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The Elastic Wave Equation Equationsequations Ofof Motionmotion TheThe ElasticElastic WaveWave EquationEquation • Elastic waves in infinite homogeneous isotropic media Numerical simulations for simple sources • Plane wave propagation in infinite media Frequency, wavenumber, wavelength • Conditions at material discontinuities Snell’s Law Reflection coefficients Free surface • Reflection Seismology: an example from the Gulf of Mexico Seismology and the Earth’s Deep Interior The elastic wave equation EquationsEquations ofof motionmotion 2 ρ∂t ui = fi + ∂ jσ ij What are the solutions to this equation? At first we look at infinite homogeneous isotropic media, then: σ ij = λθδ ij + 2µε ij σ ij = λ∂ kukδ ij + µ(∂iu j + ∂ jui ) ρ∂ 2u = f + ∂ λ∂ u δ + µ(∂ u + ∂ u ) t i i j ( k k ij i j j i ) ρ∂ 2u = f + λ∂ ∂ u + µ∂ ∂ u + µ∂ 2 u t i i i k k i j j j i Seismology and the Earth’s Deep Interior The elastic wave equation EquationsEquations ofof motionmotion –– hhomogeneousomogeneous mediamedia ρ∂ 2u = f + λ∂ ∂ u + µ∂ ∂ u + µ∂ 2u t i i i k k i j j j i We can now simplify this equation using the curl and div operators 2 2 2 2 ∇ •u = ∂iui ∇ = ∆ = ∂ x + ∂ y + ∂ z and ∆u= ∇∇ •u-∇×∇×u ρ∂ 2u= f+ (λ + 2µ)∇∇ •u- µ∇×∇×u t … this holds in any coordinate system … This equation can be further simplified, separating the wavefield into curl free and div free parts Seismology and the Earth’s Deep Interior The elastic wave equation EquationsEquations ofof motionmotion –– PP waveswaves ρ∂ 2u= (λ + 2µ)∇∇ •u- µ∇×∇×u t Let us apply the div operator to this equation, we obtain ρ∂ 2θ = (λ + 2µ)∆θ t where Acoustic wave equation θ = ∇u= ∂iui P wave velocity or 2 1 λ + 2µ ∂ θ = ∆θ α = t α 2 ρ Seismology and the Earth’s Deep Interior The elastic wave equation EquationsEquations ofof motionmotion –– sshearhear waveswaves ρ∂ 2u= (λ + 2µ)∇∇ •u- µ∇×∇×u t Let us apply the curl operator to this equation, we obtain ρ∂ 2∇×u = (λ + µ)∇×∇θ + µ∆(∇×u ) t i i we now make use of ∇×∇θ = 0 and define ∇×ui = ϕi to obtain Shear wave Wave equation for velocity shear waves 2 1 µ ∂ ϕ = ∆ϕ β = t β 2 ρ Seismology and the Earth’s Deep Interior The elastic wave equation ElastodynamicElastodynamic PotentialsPotentials Any vector may be separated into scalar and vector potentials u = ∇Φ + ∇× Ψ where P is the potential for Φ waves and Ψ the potential for shear waves ⇒ θ = ∆Φ ⇒ ϕ = ∇×u = ∇×∇× Ψ = −∆Ψ P-waves have no rotation Shear waves have no change in volume ∂ 2θ = α 2∆θ ∂ 2ϕ = β 2∆ϕ t t Seismology and the Earth’s Deep Interior The elastic wave equation SeismicSeismic VelocitiesVelocities Material and Source P-wave velocity (m/s) shear wave velocity (m/s) Water 1500 0 Loose sand 1800 500 Clay 1100-2500 Sandstone 1400-4300 Anhydrite, Gulf Coast 4100 Conglomerate 2400 Limestone 6030 3030 Granite 5640 2870 Granodiorite 4780 3100 Diorite 5780 3060 Basalt 6400 3200 Dunite 8000 4370 Gabbro 6450 3420 Seismology and the Earth’s Deep Interior The elastic wave equation SolutionsSolutions toto thethe wavewave equationequation -- ggeneraleneral Let us consider a region without sources ∂ 2η = c2∆η t Where n could be either dilatation or the vector potential and c is either P- or shear-wave velocity. The general solution to this equation is: η(xi ,t) = G(a j x j ± ct) Let us take a look at a 1-D example Seismology and the Earth’s Deep Interior The elastic wave equation SolutionsSolutions toto thethe wavewave equationequation -- hharmonicarmonic Let us consider a region without sources ∂ 2η = c2∆η t The most appropriate choice for G is of course the use of harmonic functions: ui (xi ,t) = Ai exp[ik(a j x j − ct)] Seismology and the Earth’s Deep Interior The elastic wave equation SolutionsSolutions toto thethe wavewave equationequation -- hharmonicarmonic … taking only the real part and considering only 1D we obtain u(x,t) = Acos[k(x − ct)] 2π 2π 2π k(x − ct) = kx − kct = x −ωt = x − t λ λ T c wave speed kwavenumber λ wavelength Tperiod ω frequency A amplitude Seismology and the Earth’s Deep Interior The elastic wave equation SphericalSpherical WavesWaves 2 2 Let us assume that η is a function ∂ η = c ∆η of the distance from the source t 2 1 r ∆η = ∂ 2η + ∂ η = ∂t η r r r c2 2 where we used the definition of the Laplace operator in spherical coordinates let us define η η = to obtain r 2 2 ∂ η = c ∆η with the known solution η = f (r −αt) t Seismology and the Earth’s Deep Interior The elastic wave equation GeometricalGeometrical spreadingspreading so a disturbance propagating away with spherical wavefronts decays like 1 1 η = f (r −αt) η ≈ r r r ... this is the