Shear Wave Velocity Calculator

Background.

Push a solid and it pushes back; twist or slide one layer of it past the next and it resists that too. The second kind of stiffness, resistance to shearing, is what lets a solid carry a shear wave — a disturbance in which the material moves side to side, at right angles to the direction the wave travels. Seismologists call these S waves, and they are the second arrival on a seismogram, after the faster compressional P waves.

Their speed depends on a contest between two properties. The shear modulus G measures how strongly the material snaps back when sheared; the density ρ measures how much mass must be set moving. Stiffer means faster, heavier means slower, and the speed is the square root of their ratio. Because of the square root, doubling the stiffness raises the speed by only about 41%.

A fluid has no shear stiffness at all, so shear waves cannot cross it. That single fact is how seismologists worked out that the Earth's outer core is liquid: S waves from distant earthquakes fail to appear on the far side of the planet, leaving a shadow where the core blocks them. The same physics runs in reverse in engineering and medicine. Geotechnical engineers measure shear-wave speeds in the ground to infer soil stiffness for earthquake design, and shear-wave elastography sends a gentle push through soft tissue and times the resulting wave to judge how stiff an organ has become.

The formula assumes an ideal solid — uniform, the same in every direction, and perfectly elastic. Layered rock, fibre-reinforced composites, wet soils and lossy materials behave differently, and their measured speeds can depart from this value.

What is shear wave velocity calculator?

Shear-wave velocity is the speed at which a transverse elastic disturbance travels through a solid, the particles moving perpendicular to the direction of travel. For a homogeneous, isotropic, linearly elastic material it depends only on the shear modulus and the density: v = √(G/ρ). It is always slower than the compressional (P-wave) speed in the same material, and it is zero in a fluid, which has no shear stiffness.

How to use this calculator.

1. 01Enter the shear modulus G in pascals. Tables usually give it in gigapascals, so multiply by 10⁹: 79 GPa becomes 79,000,000,000 Pa. If you only have Young's modulus E and Poisson's ratio ν, use G = E ÷ (2(1 + ν)).
2. 02Enter the mass density in kilograms per cubic metre. Convert g/cm³ by multiplying by 1,000: 2.7 g/cm³ is 2,700 kg/m³.
3. 03Read the shear-wave speed in metres per second.
4. 04For soils, layered rock or composites, treat the answer as an idealised reference; measured speeds in those materials depend on direction, frequency and water content.

The formula.

v = √(G ÷ ρ). G is the shear modulus in pascals — the shear stress needed per unit of shear strain — and ρ is the mass density in kg/m³. Since a pascal is a kg/(m·s²), the ratio G/ρ comes out in m²/s², and its square root in metres per second. The formula follows from Newton's second law applied to a thin slab of an elastic solid: the restoring force from shear stiffness balances the slab's inertia, giving a wave equation whose speed is √(G/ρ). Because speed varies with the square root, doubling G raises v by a factor of √2, about 41%, and doubling ρ lowers it by about 29%. Run it backwards and G = ρv², which is how a measured shear-wave speed is turned into a stiffness.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Aki and Richards, Quantitative Seismology, 2nd ed. (PRINT), chapter 2, elastic-wave velocities.
2. [2]USGS, Earthquake Glossary, S wave definition and propagation through solids.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.