Surface Waves and Free Oscillations Thethe Wavewave Equation:Equation: Potentialspotentials

SurfaceSurface WavesWaves andand FreeFree OscillationsOscillations

SurfaceSurface waveswaves inin anan elasticelastic halfhalf spaces:spaces: RayleighRayleigh waves waves

-Potentials-Potentials -- Free Free surfacesurface boundaryboundary conditionsconditions -- Solutions Solutions propagatingpropagating alongalong thethe surface,surface, decayingdecaying withwith depthdepth -- Lamb’s Lamb’s problemproblem

SurfaceSurface waveswaves inin mediamedia withwith depth-depth-dependentdependent properties:properties: LoveLove waveswaves

-- Constructive Constructive interferenceinterference inin aa low-velocitylow-velocity layerlayer -- Dispersion Dispersion curvescurves -Phase-Phase andand GroupGroup velocityvelocity

FreeFree OscillationsOscillations

-- Spherical Spherical HarmonicsHarmonics -- Modes Modes ofof thethe EarthEarth -- Rotational Rotational SplittingSplitting

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations TheThe WaveWave Equation:Equation: PotentialsPotentials

Do solutions to the wave equation exist for an elastic half space, which travel along the interface? Let us start by looking at potentials:

These potentials are solutions to the wave equation

u = ∇Φ ×∇+ Ψ 2 22 i t α Φ∇=Φ∂ 2 22 ∂∂∂=∇ zyx ),,( it β Ψ∇=Ψ∂ i

ui displacement α P-wave speed

Φ scalar potential β Shear wave speed

Ψi vector potential

What particular geometry do we want to consider?

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations RayleighRayleigh Waves Waves

SV waves incident on a free surface: conversion and reflection

An evanescent P-wave propagates along the free surface decaying exponentially with depth. The reflected post- crticially reflected SV wave is totally reflected and phase-shifted. These two wave types can only exist together, they both satisfy the free surface boundary condition:

-> Surface waves

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations SurfaceSurface waves:waves: GeometryGeometry

We are looking for plane waves traveling along one horizontal coordinate axis, so we can - for example - set

∂ y = 0(.)

And consider only wave motion in the x,z plane. Then

u Φ∂= − ∂ Ψyzxx Wavefront y

u Ψ∂+Φ∂= yxzz

As we only require Ψy we x set Ψy=Ψ from now on. Our trial solution is thus

Φ = ± − xazctikA )](exp[

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations SurfaceSurface waves:waves: DisperionDisperion relation relation

With this trial solution we obtain for example coefficients a for which travelling solutions exist c2 a −±= 1 α 2

In order for a plane wave of that form to decay with depth a has to be imaginary, in other words c < α Together we obtain

=Φ α 22 −−± xzcctikA )]1/(exp[ =Ψ β 22 −−± xzcctikB )]1/(exp[

So that c < β < α

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations SurfaceSurface waves:waves: BoundaryBoundary ConditionsConditions

Analogous to the problem of finding the reflection- transmission coefficients we now have to satisfy the boundary conditions at the free surface (stress free)

σ = σ zzxz = 0

In isotropic media we have

σ zz = λ(∂ + ∂ zzxx + 2) μ∂ uuu zz u = ∂ Φ − ∂ Ψyzxx where xz 2μσ∂= uzx u Ψ∂+Φ∂= yxzz and

=Φ α 22 −−± xzcctikA )]1/(exp[ =Ψ β 22 −−± xzcctikB )]1/(exp[

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations RayleighRayleigh waves: waves: solutionssolutions

This leads to the following relationship for c, the phase velocity:

c β 222 −=− c α − c β )/1()/1(4)/2( 2/1222/122

For simplicity we take a fixed relationship between P and shear-wave velocity = 3βα

…to obtain cc ββ4466 +− c β 22 =− 02/32/3/56/8/

… and the only root which fulfills the condition c < β is c = 9194.0 β

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations DisplacementDisplacement

Putting this value back into our solutions we finally obtain the displacement in the x-z plane for a plane harmonic surface wave propagating along direction x

− 8475.0 kz − 3933.0 kz x = eCu − 5773.0( e − xctk )(sin) − 8475.0 kz − 3933.0 kz z −= 8475.0( eCu + 4679.1 e − xctk )(cos)

This development was first made by Lord Rayleigh in 1885. It demonstrates that YES there are solutions to the wave equation propagating along a free surface!

Some remarkable facts can be drawn from this particular form:

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations Lamb’sLamb’s ProblemProblem

theoretical -the two components are out of phase by π − for small values of z a particle describes an ellipse and the motion is retrograde - at some depth z the motion is linear in z - below that depth the motion is again elliptical but prograde - the phase velocity is independent of k: there is no dispersion for a homogeneous half space - the problem of a vertical point force at the experimental surface of a half space is called Lamb‘s problem (after Horace Lamb, 1904). - Right Figure: radial and vertical motion for a source at the surface

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations ParticleParticle MotionMotion (1)(1)

How does the particle motion look like?

theoretical experimental

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations DataData ExampleExample

theoretical experimental

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations DataData ExampleExample

Question:

We derived that Rayleigh waves are non-dispersive!

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

We also see dispersive wave motion on both horizontal components!

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

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations LoveLove Waves:Waves: GeometryGeometry

In an elastic half-space no SH type surface waves exist. Why? Because there is total reflection and no interaction between an evanescent P wave and a phase shifted SV wave as in the case of Rayleigh waves. What happens if we have layer over a half space (Love, 1911) ?

