Geophysics/Tectonics

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Geophysics/Tectonics

• GLY 325

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Elastic Waves, as waves in general, can be
described spatially...
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or temporally.
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Elastic Waves Lames Constant (l) --
interrelates all four elastic constants and is
very useful in mathematical computations, though
it doesnt have a good intuitive meaning.
Its important for you to know the terms and what
they represent (when appropriate) because we will
be using them in labs.
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The Wave Equation Well look at the scalar wave
equation to mathematically express how elastic
strain (dilatation, D) propagates through a
material r d2D (k 2m) 2D dt2
where D exx eyy ezz and 2 is the
Lapacian of D, or d2D/dx2 d2D/dy2 d2D/dz2
D
D
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Elastic Waves When solving the wave equation
(which describes how energy propagates through an
elastic material), there are two solutions that
solve the equation, Vp and Vs . These solutions
relate to our elastic constants by the following
equations
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Elastic Waves It turns out that Vp and Vs are
probably familiar to you from your introductory
earthquake knowledge, since they are the
velocities of P-waves and S-waves,
respectively. So, now you know why there are P-
and S-waves--because they are two solutions that
both solve the wave equation for elastic media.
S
P
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The Wave Equation The wave equation can be
rewritten as 1 d2D 2D a2
dt2 where a2 (k 2m)/r, or alternatively
as 1 d2Q 2Q b2 dt2 where b2
m/r And youll recognize the physical
realization of these equations as a P-wave and
b S-wave velocity.
D
D
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The Wave Equation Since the elastic constants
are always positive, a is always greater than b,
and b/a m/(l2m)1/2 (0.5-s)/(1-s)1/2 S
o, as Poissons ratio, s, decreases from 0.5 to
0, b/a increases from 0 to its maximum value
1/v2 thus, S-wave velocity must range from 0 to
70 of the P-wave velocity of any material.
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The Wave Equation These first types of
solutionsP-waves and S-wavesare called body
waves. Body waves propagate directly through
material (i.e. its body). I. Body Waves
a. P-Waves 1. Primary wave (fastest
arrive first) 2. Typically smallest in amplitude
3. Vibrates parallel to the direction of
wave propagation. b. S-Waves 1.
Secondary waves (moderate speed arrives
second) 2. Typically moderate amplitude 2.
Vibrates perpendicular to the direction of
wave propagation.
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The Wave Equation The other types of solutions
are called surface waves. Surface waves travel
only under specific conditions at an interface,
and their amplitude exponentially decreases away
from the interface. II. Surface waves
(slowest) 1. Arrives last 2. Typically
largest amplitude 2. Vibrates in vertical,
reverse elliptical motion (Rayleigh) or
shear elliptictal motion
(Love)
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The Wave Equation The three types of surface
waves are 1) Rayleigh Wavesform at a
free-surface boundary. Air closely approximates
a vacuum (when compared to a solid), and thus
satisfies the free-surface boundary condition.
Rayleigh waves are also called ground roll. 2)
Love Wavesform in a thin layer when the layer is
bound below by a seminfinite solid layer and
above by a free surface. 3) Stonely Wavesform
at the boundary between a solid layer and a
liquid layer or between two solid layers under
specific conditions.
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The Wave Equation For a typical homogeneous
earth material, in which Poissons ratio s 0.25
(also called a Poisson solid), the following
relationship should be remembered between P-wave,
S-wave, and Rayleigh wave velocities VP VS
VR 1 0.57 0.52 In other words, VS is
about 60 of VP, and VR is about 90 of VS. But
remember, this only is a guide...
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The Wave Equation Modeled The wave equation
explains how displacements elastically propagate
through material. In models, colors represent
the displacement of discrete elements (below
yellowpositive, purplenegative) away from their
equilibrium position.
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The Wave Equation Modeled As displacements
propagate away from the initial source of
displacement (i.e., the source), a spherical
wavefront is observed. Seismologists define
raypaths showing the direction of propagation
away from the source. Raypaths are always
perpendicular to the wavefront, analogous to
flowpaths in hydrology.
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The Wave Equation Modeled Boundary
Conditions Weve seen how body-wave
displacements propagate through a homogeneous
material, but what happens at boundaries? At
boundaries (defined as a place where material
elastic properties change), body waves refract
(following Snells Law) and reflect. Without
going into details, the potential ENERGY
expressed in the propagating displacements is
partitioned at every interface into REFRACTED (or
transmitted) and REFLECTED energy as stated by
the complex Zoeppritz Equations.
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The Wave Equation Modeled Boundary
ConditionsREFRACTION Snells Law states that
an incident raypath will refract at an interface
to a degree related to the difference in
velocities
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The Wave Equation Modeled Boundary
ConditionsREFRACTION Note that by definition,
if the propagation velocity increases across an
interface, the ray will refract toward the
interface. In the example below, the diagram is
drawn such that v2 gt v1.
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The Wave Equation Modeled Boundary
ConditionsREFLECTION At an interface, body
wave displacements also reflect. Simply, waves
reflect at an interface with an angle equal to
the incidence angle, regardless of the
propagation velocities of the layers
i2
i1
i1 i2
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The Wave Equation Modeled So, at any interface,
some energy is reflected (at the angle of
incidence) and some is refracted (according to
Snells Law). Lets look at a simple model and
just watch what happens to the P-wave energy...
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The Wave Equation Modeled FYI, if we used the
same model, but only looked at the surface waves,
not surprisingly we would just see them move out
from the source at a constant velocity.
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The Wave Equation Modeled
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The Wave Equation Modeled
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The Wave Equation Modeled
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The Wave Equation Modeled Each of
the things labeled is called a phase. Phases, in
layman's terms, represent a part of the original
source energy that has done something.
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The Wave Equation Modeled One thing we didnt
check out is what happens when the variables are
set up just right so that i2 90. That means
the energy will travel right along the interface.
It turns out that this phenomenon generates the
interesting head wave phase. An important term
to know is critical angle. The critical angle
(ic)is the incidence angle at which the energy
refracts directly along the interface.
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The Wave Equation Modeled Head waves are
generated by energy refracting along an
interface, and along the way leaking some
energy back toward the surface at the critical
angle.
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The Wave Equation Modeled As displacements
propagate away from the initial source of
displacement (i.e., the source), a spherical
wavefront is observed. Seismologists define
raypaths showing the direction of propagation
away from the source. Raypaths are always
perpendicular to the wavefront, analogous to
flowpaths in hydrology.