Elasticity and Seismic Wave Propagation

What determines the speed, mode, direction and amplitude of seismic waves?

Fictional unit investigation. A fictional layered crustal section provides densities, elastic moduli and interface depths. Different wavefront sketches are supplied and students diagnose which inferences depend on homogeneity.

Before beginning, distinguish the scientific question, the measured quantity and the latent parameter that the student hopes to infer. Assign physical units, name a plausible alternate hypothesis, and state what data are fictional. For wave propagation, distinguish phase, group speed, mode conversion and energy transport. State the isotropic homogeneous approximations before using scalar wave speeds; compare the consequence of a layered model. The experiments are deliberately simplified so their calculations can be independently checked. They are not validated field instruments, real-time forecasts, or site-specific hazard assessments.

Learning goals. Explain the mechanism, translate the problem into measurable quantities, test an alternative interpretation and defend an uncertainty statement.

Physical foundation. Small recoverable deformation can be described using stress and strain tensors and elastic parameters. Bulk modulus concerns compression; shear modulus concerns shape change; the moduli cannot be substituted casually. In the broader unit, Seismic waves are vector disturbances traveling through heterogeneous elastic media. Coordinate systems, phase conventions and the validity of linear elasticity matter when comparing a simple model with measured records. A wave-speed estimate is conditional on the medium and on which phase was actually picked.

Observational method. Translate simple uniaxial tests into stress–strain points and check whether a linear elastic approximation fits at small strain. Specify the coordinate frame, observation period, relevant units, acquisition and calibration limitations before interpreting the resulting figure.

Common interpretive trap. Elastic deformation of intact rock in a laboratory does not reproduce fracture, anisotropy or nonlinear soil response during intense shaking. A neat numeric solution does not remove this interpretive ambiguity. Ask which further independent measurement could reveal whether the preferred explanation is incorrect.

Applied investigation. Estimate a toy shear strain under a prescribed stress and explain why a failure criterion is additional information. Provide a labeled sketch, a short reproducible table and a paragraph separating observed evidence from inferred process.

Interpretation and quantitative example

Worked case 2.1: Shear wave speed in an isotropic solid. Synthetic input: 21.75 GPa. Apply the explicitly declared relationship Math.sqrt(x*1e9/2700)/1000 to obtain 2.83823 km/s. Restrictions: Assumes density 2700 kg/m³ and homogeneous isotropic linear elasticity. For this worked exercise, treat the input as a declared hypothetical value rather than a measurement. Copy the formula into a fresh calculation and check the unit conversion independently. Vary the input by ten percent and decide whether the output changes proportionally or nonlinearly. Explain which real-world conditions violate the assumptions listed. If the numeric result seems implausibly precise, round it appropriately and describe a physically meaningful range instead of inventing calibration. The purpose is transparent reasoning, not an official estimate. Test the wave-mode assignment with boundary conditions. A fast first arrival can travel far within a faster layer, and a recorded horizontal component can mix polarizations. Examine whether a linear-elastic, constant-speed assumption applies over the selected frequency band. State what a path-integrated travel time can establish about velocity, and what it cannot reveal without more rays.

Evidence design and falsification

Choose one prediction that would differ under a competing mechanism. A useful prediction has a measurable sign, timing, spectral or spatial signature rather than an undefined expectation that a map should look different. Explain how the predicted quantity would be calculated before looking at the fictional test record. If both explanations remain consistent with the evidence, report the unresolved alternatives and the next practical observation needed. Here, the principal source of ambiguity is stated explicitly: Elastic deformation of intact rock in a laboratory does not reproduce fracture, anisotropy or nonlinear soil response during intense shaking. Keep a processing log with coordinate reference, UTC times, units, processing order and exclusion rules. Compare pre- and post-processing records and explain discrepancies. In the conclusion, identify which claim survives the most consequential alternative parameterization. Reproducibility means another researcher could regenerate the result and find the same limitations, not simply the same decimal answer. End the notebook by explaining how the proposed investigation (Estimate a toy shear strain under a prescribed stress and explain why a failure criterion is additional information.) addresses at least one weakness of the primary model.

