Connect elastic moduli to P and S wave support

1. Define the evidence and frame

An isotropic elastic medium resists volume change through its bulk modulus K and shape change through its shear modulus μ. Small-amplitude wave speeds obey Vp²=(K+4μ/3)/ρ and Vs²=μ/ρ. Modulus in pascals divided by density in kg/m³ produces squared speed in m²/s².

2. Connect mechanism and model

The model fixes K at 100 GPa and varies μ from 0 to 60 GPa and density from 3000 to 5000 kg/m³. At zero shear modulus the ideal liquid supports compressional motion but no static shear-wave propagation. A solid can have both wave types; increasing density at fixed stiffness lowers their speeds.

Global evidence for a liquid outer core combines absence of direct S transmission with P-wave refraction and shadow-zone structure. A single absent pick is not enough: instrument noise, unfavorable geometry or phase misidentification could also explain it. The solid inner core is inferred through additional seismic evidence, not by assuming all material below the mantle has the same phase.

The mantle is mostly solid while accommodating slow deformation and convection. Mechanical behavior depends on timescale: elastic response over seconds and viscous deformation over millions of years are compatible. These supplied elastic constants are instructional parameters, not a fit to a named mantle or core material.

3. Reconstruct the worked packet

At μ=40 GPa and ρ=4000 kg/m³, Vs=√10 km/s≈3.16228 and Vp=√(38.3333) km/s≈6.19139. At the same density with μ=0, Vs=0 while Vp=5 km/s.

4. Predict and investigate

Predict first: Does loss of shear rigidity require compressional speed to vanish?

Compare μ=0 with positive μ while holding density fixed. Then vary density with both moduli fixed. State which global observations support a phase-state inference and which alternative explanations affect an individual missing arrival.

5. Test a changed case

Apply the evidence to a changed case

At μ=0 and density 5000 kg/m³, what are Vp and Vs?

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Vp=√20≈4.47214 km/s and Vs=0. Compression still has a restoring bulk modulus; the result does not prove a particular composition.

6. Retain a defensible evidence notebook

Preserve first predictions, full units, geometry, source provenance, assumptions and rejected alternatives. Synthetic models are not field measurements. The explicitly labeled USGS packet is actual catalog evidence, but working through a public example is not independent practical competence or an operational hazard assessment.

Mechanism and data reference, checked 30 October 2026. Original teaching prose, diagrams and exercises; actual USGS rows retain their agency and provenance.

Primary USGS-hosted seismic elastic-constant reference, equations for compressional and shear speed; checked 30 October 2026. The numerical moduli in this lesson are original instructional values.