S-Wave Velocity from Shear Modulus

The S wave is the destructive one. It travels more slowly than the P wave, arrives second, carries more energy, and shakes the ground sideways rather than along the direction of travel — which is precisely the motion buildings are least able to resist. Its speed depends on one modulus only: $V_s = \sqrt{G/\rho}$, because a shear wave changes shape without changing volume and the bulk modulus never enters.

That simplicity is why $V_s$ has become the single most-used number in earthquake engineering. Site classification in ASCE 7, in Eurocode 8 and in the National Building Code of Canada is written around $V_{s30}$, the time-averaged shear velocity over the top thirty metres. Site Class A is hard rock above 1500 m/s; Class E is soft soil below 180 m/s; the design spectrum a building is checked against changes with the class, and on a soft site it can be more than double the rock value at the periods that matter.

Two subtleties trip people up. The first is that the $G$ measured by a shear wave is $G_{max}$, the small-strain modulus — the wave strains the soil by perhaps $10^{-5}$, while a strong earthquake strains it by $10^{-3}$ or more. Soil softens as it is worked, and under severe shaking the operative modulus can fall to a fifth of $G_{max}$, which lengthens the site period as the event proceeds. The second is that denser does not mean faster. Density sits in the denominator, so for a given stiffness a heavier material is slower; rock is fast because $G$ climbs far more steeply than $\rho$ does across the same range of materials.

Measuring $V_s$ in the field is done without an earthquake. Crosshole and downhole tests use boreholes and a hammer source; seismic cone penetration adds a geophone to a CPT probe; and surface-wave methods such as MASW and its passive cousins infer the profile from the dispersion of Rayleigh waves, needing no borehole at all. The last of these is why a $V_{s30}$ can now be obtained for a site in an afternoon.