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations LoveLove Waves:Waves: TrappingTrapping

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 there is a critical angle beyon which SH reverberations will be totally trapped.

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations LoveLove Waves:Waves: TrappingTrapping

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

1. Free surface condition 2. Continuity of stress on the boundary 3. Continuity of displacement on the boundary

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

μ c2 − /1/1 β 2 H βω2 c2 )/1/1tan( =− 2 2 1 2 2 βμ11 − /1/1 c

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

β < c < β2

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations LoveLove Waves:Waves: SolutionsSolutions

Graphical solution of the previous equation. Intersection of dashed and solid lines yield discrete modes.

Is it possible, now, to explain the observed dispersive behaviour?

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations LoveLove Waves:Waves: modesmodes

Some modes for Love waves

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations WavesWaves aroundaround thethe globeglobe

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations DataData ExampleExample

Surface waves travelling around the globe

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations LiquidLiquid layerlayer overover aa halfhalf spacespace

Similar derivation for Rayleigh type motion leads to dispersive behavior

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations AmplitudeAmplitude AnomaliesAnomalies

What are the effects on the amplitude of surface waves?

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

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations Group-velocitiesGroup-velocities

Interference of two waves at two positions (1)

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations VelocityVelocity

Interference of two waves at two positions (2)

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations DispersionDispersion

The typical dispersive behavior of surface waves solid – group velocities; dashed – phase velocities

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations WaveWave PacketsPackets

Seismograms of a Love wave train filtered with different central periods. Each narrowband trace has the appearance of a wave packet arriving at different times.

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations WaveWave PacketsPackets

Group and phase velocity measurements peak-and-trough method

Phase velocities from array measurement

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations DispersionDispersion

Stronger gradients cause greater dispersion

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations DispersionDispersion

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

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations ObservedObserved GroupGroup VelocitiesVelocities (T<(T< 80s)80s)

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations LoveLove wavewave dispersion dispersion

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations ModalModal SummationSummation

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations FreeFree oscillationsoscillations -- Data Data

20-hour long recording of a gravimeter recordind the strong earthquake near Mexico City in 1985 (tides removed). Spikes correspond to Rayleigh waves.

Spectra of the seismogram given above. Spikes at discrete frequencies correspond to eigenfrequencies of the Earth

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations EigenmodesEigenmodes of of aa stringstring

Geometry of a string undern tension with fixed end points. Motions of the string excited by any source comprise a weighted sum of the eigenfunctions (which?).

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations EigenmodesEigenmodes of of aa spheresphere

Eigenmodes of a homogeneous sphere. Note that there are modes with only volumetric changes (like P waves, called spheroidal) and modes with pure shear motion (like shear waves, called toroidal).

- pure radial modes involve no nodal patterns on the surface - overtones have nodal surfaces at depth - toroidal modes involve purely horizontal twisting - toroidal overtones have nodal surfaces at constant radii.

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations EigenmodesEigenmodes of of aa spheresphere

Compressional (solid) and shear (dashed) energy density for fundamental spheroidal modes and some overtones, sensitive to core structure.

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations BesselBessel andand LegendreLegendre

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

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations SphericalSpherical HarmonicsHarmonics

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

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations TheThe Earth’sEarth’s EigenfrequenciesEigenfrequencies

Spheroidal mode eigenfrequencies

Toroidal mode eigenfrequencies

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations EffectsEffects ofof Earth’sEarth’s RotationRotation

non-polar latitude

polar latitude

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations EffectsEffects ofof Earth’sEarth’s Rotation:Rotation: seismogramsseismograms

observed

synthetic

synthetic no splitting

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations LateralLateral heterogeneityheterogeneity

Illustration of the distortion of standing-waves due to heterogeneity. The spatial shift of the phase perturbs the observed multiplet amplitude

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations ExamplesExamples

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations SumatraSumatra M9,M9, 26-12-0426-12-04

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations SurfaceSurface Waves:Waves: SummarySummary

RayleighRayleigh waves wavesare are solutionssolutions toto thethe elaselastictic wavewave equationequation givengiven aa halfhalf spacespace andand aa freefree surface.surface. TheirTheir amplitudeamplitude decaysdecays exponentiallyexponentially withwith depth.depth. TheThe partparticleicle motionmotion isis ellipticalelliptical andand consistsconsists ofof motionmotion inin thethe planeplane throughthrough sourcesource andand receiver.receiver.

SH-typeSH-type surfacesurface waveswaves dodo notnot existexist inin aa halfhalf space.space. HoweverHowever inin layeredlayered media,media, particularlyparticularly ifif therethere isis aa low-velocitylow-velocity surfacesurface layer,layer, so-calledso-called LoveLove waveswaves existexist whichwhich areare dispersive,dispersive, propagatepropagate alongalong thethe surface.surface. TheirTheir amplitudeamplitude alsoalso decaysdecays exponentiallyexponentially withwith depth.depth.

FreeFree oscillationsoscillationsare are standingstanding waveswaves whwhichich formform afterafter bigbig earthquakesearthquakes insideinside thethe Earth.Earth. SpheroidalSpheroidal and and toroidaltoroidal eigenmodeseigenmodes correspond correspond areare analogousanalogous conceptsconcepts toto PP andand shearshear waves.waves.

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations