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Seismic Surface Waves: Rayleigh and Love Waves

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Elastic Wave Propagation in SolidsSeismic P and S WavesSeismic Anisotropy and Shear Wave Splitting
seismology surface-waves dispersion wave-propagation

Core Idea

Surface waves are confined to the upper layers of the Earth and decay exponentially with depth. Rayleigh waves involve coupled P and S motion in a retrograde elliptical pattern; Love waves are horizontally polarized shear waves. Both travel slower than body waves but dominate earthquake damage and global seismograms because they carry large amplitude and propagate with minimal attenuation.

Explainer

From elastic wave propagation and seismic body waves, you know that P-waves compress and expand material along the direction of travel, while S-waves shear it perpendicular to the direction of travel. Both are body waves — they radiate outward through the interior of the Earth in all directions. Surface waves are fundamentally different: they are guided by the free surface of the Earth (or by internal boundaries), and their energy is concentrated in the shallow subsurface rather than spreading through the full volume. This confinement is why surface waves carry more energy at a given distance from the source and why they dominate seismograms of distant earthquakes.

Rayleigh waves exist wherever there is a free surface — they require no layering. The particle motion is a retrograde ellipse in the vertical plane containing the propagation direction: at the surface, particles move backward (opposite to the wave's travel direction) at the top of their elliptical path, much like a point on the surface of an ocean wave but in reverse. This coupled vertical and horizontal motion involves both P and SV (vertically polarized shear) wave components interacting at the free surface. Rayleigh wave velocity in a uniform half-space is about 0.92 times the shear wave velocity, making them slower than both P and S body waves. In the real Earth, where velocity increases with depth, Rayleigh waves exhibit dispersion: longer-period waves penetrate deeper, sampling faster material, and therefore travel faster than shorter-period waves. This dispersion is not a nuisance — it is an extraordinarily useful property, because measuring how Rayleigh wave velocity varies with period reveals how shear velocity varies with depth.

Love waves require a low-velocity layer overlying a higher-velocity substrate — a condition easily met by Earth's crust over the mantle. They are horizontally polarized shear waves (SH motion) that become trapped in the low-velocity layer through total internal reflection. The particle motion is horizontal and perpendicular to the propagation direction — purely side-to-side, with no vertical component. Like Rayleigh waves, Love waves are dispersive in a layered Earth: longer periods sample deeper, faster material. Love waves are typically faster than Rayleigh waves and often arrive first in the surface-wave train on a seismogram.

The dispersion of surface waves makes them one of the most powerful tools in global seismology for imaging Earth's interior. By measuring the group velocity (the speed of the wave packet's envelope) and phase velocity (the speed of individual wave crests) as functions of period, seismologists construct dispersion curves that are then inverted for shear-velocity structure as a function of depth. Short-period surface waves (5–20 seconds) constrain crustal structure; intermediate periods (20–100 seconds) resolve the lithosphere and asthenosphere; long periods (100–300 seconds) sense the upper mantle transition zone. Surface-wave tomography using earthquake data from global seismic networks has produced detailed three-dimensional maps of shear velocity in the upper mantle, revealing features like cratonic roots, subducting slabs, and mantle plumes. For engineers, surface waves matter for a different reason: their large amplitudes and low frequencies couple efficiently with buildings and infrastructure, making them the primary cause of earthquake damage at moderate to large distances from the epicenter.

