Formula and Method for Shear Wave Velocity
Shear wave velocity (Vs) is the speed at which a shear (S-) wave — a body wave that displaces material perpendicular to its direction of travel — propagates through an elastic solid. For an isotropic, linear-elastic material it follows directly from the material's stiffness and mass: Vs = √(G / ρ), where G is the shear modulus (also called the modulus of rigidity, in pascals) and ρ is the mass density (in kg/m³). Stiffer, lighter materials transmit shear waves faster; softer, denser materials transmit them more slowly. This calculator also converts the result to ft/s, classifies the material against IBC/NEHRP seismic site categories, and — if you provide a layer thickness — estimates how long the wave takes to cross it.
How the calculation works
Enter the shear modulus and its unit, and the density and its unit; the calculator converts both to SI base units (pascals and kg/m³) before applying Vs = √(G/ρ), so mixed units never distort the result. If you only have Young's modulus (E) and Poisson's ratio (ν) from a lab report, first convert to shear modulus with G = E / (2(1 + ν)), then enter that G here. For the seismic site class, the tool compares your computed Vs against the standard NEHRP/IBC (ASCE 7) thresholds: Class A > 1,500 m/s (hard rock), Class B 760-1,500 m/s (rock), Class C 360-760 m/s (very dense soil/soft rock), Class D 180-360 m/s (stiff soil), and Class E < 180 m/s (soft soil). If you enter a layer thickness, the tool also reports the one-way travel time t = thickness / Vs, which is the same relationship used to interpret seismic refraction and downhole/crosshole Vs survey data.
Practical notes and typical values
- This is a single-material estimate, not a Vs30 profile: true seismic site classification averages travel time through the actual top 30 m of layered soil and rock (Vs30), which usually requires several layers with different G and ρ values, not one uniform block.
- Keep units consistent going in: convert G and ρ into the same unit family before comparing results across sources — a shear modulus reported in psi should be paired with a density reported in a compatible imperial unit, not left mixed with metric values.
- Typical shear modulus values: soft clay ≈5-30 MPa, dense sand/gravel ≈100-250 MPa, weathered rock ≈1-10 GPa, sound bedrock ≈10-60 GPa — density is comparatively narrow, roughly 1,400-2,200 kg/m³ for soils and 2,200-3,000 kg/m³ for rock.
- Shear waves need shear strength: because fluids have essentially zero shear modulus, Vs collapses toward zero in water, air, or molten rock — a useful check that flags a bad G or ρ input if your result comes out near zero for a solid material.