geometrical spreading for spherical waves, the amplitude decays proportional to 1/r. If we had looked at cylindrical waves the result would have been that the waves decay as (e.g. surface waves) 1 η ≈ r Seismology and the Earth’s Deep Interior The elastic wave equation PlanePlane waveswaves ... what can we say about the direction of displacement, the polarization of seismic waves? u = ∇Φ + ∇× Ψ ⇒ u = P + S P = ∇Φ S = ∇× Ψ ... we now assume that the potentials have the well known form of plane harmonic waves Φ = Aexpi(k• x−ωt) Ψ = B expi(k• x−ωt) P = Akexpi(k• x−ωt) S = k× B expi(k• x−ωt) P waves are longitudinal as P is shear waves are transverse parallel to k because S is normal to the wave vector k Seismology and the Earth’s Deep Interior The elastic wave equation HeterogeneitiesHeterogeneities .. What happens if we have heterogeneities ? Depending on the kind of reflection part or all of the signal is reflected or transmitted. • What happens at a free surface? • Can a P wave be converted in an S wave or vice versa? • How big are the amplitudes of the reflected waves? Seismology and the Earth’s Deep Interior The elastic wave equation BoundaryBoundary ConditionsConditions ... what happens when the material parameters change? ρ1 v1 welded interface ρ v 2 2 At a material interface we require continuity of displacement and traction A special case is the free surface condition, where the surface tractions are zero. Seismology and the Earth’s Deep Interior The elastic wave equation ReflectionReflection andand TransmissionTransmission –– SSnell’snell’s LawLaw What happens at a (flat) material discontinuity? sin i1 v1 Medium 1: v1 i1 = sin i2 v2 i2 Medium 2: v2 But how much is reflected, how much transmitted? Seismology and the Earth’s Deep Interior The elastic wave equation ReflectionReflection andand TransmissionTransmission coefficientscoefficients Let’s take the most simple example: P-waves with normal incidence on a material interface R A R ρ α − ρ α = 2 2 1 1 Medium 1: r1,v1 A ρ2α2 + ρ1α1 Medium 2: r2,v2 T 2ρ α = 1 1 T A ρ2α2 + ρ1α1 At oblique angles conversions from S-P, P-S have to be considered. Seismology and the Earth’s Deep Interior The elastic wave equation ReflectionReflection andand TransmissionTransmission –– AAnsatznsatz How can we calculate the amount of energy that is transmitted or reflected at a material discontinuity? We know that in homogeneous media the displacement can be described by the corresponding potentials u = ∇Φ + ∇× Ψ in 2-D this yields ux = ∂ xΦ − ∂ y Ψz u y = ∂ z Ψx − ∂ xΨz uz = ∂ zΦ + ∂ xΨy an incoming P wave has the form ω Φ = A expi (a x −αt) 0 α j j Seismology and the Earth’s Deep Interior The elastic wave equation ReflectionReflection andand TransmissionTransmission –– AAnsatznsatz ... here ai are the components of the vector normal to the wavefront : ai=(cos e, 0, -sin e), where e is the angle between surface and ray direction, so that for the free surface Φ = A0 expik(x1 − x3 tan e − ct) + Aexpik(x1 + x3 tan e − ct) Ψ = B expik'(x1 + x3 tan f − c't) α β e f where c = c'= cose cos f Pr ω ω ω k = cose = k'= cos f α c β P SVr what we know is that σ xz = 0 σ zz = 0 Seismology and the Earth’s Deep Interior The elastic wave equation ReflectionReflection andand TransmissionTransmission –– CCoefficientsoefficients ... putting the equations for the potentials (displacements) into these equations leads to a relation between incident and reflected (transmitted) amplitudes A 4 tan e tan f − (1− tan 2 f )2 RPP = = 2 2 A0 4 tan e tan f + (1− tan f ) A 4 tan e − (1− tan 2 f ) RPS = = 2 2 A0 4 tan e tan f + (1− tan f ) These are the reflection coefficients for a plane P wave incident on a free surface, and reflected P and SV waves. Seismology and the Earth’s Deep Interior The elastic wave equation CaseCase 1:1: ReflectionsReflections atat aa freefree surfacesurface A P wave is incident at the free surface ... i j P P SV The reflected amplitudes can be described by the scattering matrix S ⎛ Pu Pd Su Pd ⎞ S = ⎜ ⎟ ⎝ Pu Sd Su Sd ⎠ Seismology and the Earth’s Deep Interior The elastic wave equation CaseCase 2:2: SHSH waveswaves For layered media SH waves are completely decoupled from P and SV waves SH There is no conversion only SH waves are reflected or transmitted ⎛ Su Sd Su Sd ⎞ S = ⎜ ⎟ ⎝ Su Sd Su Sd ⎠ Seismology and the Earth’s Deep Interior The elastic wave equation CaseCase 3:3: Solid-solidSolid-solid interfaceinterface SVr P P r SVt Pt To account for all possible reflections and transmissions we need 16 coefficients, described by a 4x4 scattering matrix.
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