Knowledge check. (1) Which claim is directly observed and which is inferred? (2) How would an alternate model change the expected observation? (3) Which number must be accompanied by uncertainty? (4) Why is the worked result unsuitable as a real-time safety conclusion?

Learning goals. Explain the mechanism, translate the problem into measurable quantities, test an alternative interpretation and defend an uncertainty statement.

Physical foundation. An isotropic elastic solid supports P waves involving volumetric and shear deformation, and S waves involving shear distortion. Their speeds depend on density and moduli, and fluid media lack propagating bulk shear waves. In the broader unit, Seismic waves are vector disturbances traveling through heterogeneous elastic media. Coordinate systems, phase conventions and the validity of linear elasticity matter when comparing a simple model with measured records. A wave-speed estimate is conditional on the medium and on which phase was actually picked.

Observational method. Use synthetic density, bulk modulus and rigidity values to compute the two velocities, and check if the resulting ratio is physically consistent. Specify the coordinate frame, observation period, relevant units, acquisition and calibration limitations before interpreting the resulting figure.

Common interpretive trap. An arrival labelled P or S can reflect conversions and path complexity; raw travel time is not a direct rock property. A neat numeric solution does not remove this interpretive ambiguity. Ask which further independent measurement could reveal whether the preferred explanation is incorrect.

Applied investigation. Predict P and S arrival separation over two path lengths in an ideal homogeneous medium. Provide a labeled sketch, a short reproducible table and a paragraph separating observed evidence from inferred process.

Interpretation and quantitative example

Worked case 2.2: Wave wavelength from frequency. Synthetic input: 4.448 Hz. Apply the explicitly declared relationship 3.5/x to obtain 0.786871 km. Restrictions: Assumes nondispersive phase speed 3.5 km/s. For this worked exercise, treat the input as a declared hypothetical value rather than a measurement. Copy the formula into a fresh calculation and check the unit conversion independently. Vary the input by ten percent and decide whether the output changes proportionally or nonlinearly. Explain which real-world conditions violate the assumptions listed. If the numeric result seems implausibly precise, round it appropriately and describe a physically meaningful range instead of inventing calibration. The purpose is transparent reasoning, not an official estimate. Test the wave-mode assignment with boundary conditions. A fast first arrival can travel far within a faster layer, and a recorded horizontal component can mix polarizations. Examine whether a linear-elastic, constant-speed assumption applies over the selected frequency band. State what a path-integrated travel time can establish about velocity, and what it cannot reveal without more rays.

Evidence design and falsification

Reconstruct the inference backwards. Start with the headline result and locate every conversion, classification and assumption required to obtain it. An estimate based on a chosen velocity, phase pick or site parameter should be labelled conditional on those inputs. Review the same steps after making one intentional mistake in the input metadata; document which output would become misleading and why. Here, the principal source of ambiguity is stated explicitly: An arrival labelled P or S can reflect conversions and path complexity; raw travel time is not a direct rock property. Method A: compute a baseline result using the stated relation, preserving all units. Method B: perturb one relevant input and record the change. Method C: replace the simplest assumption with an alternative that would be plausible at the study scale; if you cannot recalculate it with the available information, state what is missing. Method D: compare with an independent kind of observation, if available. This order avoids hiding assumptions under additional significant figures. End the notebook by explaining how the proposed investigation (Predict P and S arrival separation over two path lengths in an ideal homogeneous medium.) addresses at least one weakness of the primary model.

Knowledge check. (1) Which claim is directly observed and which is inferred? (2) How would an alternate model change the expected observation? (3) Which number must be accompanied by uncertainty? (4) Why is the worked result unsuitable as a real-time safety conclusion?

Learning goals. Explain the mechanism, translate the problem into measurable quantities, test an alternative interpretation and defend an uncertainty statement.

Physical foundation. At material boundaries, wave energy partitions among reflected, transmitted and converted modes. Ray directions approximately follow slowness relations at wavelengths small compared with heterogeneity scales. In the broader unit, Seismic waves are vector disturbances traveling through heterogeneous elastic media. Coordinate systems, phase conventions and the validity of linear elasticity matter when comparing a simple model with measured records. A wave-speed estimate is conditional on the medium and on which phase was actually picked.

