Hong Life CalculatorsUnderstanding the shear wave velocity calculator — concept and principle
This calculator finds the shear-wave (S-wave) velocity of a material from its shear modulus and density, or from Young's modulus and Poisson's ratio, and also the P-wave velocity and the Vp/Vs ratio. It can also work backwards from a measured Vs to the small-strain shear modulus. Geophysicists, geotechnical engineers and seismologists use it.
How it is calculated
V_s = √(G ÷ ρ)
V_p = √((K + 4G/3) ÷ ρ)
V_p ÷ V_s = √((2 − 2ν) ÷ (1 − 2ν))
From a measured V_s: G_max = ρ V_s²
- G in GPa, ρ in kg/m³; E and ν give G = E/(2(1 + ν)) and K = E/(3(1 − 2ν)).
The principle
A wave travels through a solid because each part pulls on its neighbours. For shear waves the restoring force comes from the shear stiffness G, and the inertia from the density ρ, so the speed is √(G/ρ), just as a wave on a string travels at √(tension/mass per length). P-waves compress the material and are resisted by both bulk and shear stiffness, so they are always faster. Fluids have no shear stiffness and so carry no S-waves.
Worked example
Default inputs: from G, G = 30 GPa, ρ = 2,700 kg/m³, ν = 0.25.
- V_s = √(30 × 10⁹ ÷ 2,700) = 3,333 m/s = 10,936 ft/s.
- V_p ÷ V_s = √(1.5 ÷ 0.5) = √3 = 1.732.
- V_p = 3,333 × 1.732 = 5,774 m/s.
- G = 30,000 MPa.
| Density ρ (kg/m³) | Shear-wave velocity Vs (m/s) |
|---|---|
| 675 | 6,667 |
| 1,350 | 4,714 |
| 2,025 | 3,849 |
| 2,700 | 3,333 |
| 3,375 | 2,981 |
| 4,050 | 2,722 |
| 5,400 | 2,357 |
How to read the result
Hard rock has shear-wave velocities of thousands of metres per second, while soft soils may be a hundred or two. In geotechnics, measured Vs gives the small-strain stiffness used in earthquake and foundation design. The arrival-time gap between P- and S-waves on a seismogram also gives the distance to an earthquake.
Common mistakes
- Using g/cm³ for density instead of kg/m³.
- Using large-strain moduli from load tests for wave speeds.
- Assuming Vp/Vs = √3 for every material; it depends on ν.
Frequently asked questions
Why are S-waves slower than P-waves?
P-waves are resisted by both bulk and shear stiffness; S-waves only by shear stiffness.
Why can't S-waves travel through liquids?
Liquids have no shear stiffness.
What is G_max?
The small-strain shear modulus, ρVs², used in soil dynamics.
What does Vp/Vs tell you?
It depends on Poisson's ratio and helps identify rock type and fluid content.
Sources: MIT OpenCourseWare 2.001 Mechanics & Materials I (Fall 2006), lecture 13 notes; standard elastic-wave speed relations