・ 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

Seismic waves Faulting

Homogeneous Earth

Structure in the Earth Crust-Mantle Core-Mantle 440 km 660 km

Snell’s Law Fermat’s Principle 11 22 sin  1 / sin  2 = n 21 Air Water Rays

11 ２２  1 <  2   Ray Paths in a Layered Medium 11 ２２  1 >  2   Ｓ lower Faster Slower

11 22 33 Ray Paths in a Layered Medium 1/  1 1/  2 1/  3 Distance Time

Andrija Mohorovicic ( ) Found seismic discontinuity at 30 km depth in the Kupa Valley (Croatia). Mohorovicic discontinuity or ‘Moho’ Boundary between crust and mantle Moho

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 105 o to 143 o ・ 1939 Jeffreys fixed depth of core at 2898 km (using PcP) Forward Branch Backward Branch Forward Branch P PcP PKP Shadow Zone

PcP Core Reflections

Aspects of Waves not Explained by Ray Theory ・ Different types of waves (P, S) ・ Surface Waves ・ Static Displacements ・ Frequency content Seismic Waves

Wave Equation 1-D wave equation c = propagation speed Slinky: constant velocity wave propagation, no mass transfer, different from circulation eq.

1-D Wave Equation LW T = wave period  = angular frequency Solution

Wave Period and Wavelength wavelength 300 km Velocity 6 km/s Velocity = Wavelength / Period x t wavelength period Space Time period 50 s frequency = 1/period= 0.02 hz

Body waves （ P ・ S ） 0.01 to 50 sec50 m to 500 km Surface waves 10 to 350 sec30 to 1000 km Free Oscillations 350 to 3600 sec1000 to km Static Displacements - PeriodWavelength

3-D Wave Equation with Source sourcespatial 2nd derivative Solution 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 →t → r/  r/ 

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

Static displacements Co-seismic deformation of 2003 Tokachi-oki Earthquake (M8.0)

Generation of Tsunami from Near-field Term

Far-field Terms ・ Propagating Waves ・ No net displacement ・ P waves ・ S waves

Surface Waves S Shearer, Fig. 8.1 Period (sec) Love Rayleigh

January 26, 2001 Gujarat, India Earthquake (Mw7.7) Recorded in Japan at a distance of 57 o (6300 km) Love Waves vertical radial transverse Rayleigh Waves

Normal Modes (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)

Toroidal and Spheroidal Modes Toroidal Spheroidal Dahlen and Tromp Fig. 8.5, 8.17

Natural Vibrations of the Earth Shearer Ch.8.6 Lay and Wallace, Ch. 4.6

Houseman Free Oscillations l=1 m=1

Houseman Free Oscillations l=1 m=2

Houseman Free Oscillations l=1 m=3

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

Why are observed seismograms so messy ?

Structure: Free Surface Earth is a not homogenous whole-space Free surface causes many complications - surface waves - reflections (pP, sP, sS)