P-Wave Velocity from Elastic Moduli

A P wave is a compression travelling through rock, exactly like sound through air, and its speed follows the same rule every wave in an elastic medium follows: the square root of a stiffness over a density. What makes the seismic case interesting is that the relevant stiffness is a combination — $K + \frac{4}{3}G$ — rather than a single modulus.

The combination is there because a P wave cannot spread sideways. The rock beside the compressed zone confines it, so the material has to change shape as well as volume, and it fights back with its shear stiffness as well as its bulk stiffness. That is why the compressional wave always outruns the shear wave: it has more stiffness working for it and the same density working against it. For a Poisson solid, the idealised material with $\lambda = \mu$, the ratio comes out at exactly $V_p/V_s = \sqrt{3} \approx 1.73$, and real crustal rock sits close to that.

Set $G = 0$ and the equation tells you something profound about the planet. A fluid has no shear stiffness, so $V_p$ collapses to $\sqrt{K/\rho}$ — the ordinary speed of sound — while $V_s$ collapses to zero. Fluids carry no shear waves at all. In 1926 Harold Jeffreys used exactly this to establish that the Earth's outer core is liquid: S waves arriving from any earthquake vanish beyond about 103° of arc, leaving a shadow whose geometry outlines the core. The equation on this page is the argument.

Practical numbers: air 340 m/s, water 1500, unsaturated soil 300–800, saturated soil jumps to 1500 or more because the water now carries the compression, weathered rock 2000–3500, and intact crystalline rock 5000–6500. That jump at the water table is used deliberately — a seismic refraction survey locates groundwater by watching the P-wave velocity leap, and it is the same physics that makes a saturated site behave so differently from a dry one.