India Earthquake (Mw7.7)
Recorded in Japan at a distance of 57o (6300 km)
Love Waves
vertical
radial
transverse
Rayleigh Waves
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Amplitude and IntensitySeismic waves loose amplitude with distance traveled - attenuation A(t) = A0e -ω0t/2Q
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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I Barely felt II Felt by only few people III Felt noticeably, standing autos rock slightlyIV Felt by many, windows and walls creak V Felt by nearly everyone, some dished and windows brokenVI Felt by all, damaged plaster and chimneysVII Damage to poorly constructed buildingsVIII Collapse of poorly constructed buildings,
slight damage to well built structuresIX Considerable damage to well constructed buildings, buildings shifted off foundationsX Damage to well built wooden structures, some masonary buildings destroyed, train rails bent, landslides XI Few masonary structure remain standing, bridges destroyed, ground fissuresXII Damage total
Modified Mercalli Intensity
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Normal Modes Normal Modes
(Daishinji, Fukui Prefecture)(Daishinji, Fukui Prefecture)Free Oscillations of the Earth 1960 Chile Earthquake
Useful for studies of ・ Interior of the Earth ・ Largest earthquakes
(Stein and Gellar 1978)
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Toroidal and Spheroidal Modes
ToroidalSpheroidal
Dahlen and Tromp Fig. 8.5, 8.17
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Natural Vibrations of the Earth
Shearer Ch.8.6Shearer Ch.8.6Lay and Wallace, Ch. 4.6Lay and Wallace, Ch. 4.6
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Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html
Free Oscillations l=1 m=1
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Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html
Free Oscillations l=1 m=2
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Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html
Free Oscillations l=1 m=3
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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