Seismic body waves are elastic disturbances that travel through Earth materials. The two most important body waves are the
primary wave or P wave and the secondary wave or S wave. A P wave is a compressional wave: particles in the medium oscillate
mainly back and forth in the same general direction as propagation. An S wave is a shear wave: particles move perpendicular to the direction of travel.
Because the restoring physics is different in the two cases, the two waves do not travel at the same speed.
In an isotropic elastic medium, the P-wave speed depends on the material’s resistance to both compression and shear. If the bulk modulus is \(K\),
the shear modulus is \(\mu\), and the density is \(\rho\), then the standard formulas are
\[
\begin{aligned}
v_p &= \sqrt{\frac{K+\tfrac{4}{3}\mu}{\rho}} \\
v_s &= \sqrt{\frac{\mu}{\rho}}
\end{aligned}
\]
These formulas show an important physical idea. The S-wave speed depends only on shear rigidity \(\mu\), because an S wave changes shape but not volume very much.
The P-wave speed includes both \(K\) and \(\mu\), because a compressional disturbance involves volume change as well as elastic resistance to deformation.
Since \(K + \tfrac{4}{3}\mu\) is larger than \(\mu\) for ordinary solids, the P wave is faster than the S wave. This is why seismographs record the P arrival first.
Unit consistency matters. In geology and geophysics, elastic moduli are often tabulated in gigapascals, while density is usually given in kilograms per cubic meter.
To use the wave-speed equations correctly, convert the moduli to pascals:
\[
\begin{aligned}
1\ \text{GPa} &= 10^9\ \text{Pa}
\end{aligned}
\]
Once the speeds are known, a simple travel-time estimate follows from distance divided by speed. If a wave travels a path length \(d\), then
\[
\begin{aligned}
t_P &= \frac{d}{v_p} \\
t_S &= \frac{d}{v_s}
\end{aligned}
\]
The difference \(\Delta t = t_S - t_P\) is especially useful in earthquake seismology. If a station records both arrivals, the lag between them grows with distance.
This is the basic reason that regional earthquake-location methods can use P–S arrival gaps to estimate how far the station is from the source. A larger gap usually means
a larger source-to-station distance, assuming the path averages are known well enough.
Consider the sample values used in this calculator: \(K = 50\ \text{GPa}\), \(\mu = 30\ \text{GPa}\), and \(\rho = 2700\ \text{kg/m}^3\). Substituting into the formulas gives
a P-wave speed of about \(6.31\ \text{km/s}\) and an S-wave speed of about \(3.33\ \text{km/s}\). These numbers are realistic for many crustal rocks. If the travel distance is
\(100\ \text{km}\), then the P arrival occurs after roughly \(15.9\ \text{s}\), while the S arrival occurs after roughly \(30.0\ \text{s}\). The lag is therefore about \(14.1\ \text{s}\).
This model is intentionally simple. Real Earth structure is layered, anisotropic in some regions, and temperature- and pressure-dependent. Paths bend because wave speed changes with depth,
and fluids eliminate S-wave propagation entirely because fluids cannot sustain static shear stress. Even so, the elastic formulas in this tool are the standard first step for understanding why
P and S waves move differently and why the P wave always arrives first in ordinary solid Earth materials.
