Shear Wave Velocity Calculator

Use the shear wave velocity calculator to determine the velocity of a shear wave propagating in the body.

Shear wave velocity3,130.354 m/s√(G/ρ) for Mild steel — shear waves cannot travel through a liquid at all, because a liquid has no shear modulus. That absence is how the Earth's liquid outer core was discovered

Pressure wave velocity5,856.357 m/salways the FASTER of the two, which is why the P in P-wave is usefully read as "primary" — it arrives first

P to S velocity ratio1.870829√(2(1−ν)/(1−2ν)) — this depends ONLY on Poisson's ratio. Neither stiffness nor density survives the ratio, so measuring both speeds gives ν directly, which is how seismology infers composition at depths nobody can sample

Thin-rod velocity5,047.545 m/s√(E/ρ) — SLOWER than the bulk P-wave, because a thin rod is free to contract sideways while bulk material is constrained by its surroundings

Rayleigh surface wave2,899.19 m/s92.615% of the shear speed. These travel along the surface, decay slowly with distance, and do most of the damage in an earthquake

Shear modulus G76.92308 GPa

Bulk modulus K166.66667 GPa

Poisson's ratio ν0.3

Stiffness C₁₁269.23077 GPaλ + 2G — the diagonal term relating direct stress to direct strain in the same direction

Stiffness C₁₂115.38462 GPaLamé's λ — the coupling term, which is what makes stretching one way contract the others

Stiffness C₄₄76.92308 GPathe shear term, equal to G

Check: C₄₄ = (C₁₁ − C₁₂)/276.92308 GPathis identity holds ONLY for an isotropic material. A cubic crystal has three genuinely independent constants and breaks it, and the amount by which it is broken is the Zener anisotropy ratio

Two constants, not threeisotropy fixes the thirdan isotropic solid has only TWO independent elastic constants. Give any two of E, G, K, ν or λ and the rest follow

The formula

v_s = √(G/ρ); v_p = √(E(1−ν)/ρ(1+ν)(1−2ν)); v_p/v_s = √(2(1−ν)/(1−2ν))

Two speeds, one ratio

A solid carries two bulk waves. The pressure wave compresses and dilates the material and travels faster; the shear wave distorts it without changing volume and travels slower. Their ratio is √(2(1−ν)/(1−2ν)), which contains Poisson's ratio and nothing else — no stiffness, no density.

That is a remarkably useful accident. Timing both arrivals from an earthquake gives Poisson's ratio for material thousands of kilometres down, without knowing its density or stiffness separately. It is the basis of how the Earth's internal structure was mapped.

Shear waves cannot cross a liquid

A liquid has no shear modulus, so v_s is zero in one: there is nothing to restore a shear distortion. The resulting S-wave shadow zone on the far side of the Earth is what revealed that the outer core is liquid — an inference drawn entirely from waves that did not arrive.

Isotropy leaves only two constants

An isotropic solid is fully described by any two of Young's modulus, the shear modulus, the bulk modulus, Poisson's ratio and Lamé's λ. The stiffness matrix has three non-zero distinct entries but they satisfy C₄₄ = (C₁₁ − C₁₂)/2, so only two are free. A cubic crystal breaks that identity and needs all three; how badly it is broken is the Zener anisotropy ratio.