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Surface waves
Rayleigh and Love waves
Particle motionPhase and group velocity
Dispersion
Reading: Shearer, chapter 8
Telford et al., 4.2.4, 4.2.6
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Mechanism
Surface waves are always associatedwith a boundary.
The (e.g., horizontal) boundarydisrupts vertical wave propagationbutprovides for special wave modespropagating along it.
Because there are 2 or 4 boundaryconditions to satisfy (e.g.,displacement and stress continuity),surface waves are always build up of2 or 4 interacting wave modes:
Pand SVwave modes (Rayleigh orStoneleywaves);
Two SHmodes (Love waves).
Such waves are tied to the surface andexponentially decrease away from it.
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Surface wave
potentials
General wave equations for potentials:
Surface waves are combinations ofsolutions with complex(e.g., pureimaginary) wavenumbers alongz.
e.g., for Rayleigh wave:
Question: why are such solutions not
allowed without a boundary?
2
=1
VP2
2
t2 , P-wave
2V=1
VS2
2V
t2, SV-wave
2
H=
1
VS2
2H
t2 . SH-wave
=Aemz
ei kxt
,
V=Benz
ei kx t
.
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Depth dependence
To satisfy the wave equations for any kand , m and n must equal (show this):
note that therefore, for anysurfacewave:
and so
To further describe the solution, we needto:
1)consider andA as free variables;
2)determine B and k() from the boundaryconditions.
P-wave component
in Rayleigh wave
SV-wave component
m=k2
2
VP2,
n=
k
22
VS2.
k
VS
VP
,
VSurface=kVS.
Dispersion relation
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Rayleigh waves
(ground roll)
Rayleigh waves propagate along the freesurface
The displacements are as usual:
and traction:
For a free surface, the boundaryconditions read:
xz=
zz= 0,
Solution:
P-wave
SV-wave
uPx ,z =
x,0,
z,
uSx ,z =
z,0,
x,
FPx ,z =22x z
,0,222
z2,
FSx ,z =2
x22
z2,0,22
2xz
,
=emz e ikx t,
V=Benz ei kx t.
we can setA = 1 and
seekB and k()
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Rayleigh waves
(ground roll)
Result (for =0.25): relative P- and S-waveamplitudes:
...and dispersion relation:
Rayleigh wave velocity:
For varying :
P-wave
SV-wave
=e0.848kz ei kx t,
V=1.468ie0.393z
eikxt
.
k=VR .VR=0.919VS
(This means no dispersion!)
VR/VP VR/V
S
VS/VP
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Rayleigh waves
(ground roll)
Particle motion is elliptical and retrograde(counter-clockwise when the wave ismoving left to right):
How does it follow from the
equations for potentials
and displacements?
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Rayleigh waves
(ground roll)
Real Earth is never auniform half-space,
and thus in Rayleighwaves:
Particle motion pathsare tilted andcomplex;
Retrograde motion
may change intoprograde at somedepth;
Normal dispersion ispresent.
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Rayleigh-wave
dispersion Ideal Rayleigh wave (in a uniform half-
space) is non-dispersive
However, all realsurface waves exhibitdispersion
It is because the subsurface is alwayslayered
Phase velocity
Group velocity
Note the change
in wavelet shape
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Stoneley waves
These waves propagate along thecontact oftwo semi-infinite
media They are P/SVin nature, like
Rayleigh waves;
They always exist when one ofthe media is a fluid;
An important example is tube wavepropagating along a fluid-filledborehole.
If both media are solids, Stoneleywaves exist only when V
S1V
S2
and and lie within narrowlimits:
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Love waves
These are SH-type wavespropagating along the free surface
particle motion is transverse andparallel to the surface;
Because there is just one SHpotential, two modes are required tosatisfy the two boundary conditions(xz
=zz
=0 on the free surface)
Thus, Love waves exist when thesemi-infinite medium is overlain witha layer with different elasticproperties.
... this situation is quite common.
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Love-wave
dispersion Love waves are dispersive:
At high frequencies, its velocity
approaches the S-wave velocity in thesurface layer
At low frequencies, velocity is close tothat of the lower layer.
Displacement and stress in Love wave
Period [s]
Depth of
sampling
increases
with period.This is
common to
all surface
waves.
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Surface-wave
branches In layered structures, multiple surface-
wave branches, or modes exist for the
same frequency The branch with the lowest phase
velocity (longest wavelength) is calledthe fundamental mode
Theoreticalcurves Numericalsynthetics
Earth
model
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Surface-wave
branches (theory)
To see that
multiplebranches existand determinetheir parameters,the followingmatrix method
can be used:
1) Chose a set ofNbasis functions indepth, f
i(z), and
express the
potential (ordisplacement)
through them:
x , z , t=e
i tkx
i=1
N
ci f iz
Example of basis functions
for global Love waves
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Surface-wave
branches (theory, cont.)
2) For a given k, express the total kinetic andpotential energies; they will be quadraticmatrix products ofc
i
:
3) Note that in a wave, Ekin
= Eel, and therefore:
4) This means that ciis an eigenvector of
matrix A-1B, and 2 is the correspondingeigenvalue:
Ekin=2 ui u idz=2ci Aij c j
Eel=
2ii jjijij
dz=c iB ijc j
2 ci Aij c j=c iB ijc j
A1B2I c=0
There may be up toN
positive eigenvalues.
They give frequencies
of up toNmodes.
Note thatEkin is alwaysproportional to 2
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MASW (SASW)
method Multichannel (or Spectral) Analysis
of Surface Waves
Uses dispersion V(f) measurementsto invert for V
S(z) and (z)
Geotechnical applications
Kansas Geological Survey
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MASW method
Kansas Geological Survey
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Lamb (plate) wave(e.g., in road pavement)
Ka
nsasGeologicalSurvey
The case of surface waves in a thin elasticplate is called the Lamb's problem
ote the difference
dispersion curves
for symmetric
and asymmetric
deformations
of the plate
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