Observational method. Trace a set of incident and refracted rays for a layered velocity section and document the convention used for angles. Specify the coordinate frame, observation period, relevant units, acquisition and calibration limitations before interpreting the resulting figure.

Common interpretive trap. A single sharp arrival does not imply one interface: multiples, converted phases or processing artifacts can appear similar. A neat numeric solution does not remove this interpretive ambiguity. Ask which further independent measurement could reveal whether the preferred explanation is incorrect.

Applied investigation. Construct alternative layered profiles that yield similar first-arrival times. Provide a labeled sketch, a short reproducible table and a paragraph separating observed evidence from inferred process.

Interpretation and quantitative example

Worked case 2.3: Shear wave speed in an isotropic solid. Synthetic input: 34.25 GPa. Apply the explicitly declared relationship Math.sqrt(x*1e9/2700)/1000 to obtain 3.56163 km/s. Restrictions: Assumes density 2700 kg/m³ and homogeneous isotropic linear elasticity. For this worked exercise, treat the input as a declared hypothetical value rather than a measurement. Copy the formula into a fresh calculation and check the unit conversion independently. Vary the input by ten percent and decide whether the output changes proportionally or nonlinearly. Explain which real-world conditions violate the assumptions listed. If the numeric result seems implausibly precise, round it appropriately and describe a physically meaningful range instead of inventing calibration. The purpose is transparent reasoning, not an official estimate. Test the wave-mode assignment with boundary conditions. A fast first arrival can travel far within a faster layer, and a recorded horizontal component can mix polarizations. Examine whether a linear-elastic, constant-speed assumption applies over the selected frequency band. State what a path-integrated travel time can establish about velocity, and what it cannot reveal without more rays.

Evidence design and falsification

A strong evidence chain identifies what the observer physically controls. Specify waveform or map provenance, the acquisition time standard, the instrument or field method, the adopted calibration, and an independent comparison. Is the observed pattern repeatable across genuinely separate records or did it arise after multiple views of the same data? Where an observation is transformed into a calculated quantity, retain its original units and values. Write an audit note on which data would be decisive and which are merely compatible. Here, the principal source of ambiguity is stated explicitly: A single sharp arrival does not imply one interface: multiples, converted phases or processing artifacts can appear similar. Create a four-row evidence matrix: observed quantity, inferred physical quantity, principal confounder, and proposed falsifying observation. Mark an entry as measured only when acquisition metadata is known. Repeat the matrix for a second hypothesis instead of dismissing it without calculation. This discipline prevents a tidy narrative from substituting for testable physical interpretation. End the notebook by explaining how the proposed investigation (Construct alternative layered profiles that yield similar first-arrival times.) addresses at least one weakness of the primary model.

Knowledge check. (1) Which claim is directly observed and which is inferred? (2) How would an alternate model change the expected observation? (3) Which number must be accompanied by uncertainty? (4) Why is the worked result unsuitable as a real-time safety conclusion?

Learning goals. Explain the mechanism, translate the problem into measurable quantities, test an alternative interpretation and defend an uncertainty statement.

Physical foundation. Rayleigh and Love waves propagate along a free surface and commonly sample depths dependent on their wavelengths. Dispersion reflects stratification, not a single universal surface-wave speed. In the broader unit, Seismic waves are vector disturbances traveling through heterogeneous elastic media. Coordinate systems, phase conventions and the validity of linear elasticity matter when comparing a simple model with measured records. A wave-speed estimate is conditional on the medium and on which phase was actually picked.

Observational method. Make a synthetic phase-velocity-versus-period plot and compare its inferred depth sensitivity with the survey dimensions. Specify the coordinate frame, observation period, relevant units, acquisition and calibration limitations before interpreting the resulting figure.

Common interpretive trap. Surface-wave amplitude or ellipticity depends on path, source and local structure; polarization alone is not an absolute depth profile. A neat numeric solution does not remove this interpretive ambiguity. Ask which further independent measurement could reveal whether the preferred explanation is incorrect.

Applied investigation. Estimate wavelength at two frequencies and discuss how that changes the range of sampled structures. Provide a labeled sketch, a short reproducible table and a paragraph separating observed evidence from inferred process.

