・ 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

II.1 Theoretical Seismology 2: Wave Propagation

Magnitude for Local Tsunami(Example)JMA Magnitude (Tsuboi, 1954) M=log 10A + 1.73 log10 Δ － 0.83 A : Half of maximum total amplitude [μm] Δ : Epicentral distance [km]

Seismic waves

Faulting

Travel Time and Distance

Homogeneous Earth

1

２

1 < 2

Ray Paths in a Layered Medium

1

２

1 > 2

Ｓ lower

Faster

Faster

Slower

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

1

2

3

Ray Paths in a Layered Medium

1/1

1/2

1/3

Distance

Time

Structure in the Earth 　

Crust-MantleCore-Mantle

440 km660 km

Forward Branch

Backward Branch

Forward Branch

Backward Branch

Forward Branch

Shadow Zone

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

PcP

Core Reflections

Why are observed seismograms so complicated ?

Structure: Free Surface　Earth is a not homogenous whole-space　

Free surface causes many complications　 - surface waves 　 - reflections (pP, sP, sS) depth phase 　　　　　

Surface Wave and Maximum Amplitude

Observed in Japan.Δ=57(deg)

Max Amp., 40 min after occurrence.

(Ms, 20 deg Δ 160 deg)≦ ≦

Seismogram of a distant earthquakeSeismogram of a distant earthquake

( LR: Rayleigh wave, LQ: Love wave )Fig.16

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

Aspects of Waves not Explained by Ray Theory

・ Different types of waves (P, S) ・ Surface Waves ・ Static Displacements ・ Frequency content

Seismic Waves

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 　　 (6 min to 1 hour)

1000 to 10000 km

Static Displacements -

Period Wavelength

Static Displacements

Bei-Fung Bridge near Fung-Yan city, 1999 Chi-Chi, Taiwan earthquake

Static displacements

Co-seismic deformationof 2003 Tokachi-okiEarthquake (M8.0)

Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html

Free Oscillations l=1 m=1

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

Thank you for your attention

Snell’s LawFermat’s Principle

1

2 sin 1 / sin 2 = n21

Air

Water

Rays

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.

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

Wave Period and Wavelength 　

wavelength 300 km

Velocity 6 km/s

Velocity = Wavelength / Period

x

t

wavelength　　

period

Space

Time

period 50 sfrequency = 1/period= 0.02 hz

)()()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)

Near-field terms

・ Static displacements

・ Only significant close to the fault 　　

・ Source of tsunamis　

r/ r/

t 　→r/ r/

Far-field Terms

・ Propagating Waves

・ No net displacement 　

・ P waves

・ S waves

　　　　　　　　　　　　　　　　　　　　　　

)(14

1)(14

10303

rtMr

ArtMr

A FSFP

Surface Waves

S

Shearer, Fig. 8.1

Period (sec)

Love

RayleighGr

oup

　Velo

city

(km

/sec

)

Generation of Tsunami from Near-field Term

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)

Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html

Free Oscillations l=1 m=2

Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html

Free Oscillations l=1 m=3

Toroidal and Spheroidal Modes

ToroidalSpheroidal

Dahlen and Tromp Fig. 8.5, 8.17

Natural Vibrations of the Earth

Shearer Ch.8.6Shearer Ch.8.6Lay and Wallace, Ch. 4.6Lay and Wallace, Ch. 4.6