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Surface Waves and Free OscillationsSeismolo and the Earths Dee Interior

Surface waves in an elastic half spaces: Rayleigh waves

- Potentials- Free surface boundary conditions- Solutions propagating along the surface, decaying with depth- Lambs problem

Surface waves in media with depth-dependent properties: Love waves

- Constructive interference in a low-velocity layer- Dispersion curves- Phase and Group velocity

Free Oscillations

- Spherical Harmonics- Modes of the Earth- Rotational Splitting

Surface Waves and Free Oscillations


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The Wave Equation: Potentials

Do solutions to the wave equation exist for an elastic half space, which

travel along the interface? Let us start by looking at potentials:

i

i

zyx

i

u

u

),,(

scalar potential

vector potential

displacement

These potentials are solutions to the wave equation

iit

t

222

222

P-wave speed

Shear wave speed

What particular geometry do we want to consider?


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Rayleigh Waves

SV waves incident on a free surface: conversion and reflection

An evanescent P-wavepropagates along the freesurface decaying

exponentially with depth.The reflected post-crticially reflected SVwave is totally reflectedand phase-shifted. Thesetwo wave types can onlyexist together, they both

satisfy the free surfaceboundary condition:

-> Surface waves


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Surface waves: Disperion relation

With this trial solution we obtain for example coefficients

a for which travelling solutions exist

12

2

ca

Together we obtain

In order for a plane wave of that form to decay with depth a has to beimaginary, in other words

c

)]1/(exp[

)]1/(exp[

22

22

xzcctikB

xzcctikA

So that

c


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Surface waves: Boundary Conditions

Analogous to the problem of finding the reflection-

transmission coefficients we now have to satisfy theboundary conditions at the free surface (stress free)

0 zzxz

In isotropic media we have

)]1/(exp[

)]1/(exp[

22

22

xzcctikB

xzcctikA

and

yxzz

yzxx

u

u

zxxz

zzzzxxzz

u

uuu

2

2)(where


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Rayleigh waves: solutions

This leads to the following relationship forc, the phase velocity:

2/1222/122222 )/1()/1(4)/2( ccc

For simplicity we take a fixed relationship between P andshear-wave velocity

3

to obtain

02/32/3/56/8/ 224466 ccc

and the only root which fulfills the condition

is

c

9194.0c


8/46Surface Waves and Free OscillationsSeismolo and the Earths Dee Interior

Displacement

Putting this value back into our solutions

we finally obtain the displacement in thex-z plane for a plane harmonic surfacewave propagating along direction x

)(cos)4679.18475.0(

)(sin)5773.0(

3933.08475.0

3933.08475.0

xctkeeCu

xctkeeCu

kzkz

z

kzkz

x

This development was first made by Lord Rayleigh in 1885. Itdemonstrates that YES there are solutions to the wave

equation propagating along a free surface!

Some remarkable facts can be drawn from this particular form:



Lambs Problem

-the two components are out of phase by p

for small values of z a particle describes anellipse and the motion is retrograde

- at some depth z the motion is linear in z

- below that depth the motion is again ellipticalbut prograde

- the phase velocity is independent of k: thereis no dispersion for a homogeneous half space

- the problem of a vertical point force at thesurface of a half space is called Lambs

problem (after Horace Lamb, 1904).

- Right Figure: radial and vertical motion for asource at the surface

theoretical

experimental



Particle Motion (1)

How does the particle motion look like?

theoretical experimental



Data Example

theoretical experimental



Data Example

Question:

We derived that Rayleigh waves are non-dispersive!

But in the observed seismograms we clearly see ahighly dispersed surface wave train?

We also see dispersive wave motion on bothhorizontal components!

Do SH-type surface waves exist?Why are the observed waves dispersive?



Love Waves: Geometry

In an elastic half-space no SH type surface waves exist. Why?

Because there is total reflection and no interaction betweenan evanescent P wave and a phase shifted SV wave as in thecase of Rayleigh waves. What happens if we have layer over ahalf space (Love, 1911) ?



Love Waves: Trapping

Repeated reflection in a layer over a half space.

Interference between incident, reflected and transmitted SH waves.

When the layer velocity is smaller than the halfspace velocity, then thereis a critical angle beyon which SH reverberations will be totally trapped.



