How this instrument works
Shear wave velocity is the speed at which a transverse disturbance crosses a solid: material displaces sideways, perpendicular to the direction the disturbance travels, the way a plucked rope wiggles while the ripple itself runs along its length. This differs from the longitudinal, or P-wave, speed, in which material compresses and expands along the direction of travel. Only the transverse case, not the longitudinal one, requires the material to resist a change of shape at constant volume.
The formula vs = √(G ⁄ ρ) has the same shape as every non-dispersive wave speed in physics: the square root of a restoring stiffness divided by an inertia, the same pattern behind the speed of a pulse on a string, √(tension ⁄ linear density). Here the restoring term is the shear modulus G, the stress needed to distort a unit cube by a unit angle without changing its volume; the inertial term is ρ, ordinary mass density. Stiffer materials push back harder and carry the signal faster; denser ones carry more inertia per unit volume and slow it down.
The formula assumes an isotropic, linearly elastic, homogeneous solid, and it is genuinely frequency-independent under those conditions — a seismometer and an ultrasonic transducer measuring the same steel plate should agree, because neither the rock nor the metal disperses at these strain levels. The assumption breaks in layered rock, wood, or fibre composites, where shear stiffness differs by direction and a single G no longer describes the material; it also weakens in soils at large shaking strain, where the effective shear modulus itself drops as strain grows, which is exactly why geotechnical engineers distinguish a small-strain vs from the softer value a soil shows in a strong earthquake.
- Enter the Shear modulus (G) of the material in pascals — switch the unit menu to GPa if that is how your handbook or datasheet lists it.
- Enter the Density (rho) of the same material in kilograms per cubic metre.
- Read the Shear (S-)wave velocity (vs) result in metres per second, or switch its unit to km/s for geophysical-scale numbers.
- Recheck the material identification if the result looks off by a large factor — a mismatched G and ρ pair, say a rock's density with a metal's modulus, is the most common input mistake.
Worked example — an S-wave through structural steel
Structural steel is commonly tabulated with a shear modulus of about 80 GPa and a density of 7,850 kg/m³, the standard design value used for mild and low-alloy steels. Convert the modulus to base units, 80,000,000,000 Pa, and divide by the density: 80,000,000,000 ⁄ 7,850 = 10,191,082.8 m²/s². The square root of that ratio is 3,192.35 m/s, the speed a shear wave — and, at the same underlying physics, a transverse ultrasonic pulse from a weld-inspection probe — travels through that steel.
That figure sits well below steel's longitudinal speed, close to 5,900 m/s, because a P-wave is resisted by both the bulk modulus and the shear modulus together, while the transverse case answers to shear stiffness alone. An inspector reading a transverse-probe echo on a steel plate is timing exactly this 3,192 m/s signal; a seismologist reading the S-wave arrival at a steel-framed building's foundation is timing the same physics in a different material.
Questions
Why does a denser material carry a shear wave more slowly?
Density sits in the denominator because it measures inertia — how much mass the disturbance has to accelerate sideways as it passes. More mass per unit volume resists that sideways motion more, for the same restoring stiffness G, so the signal takes longer to propagate. Two materials with identical shear modulus but different density will always carry the slower shear disturbance.
How does shear wave velocity differ from P-wave velocity?
A P-wave (longitudinal) speed is √((K + 4G⁄3) ⁄ ρ), driven by both the bulk modulus K and the shear modulus G, because compression changes both shape and volume. The transverse case answers only to G, so vp is always faster than vs for the same material — in steel, roughly 5,900 m/s versus 3,200 m/s, close to a factor of √3.
Can a shear wave travel through a liquid or a gas?
No. Fluids have zero static shear modulus — they offer no lasting resistance to a change of shape, only to a change of volume — so vs = √(G⁄ρ) collapses to zero. This is why seismologists know Earth's outer core is liquid: S-waves from deep earthquakes never arrive on the far side of the planet, while P-waves do.
What is Vs30 and how does it use this formula?
Vs30 is the average shear wave velocity in the top 30 metres of ground at a site, used by earthquake engineers to classify how strongly local soil will amplify shaking. It is built from measured or estimated vs = √(G⁄ρ) values through each soil layer, and a low Vs30 (soft, low-G ground) generally means stronger amplification than a high Vs30 site on bedrock.
Does shear wave velocity change with the frequency of the disturbance?
In an ideal linear-elastic solid at small strain, no — vs is a material property independent of frequency, which is why a single number describes both a low-frequency seismic S-wave and a megahertz ultrasonic pulse in the same steel. Real soils and rocks depart from this at large shaking strain, where the effective G softens and vs drops.
Where do typical shear modulus and density values come from?
Materials handbooks and standards tables list both directly for common metals, rocks, and soils — steel near 80 GPa and 7,850 kg/m³, aluminium near 26 GPa and 2,700 kg/m³. For soils, in-situ tests such as seismic cone penetration or downhole logging measure vs directly and back-calculate G from the density instead.