lecture 02 session 01 slides 1

7/31/2019 Lecture 02 Session 01 Slides 1

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Lecture 02

Seismic Waves


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Objectives

(i) to learn basic definitions related toseismology

(ii) to learn about the nature and basictypes of seismic waves which canpropagate in elastic materials;

(iii) to learn about the attenuation,refraction and multi-path propagation ofseismic waves in the Earth.


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Basic Definitions

A seismic waveis a disturbance that carrymechanical energy and propagate in an elasticmedium

ground displacement (deformation) vector ground velocity vector

ground acceleration vector

stress tensor fluid pressure


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Basic Relations

2

2

i ii i i

u va u u

t t

In the case of harmonic ground displacement 0 sin( )iu u t

2

0 0a u acceleration

0 0v uvelocity circular frequency

time


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Harmonic Wave Representation

Any seismic wave of finite duration can be represented asa superposition of harmonics

1( ) cos( ) sin( )

N

n n n nnu t a t b t

where 0

0 0

2( )cos( )

T

nn na u t t dt

T

and0

0 0

2( )sin( )

T

n nb u t t dt

T


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Frequency and Period

Frequency of oscillations (rad/sec)

Frequency of oscillations (Hertz)

Fundamental frequency

Time period of n-th harmonic

n n

02 /T

2n

n

T

/ 2n nf


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Harmonic Representation(example)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time [sec]

Displacement[m]

Original stimulus

1st

harmonic

1st

and 3rd

harmonics

1st

, 3rd

and 5th

harmonics


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Wavelength and Wavenumber

Wave velocity

Spatial period (wavelength)

Wavenumber

Attenuation

2nn

V

V

n

nk V

1

2n

n

Q V


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Governing Equations for SeismicWaves

System of coordinatesx

y

z

, ,

x y z

u u uu

Stress tensor

, , 1,2,3ij i j Displacement vector


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Mechanical Properties

Bulk modulus

Youngs modulus

Shear modulus

Poisson ratio

2/3K

(3 2 ) /( )E

G

/ 2( )


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Governing Equations for SeismicWaves

2nd Newtons law

1,1,

0,1 2 1 2

i k ik lik ik

k i l

i k lu u v uE

i kv x x v x

2

2, 1,2,3i ik

i k

uk

x x

Hooks law


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Two Types of Body Waves

The total displacement vector can be decomposed into twocomponents

p su = u + u

displacement in the direction of stress

displacement perpendicular to the

direction of stress


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Two Types of Wave (cont)

The displacement components are governedby

2 23, ,

2 21

(1 )

(1 )(1 2 )

p i p i

k k

u uE v

t v v x

2 23

, ,2 2

12 (1 )s i s i

k k

u uE

t v x

compressional

shear


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Compressional Wave

Also called primary wave, P-wave, longitudinal wave

4

(1 ) 3

(1 )(1 2 )p

E GE vVv v


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Shear Wave

Also called secondary wave, S-wave, transverse wave

2 (1 )s

E GV

v


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Difference Between P- and S-waves

P-wave is faster than S-wave, by at least

S-wave cannot propagate through gases or

liquids

Unlike P-wave, S-wave can be polarized in

vertical or horizontal planes

Curl of compressional wave displacement: 0p u

Divergence of shear wave displacement: 0s u

2


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Wave Propagation Through Earth


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Surface Waves

Surface waves are caused by theinterference of P- and S-waves with a freeboundary and are called the Love (L-wave)

and Rayleigh waves.

These surface waves are almost entirelyresponsible for the damage and destruction

associated with earthquakes.


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Love Wave

Velocity of Love wave depends on frequency, i.e. this wave is dispersive.


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Love Wave (cont)

Love wavecan only exist in a finite elastic layerand is characterised by transverse motion in thevicinity of the free boundary.

The Love wave is a type of surface waveformed by the constructive interference ofmultiple reflections of SH waves between the topfree surface and the solid-solid or solid-liquidinterface.

Love waves are faster than Rayleigh waves andtherefore arrive after the S-wave on aseismogram.


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Rayleigh Wave

Velocity of Rayleigh wave depends on frequency, i.e. this wave is dispersive.