Love Waves: Trapping

The formal derivation is very similar to the derivation of the Rayleighwaves. The conditions to be fulfilled are:

1. Free surface condition2. Continuity of stress on the boundary3. Continuity of displacement on the boundary

Similary we obtain a condition for which solutions exist. This time weobtain a frequency-dependent solution a dispersion relation

22

11

2

2

2

222

1

/1/1

/1/1)/1/1tan(

c

ccH

... indicating that there are only solutions if ...

2 c



Love Waves: Solutions

Graphical solution ofthe previous equation.Intersection of dashedand solid lines yielddiscrete modes.

Is it possible, now, toexplain the observeddispersive behaviour?



Love Waves: modes

Some modes for Love waves



Waves around the globe



Data Example

Surface waves travelling around the globe



Liquid layer over a half space

Similar derivation for Rayleigh type motion leads to dispersive behavior



Amplitude Anomalies

What are the effects on the amplitude of surface waves?

Away from source or antipode geometrical spreading is approx. prop. to (sinD)1/2



Group-velocities

Interference of two waves at two positions (1)



Velocity

Interference of two waves at two positions (2)



Dispersion

The typical dispersive behavior of surface waves

solid group velocities; dashed phase velocities



Wave Packets

Seismograms of a Lovewave train filtered with

different central periods.Each narrowband tracehas the appearance of awave packet arriving atdifferent times.



Wave Packets

Group and phasevelocity measurementspeak-and-troughmethod

Phase velocities fromarray measurement


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Dispersion

Stronger gradients cause greater dispersion


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Dispersion

Fundamental Mode Rayleigh dispersion curve fora layer over a half space.


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Observed Group Velocities (T< 80s)


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Love wave dispersion


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d l


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Modal Summation

F ll D


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Free oscillations - Data

20-hour long recording of agravimeter recordind thestrong earthquake near

Mexico City in 1985 (tidesremoved). Spikes correspond

to Rayleigh waves.

Spectra of the seismogramgiven above. Spikes atdiscrete frequencies

correspond toeigenfrequencies of the Earth

Ei d f i


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Eigenmodes of a string

Geometry of a string underntension with fixed end points.Motions of the string excited

by any source comprise aweighted sum of the

eigenfunctions (which?).

Ei d f h


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Eigenmodes of a sphere

Eigenmodes of a homogeneoussphere. Note that there are modeswith only volumetric changes (like Pwaves, called spheroidal) and modeswith pure shear motion (like shear

waves, called toroidal).

- pure radial modes involve no nodalpatterns on the surface

- overtones have nodal surfaces atdepth

- toroidal modes involve purelyhorizontal twisting

- toroidal overtones have nodalsurfaces at constant radii.

Ei d f h


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Eigenmodes of a sphere

Compressional (solid)and shear (dashed)

energy density forfundamental

spheroidal modes andsome overtones,sensitive to core

structure.

B l d L d


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Bessel and Legendre

Solutions to the wave equation on spherical coordinates: Bessel functions (left)and Legendre polynomials (right).

S h i l H i


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Spherical Harmonics

Examples of spherical surface harmonics. There are zonal,sectoral and tesseral harmonics.


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Eff t f E th R t ti


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Effects of Earths Rotation

non-polar latitude

polar latitude

Eff ts f E ths R t ti : s is s


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Effects of Earths Rotation: seismograms

observed

synthetic no splitting

synthetic

L t l h t n it


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Lateral heterogeneity

Illustration of the distortion of standing-waves

due to heterogeneity. The spatial shift of thephase perturbs the observed multiplet amplitude

E l


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Examples

Sumatra M9 26 12 04


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Sumatra M9, 26-12-04

S f W S


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Surface Waves: Summary

Rayleigh waves are solutions to the elastic wave equation givena half space and a free surface. Their amplitude decaysexponentially with depth. The particle motion is elliptical andconsists of motion in the plane through source and receiver.

SH-type surface waves do not exist in a half space. Howeverin layered media, particularly if there is a low-velocitysurface layer, so-called Love waves exist which aredispersive, propagate along the surface. Their amplitude alsodecays exponentially with depth.

Free oscillations are standing waves which form after bigearthquakes inside the Earth. Spheroidal and toroidaleigenmodes correspond are analogous concepts to P and shearwaves.

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