Practice Questions 5 questions

Prerequisite Chain

Understanding ZeroThe Number ZeroCounting to FiveCounting to 10Counting to 20Counting a Set of Objects Up to 20Cardinality: The Last Number CountedMatching Numerals to QuantitiesSubitizing Small QuantitiesAddition Within 10Number Bonds to 10Addition Within 20Doubles and Near DoublesDoubles Facts Within 10Near Doubles Facts Within 20Mental Math Strategies for AdditionMental Math: Adding and Subtracting TensAddition Within 100Repeated Addition as MultiplicationMultiplication as Equal GroupsMultiplication: ArraysBasic Multiplication Facts (0s, 1s, 2s, 5s, 10s)Multiplication Facts Within 100Division as Equal SharingDivision as Grouping (Measurement Division)Division: Grouping (Repeated Subtraction) ModelDivision: Fair Sharing ModelDivision as Equal SharingDivision as GroupingBasic Division FactsDivision Facts Within 100Multiplication and Division Fact FamiliesRelationship Between Multiplication and DivisionDivision Facts as Inverse of MultiplicationRemainders and Quotients in DivisionDivision Word ProblemsMulti-Step Word ProblemsSolving Multi-Step Word ProblemsMultiplication Word ProblemsDivision Word ProblemsIntroduction to Long DivisionFactors and MultiplesPrime and Composite NumbersEquivalent FractionsRelating Fractions and DecimalsDecimal Place ValueIntegers and the Number LineComparing and Ordering IntegersAbsolute ValueAdding IntegersSubtracting IntegersMultiplying IntegersDividing IntegersUnit RatesProportionsPercent ConceptConverting Between Fractions, Decimals, and PercentsOperations with Rational NumbersTwo-Step EquationsSolving Multi-Step EquationsEquations with Variables on Both SidesAngle Pairs: Complementary, Supplementary, and VerticalParallel Lines and TransversalsCorresponding AnglesAlternate Interior AnglesTriangle Angle Sum TheoremExterior Angle TheoremTriangle Inequality TheoremSimilar Triangles: AA SimilaritySimilar Triangles: SSS and SAS SimilarityProportions in Similar TrianglesRight Triangle Trigonometry IntroductionSine, Cosine, and Tangent RatiosTrigonometric Ratios ReviewRadian MeasureConverting Between Degrees and RadiansThe Unit CircleGraphing Sine and CosineGraphing Tangent and Reciprocal Trigonometric FunctionsDerivatives of Trigonometric FunctionsAntiderivativesIndefinite IntegralsBasic Integration RulesRiemann SumsDefinite Integral DefinitionDouble Integrals: Definition and SetupIterated Integrals and Fubini's TheoremDouble Integrals over Rectangular RegionsDouble Integrals over General RegionsApplications of Double Integrals: Area, Mass, and MomentsTriple Integrals in Cartesian CoordinatesTriple Integrals in Cylindrical and Spherical CoordinatesChange of Variables and the Jacobian DeterminantApplications of Triple Integrals: Volume and MassVector Fields and Their RepresentationsLine Integrals of Vector FieldsWork and CirculationLine Integrals of Scalar and Vector FunctionsFundamental Theorem for Line IntegralsConservative Vector FieldsConservative Vector Fields and Potential FunctionsCurl and Divergence of Vector FieldsCurl and DivergenceDivergence TheoremElectric Flux and Divergence TheoremGauss's Law: Integral Form and MeaningSolving Problems with Gauss's LawConductors in Electrostatic EquilibriumCapacitance and CapacitorsDielectricsDielectric Constant and Relative PermittivityElectric Field Inside Dielectric MaterialsDielectric Materials and PolarizationDielectric Susceptibility and PermittivityEnergy Density in Electric FieldsElectric Current and Current DensityElectrical Resistance and ResistivityOhm's Law and Circuit ElementsElectromotive Force (EMF) and BatteriesKirchhoff's Circuit Laws: Voltage and CurrentDC Circuit Network Analysis MethodsTransient Response in RC CircuitsRC CircuitsLC and RLC CircuitsAC Circuits: FundamentalsImpedance and ReactanceAC Power and ResonanceElectromagnetic WavesPostulates of Special RelativityTime DilationLength ContractionLorentz TransformationRelativistic Velocity AdditionRelativistic Momentum and EnergyMass-Energy Equivalence and E=mc²Photons as Particles with Energy and MomentumPlanck-Einstein Relation: Energy and FrequencyPhotoelectric EffectThe Photon: Light as QuantaCompton ScatteringWave-Particle Dualityde Broglie WavelengthThe Schrödinger EquationState Vectors and WavefunctionsQuantum SuperpositionQuantum EntanglementBell Theorem and Bell InequalitiesPostulates of Quantum MechanicsObservables and Quantum OperatorsCommutators and Commutation RelationsQuantum Angular MomentumQuantum Mechanical Treatment of HydrogenSolving the Schrödinger Equation for Hydrogen AtomQuantum NumbersElectron ConfigurationPeriodic TrendsCovalent BondingElectronegativity and Bond PolarityIonic BondingLewis StructuresVSEPR Theory and Molecular GeometryMolecular Geometry and Electron Pair GeometryMolecular Polarity and Dipole MomentsIntermolecular ForcesStates of Matter and Phase Changes: Melting, Boiling, and SublimationGas Laws and the Ideal Gas EquationGas Stoichiometry and Volume-Volume CalculationsThermochemistry and EnthalpyHeat Capacity and CalorimetryEntropy and Molecular DisorderSpontaneity and ΔGEntropy and Gibbs Free EnergyChemical EquilibriumStatistical Mechanics: Ensembles and the Boltzmann DistributionPartition Function: Definition and PropertiesThe Canonical Partition Function and Thermodynamic DerivationFree Energy and Thermodynamic Relations from Partition FunctionsLegendre Transformations and Thermodynamic PotentialsChemical Potential and Partial Molar PropertiesPhase Equilibrium and Coexistence ConditionsClausius-Clapeyron EquationPhase Diagrams and Phase BoundariesIgneous RocksMetamorphic RocksThe Rock CyclePlate TectonicsTectonic Plate BoundariesGeologic Structures: Folds and FaultsEarthquakes and SeismologySeismic WavesElastic Wave Propagation in SolidsSeismic P and S WavesSeismic Surface Waves: Rayleigh and Love Waves

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