Interpretation and quantitative example

Worked case 2.4: Wave wavelength from frequency. Synthetic input: 7.398 Hz. Apply the explicitly declared relationship 3.5/x to obtain 0.473101 km. Restrictions: Assumes nondispersive phase speed 3.5 km/s. For this worked exercise, treat the input as a declared hypothetical value rather than a measurement. Copy the formula into a fresh calculation and check the unit conversion independently. Vary the input by ten percent and decide whether the output changes proportionally or nonlinearly. Explain which real-world conditions violate the assumptions listed. If the numeric result seems implausibly precise, round it appropriately and describe a physically meaningful range instead of inventing calibration. The purpose is transparent reasoning, not an official estimate. Test the wave-mode assignment with boundary conditions. A fast first arrival can travel far within a faster layer, and a recorded horizontal component can mix polarizations. Examine whether a linear-elastic, constant-speed assumption applies over the selected frequency band. State what a path-integrated travel time can establish about velocity, and what it cannot reveal without more rays.

Evidence design and falsification

When comparing scenarios, preserve the same measurement geometry if possible. A change in a source parameter may look like a path effect when stations or filters differ. Design a sensitivity table with columns for adopted value, plausible range, reason for uncertainty and effect on inference. Distinguish an instrumental limitation from a physical ambiguity: improving sampling can solve the former while the latter may need an entirely different experiment. Here, the principal source of ambiguity is stated explicitly: Surface-wave amplitude or ellipticity depends on path, source and local structure; polarization alone is not an absolute depth profile. Build the result graph only after labelling its axes and quantities. Include a plausible range of inputs rather than a single favorable selection. Annotate any discontinuity, threshold or saturated measurement. Next, state whether the curve demonstrates a general law, a restricted approximation or merely the numerical behavior of an arbitrary toy formula. The caption should not promise resolution that the data cannot support. End the notebook by explaining how the proposed investigation (Estimate wavelength at two frequencies and discuss how that changes the range of sampled structures.) addresses at least one weakness of the primary model.

Knowledge check. (1) Which claim is directly observed and which is inferred? (2) How would an alternate model change the expected observation? (3) Which number must be accompanied by uncertainty? (4) Why is the worked result unsuitable as a real-time safety conclusion?

Learning goals. Explain the mechanism, translate the problem into measurable quantities, test an alternative interpretation and defend an uncertainty statement.

Physical foundation. Amplitude generally diminishes through geometric spreading and anelastic loss; scattering redistributes energy rather than necessarily removing it from the seismogram. Quality factor Q parameterizes attenuation in an approximation. In the broader unit, Seismic waves are vector disturbances traveling through heterogeneous elastic media. Coordinate systems, phase conventions and the validity of linear elasticity matter when comparing a simple model with measured records. A wave-speed estimate is conditional on the medium and on which phase was actually picked.

Observational method. Plot expected amplitude versus distance for a fixed frequency while varying Q in a labeled hypothetical velocity model. Specify the coordinate frame, observation period, relevant units, acquisition and calibration limitations before interpreting the resulting figure.

Common interpretive trap. Observed damping cannot be assigned entirely to intrinsic attenuation without examining radiation, focusing and station response. A neat numeric solution does not remove this interpretive ambiguity. Ask which further independent measurement could reveal whether the preferred explanation is incorrect.

Applied investigation. Fit two competing decay curves and explain why either may fail in a laterally complex basin. Provide a labeled sketch, a short reproducible table and a paragraph separating observed evidence from inferred process.

Interpretation and quantitative example

Worked case 2.5: Shear wave speed in an isotropic solid. Synthetic input: 46.75 GPa. Apply the explicitly declared relationship Math.sqrt(x*1e9/2700)/1000 to obtain 4.16111 km/s. Restrictions: Assumes density 2700 kg/m³ and homogeneous isotropic linear elasticity. For this worked exercise, treat the input as a declared hypothetical value rather than a measurement. Copy the formula into a fresh calculation and check the unit conversion independently. Vary the input by ten percent and decide whether the output changes proportionally or nonlinearly. Explain which real-world conditions violate the assumptions listed. If the numeric result seems implausibly precise, round it appropriately and describe a physically meaningful range instead of inventing calibration. The purpose is transparent reasoning, not an official estimate. Test the wave-mode assignment with boundary conditions. A fast first arrival can travel far within a faster layer, and a recorded horizontal component can mix polarizations. Examine whether a linear-elastic, constant-speed assumption applies over the selected frequency band. State what a path-integrated travel time can establish about velocity, and what it cannot reveal without more rays.

