Lecture 02 Session 01 Slides 1
-
Upload
elena-christodoulou -
Category
Documents
-
view
220 -
download
0
Transcript of Lecture 02 Session 01 Slides 1
-
7/31/2019 Lecture 02 Session 01 Slides 1
1/32
Lecture 02
Seismic Waves
-
7/31/2019 Lecture 02 Session 01 Slides 1
2/32
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.
-
7/31/2019 Lecture 02 Session 01 Slides 1
3/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
4/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
5/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
6/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
7/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
8/32
Wavelength and Wavenumber
Wave velocity
Spatial period (wavelength)
Wavenumber
Attenuation
2nn
V
V
n
nk V
1
2n
n
Q V
-
7/31/2019 Lecture 02 Session 01 Slides 1
9/32
Governing Equations for SeismicWaves
System of coordinatesx
y
z
, ,
x y z
u u uu
Stress tensor
, , 1,2,3ij i j Displacement vector
-
7/31/2019 Lecture 02 Session 01 Slides 1
10/32
Mechanical Properties
Bulk modulus
Youngs modulus
Shear modulus
Poisson ratio
2/3K
(3 2 ) /( )E
G
/ 2( )
-
7/31/2019 Lecture 02 Session 01 Slides 1
11/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
12/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
13/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
14/32
Compressional Wave
Also called primary wave, P-wave, longitudinal wave
4
(1 ) 3
(1 )(1 2 )p
E GE vVv v
-
7/31/2019 Lecture 02 Session 01 Slides 1
15/32
Shear Wave
Also called secondary wave, S-wave, transverse wave
2 (1 )s
E GV
v
-
7/31/2019 Lecture 02 Session 01 Slides 1
16/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
17/32
Wave Propagation Through Earth
-
7/31/2019 Lecture 02 Session 01 Slides 1
18/32
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.
-
7/31/2019 Lecture 02 Session 01 Slides 1
19/32
Love Wave
Velocity of Love wave depends on frequency, i.e. this wave is dispersive.
-
7/31/2019 Lecture 02 Session 01 Slides 1
20/32
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.
-
7/31/2019 Lecture 02 Session 01 Slides 1
21/32
Rayleigh Wave
Velocity of Rayleigh wave depends on frequency, i.e. this wave is dispersive.
-
7/31/2019 Lecture 02 Session 01 Slides 1
22/32
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.
-
7/31/2019 Lecture 02 Session 01 Slides 1
23/32
Seismic Measurements
~1/1000 30 HzSeismometer
Seismographer Geophone
http://upload.wikimedia.org/wikipedia/commons/f/f2/Seismometer_kum_hg.jpg -
7/31/2019 Lecture 02 Session 01 Slides 1
24/32
Example Seismogram
-
7/31/2019 Lecture 02 Session 01 Slides 1
25/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
26/32
Reflection and Transmission ofElastic Waves
x
z
incident wave
surface wave
reflected wave
transmitted wave
-
7/31/2019 Lecture 02 Session 01 Slides 1
27/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
28/32
Seismic Refraction
-
7/31/2019 Lecture 02 Session 01 Slides 1
29/32
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
-
7/31/2019 Lecture 02 Session 01 Slides 1
30/32
IRIS Earths Interior Structure Poster
Seismic waves through the Earth
-
7/31/2019 Lecture 02 Session 01 Slides 1
31/32
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.
-
7/31/2019 Lecture 02 Session 01 Slides 1
32/32
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.