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Rayleigh Wave (cont)

Rayleigh wave, also known as "ground roll", ischaracterised by movements parallel to the direction oftravel.

This wave is the result of an incident P- and SV-wavesinteracting at the top free surface and travelling parallelto that surface.

Rayleigh waves travel along the surface and decayexponentially with depth passing through a node wherethere is no motion at all.

Rayleigh waves are most commonly found on thevertical component of seismograms.

Most of the shaking felt from an earthquake is due to theRayleigh wave, which can be much larger than the otherwaves.


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Seismic Measurements

~1/1000 30 HzSeismometer

Seismographer Geophone
http://upload.wikimedia.org/wikipedia/commons/f/f2/Seismometer_kum_hg.jpg


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Example Seismogram


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Attenuation and Spreading ofSeismic Waves

sin( )n ix

j n iq

i

eu k x

x

attenuation due to viscous losses (friction): n1

attenuation due to geometrical spreading: ,q

ix

Amplitude of seismic wave:

1 for body waves (spherical spreading)q

0.5 for surface waves (cylindrical spreading)q


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Reflection and Transmission ofElastic Waves

x

z

incident wave

surface wave

reflected wave

transmitted wave


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Trace Wave Velocities

Trace wave wavenumbers (whether a P-or S-wave) and velocities in the media Iand II must be equal, i.e.

/ sin / sinI I II IIV V

sin sinI I II IIk k

and


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Seismic Refraction


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Seismic Transmission

The transmission coefficient for a seismic wave propagating from medium 1to medium 2 can be estimated from

1

2

m

where

1

22

1

112

sincos

cos2

nm

mw

,2

1

c

cn


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IRIS Earths Interior Structure Poster

Seismic waves through the Earth


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Summary: Body Waves

Type (and names) Particle Motion Typical Velocity Other Characteristics

P, Compressional,

Primary,

Longitudinal

Alternating compressions

(pushes) and dilations

(pulls) which are

directed in the same

direction as the wave is

propagating (along the

raypath); and therefore,

perpendicular to the

wavefront

VP ~ 5 7 km/s in

typical Earths

crust;

>~ 8 km/s in Earths

mantle and

core; 1.5 km/s in

water; 0.3 km/s in

air

P motion travels fastest in materials, so the

P-wave is the first-arriving energy on

a seismogram. Generally smaller and

higher frequency than the S and

Surface-waves. P waves in a liquid or

gas are pressure waves, including

sound waves.

S, Shear,

Secondary,

Transverse

Alternating transverse

motions (perpendicular

to the direction of

propagation, and the

raypath); commonlypolarized such that

particle motion is in

vertical or horizontal

planes

VS ~ 3 4 km/s in

typical Earths

crust;

>~ 4.5 km/s in

Earthsmantle; ~ 2.5-

3.0 km/s in (solid)

inner core

S-waves do not travel through fluids, so do

not exist in Earths outer core

(inferred to be primarily liquid iron) or

in air or water or molten rock

(magma). S waves travel slower thanP waves in a solid and, therefore,

arrive after the P wave.


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Summary: Surface Waves

Type (and names) Particle Motion Typical Velocity Other Characteristics

L, Love, Surface

waves, Long

waves

Transverse horizontal

motion, perpendicular

to the direction of

propagation and

generally parallel to the

Earths surface

VL ~ 2.0 - 4.5 km/s in

the Earth

depending on

frequency of the

propagating wave

Love waves exist because of the Earths

surface. They are largest at the

surface and decrease in amplitude

with depth. Love waves are

dispersive, that is, the wave velocity is

dependent on frequency, with low

frequencies normally propagating at

higher velocity. Depth of penetration

of the Love waves is also dependent

on frequency, with lower frequencies

penetrating to greater depth.

R, Rayleigh,

Surface

waves, Longwaves,

Ground roll

Motion is both in the

direction of propagation

and perpendicular (in avertical plane),

and phased so that

the motion is generally

elliptical either

prograde or retrograde

VR ~ 2.0 - 4.5 km/s in

the Earth

depending onfrequency of the

propagating wave

Rayleigh waves are also dispersive and the

amplitudes generally decrease with

depth in the Earth. Appearance andparticle motion are similar to water

waves.

lecture 02 session 01 slides 1

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