Evidence design and falsification

Choose one prediction that would differ under a competing mechanism. A useful prediction has a measurable sign, timing, spectral or spatial signature rather than an undefined expectation that a map should look different. Explain how the predicted quantity would be calculated before looking at the fictional test record. If both explanations remain consistent with the evidence, report the unresolved alternatives and the next practical observation needed. Here, the principal source of ambiguity is stated explicitly: Observed damping cannot be assigned entirely to intrinsic attenuation without examining radiation, focusing and station response. Keep a processing log with coordinate reference, UTC times, units, processing order and exclusion rules. Compare pre- and post-processing records and explain discrepancies. In the conclusion, identify which claim survives the most consequential alternative parameterization. Reproducibility means another researcher could regenerate the result and find the same limitations, not simply the same decimal answer. End the notebook by explaining how the proposed investigation (Fit two competing decay curves and explain why either may fail in a laterally complex basin.) addresses at least one weakness of the primary model.

Knowledge check. (1) Which claim is directly observed and which is inferred? (2) How would an alternate model change the expected observation? (3) Which number must be accompanied by uncertainty? (4) Why is the worked result unsuitable as a real-time safety conclusion?

Research scenario: A fictional layered crustal section provides densities, elastic moduli and interface depths. Different wavefront sketches are supplied and students diagnose which inferences depend on homogeneity. The goal is to produce a reasoned comparison of at least two defensible models, not to reverse-engineer a predetermined answer. Start by recording each input variable with units, a confidence grade and a plausible range. Name the forward calculation associated with each inference and outline how the inputs enter the formula. Preserve the fabricated dataset in its supplied CSV form and cite its generation method; these records do not correspond to real earthquakes or real sensor networks.

Procedure. First, draw a map or schematic that makes geometric assumptions visible. Second, check station or sample identifiers and the relationship of the observation period to the question. Third, compute baseline statistics and at least one worked relation by hand before using the browser model. Fourth, investigate a competing hypothesis using an independently chosen measurement or a perturbation of a consequential parameter. Fifth, repeat after withholding one doubtful observation. Compare whether the scientific conclusion survives that choice. Test the wave-mode assignment with boundary conditions. A fast first arrival can travel far within a faster layer, and a recorded horizontal component can mix polarizations. Examine whether a linear-elastic, constant-speed assumption applies over the selected frequency band. State what a path-integrated travel time can establish about velocity, and what it cannot reveal without more rays.

Required outputs. Submit a two-page technical brief with a hypothesis table, raw and corrected values, an equation inventory, an uncertainty budget, a labelled graphic and a short limitations section. Include all model parameters. Do not omit unsuccessful tests and do not represent computer-generated data as measured field observations. Students should record their own explanations rather than copying one another's interpretation. Discuss how different station distributions or sampling choices could cause an apparently significant spatial pattern without a corresponding physical change.

Extensions. Seismic waves are vector disturbances traveling through heterogeneous elastic media. Coordinate systems, phase conventions and the validity of linear elasticity matter when comparing a simple model with measured records. A wave-speed estimate is conditional on the medium and on which phase was actually picked. Write an explicit counterfactual: what observation would undermine the main hypothesis? Consider what a larger dataset could resolve and what it would leave uncertain. Finally, design an accessible presentation of the result with a descriptive figure caption, a plain-language explanation of each symbol, and a table of numerical values for readers who cannot interpret a colored graph. Re-evaluate the result against professional source and public-safety boundaries before suggesting a consequential application.

Peer-review rubric. Assign 0–4 points each for correctness of units, explicit assumptions, data provenance, comparison with alternatives, uncertainty treatment, computational reproducibility and ethical communication. A polished figure without units, or a convincing narrative without a falsifiable test, should not receive full credit. Explain in prose why the least certain input affects confidence, and identify at least one result that could change with additional observations.