10 Surface Waves

7/30/2019 10 Surface Waves

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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

10 Surface Waves

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