P and S Seismic Waves

Longitudinal P waves and transverse S waves in elastic solid media, velocities and seismic detection.

Conceptual context

When an earthquake releases energy inside the Earth, that energy propagates as elastic waves through rocks. There are two fundamental types of body waves: P waves (primary, compression) and S waves (secondary, shear). Understanding their velocity differences and behavior in different media is essential for seismology, exploration geophysics, and earthquake-resistant engineering.

P and S waves form the basis of the seismological method: the arrival time difference allows locating the earthquake focus, and the presence or absence of S waves reveals whether internal layers are solid or liquid.

🟢 Essential level

P waves compress and dilate the material in the same direction as the wave advances, like a spring being pushed and released. Their velocity $v_p$ is the highest of all seismic waves. S waves displace particles perpendicular to the propagation direction, like a vibrating string. Their velocity $v_s$ is always less than that of P waves.

[!NOTE] P waves always travel faster than S waves in the same medium. That is why they arrive first at a seismograph and are called "primary." The arrival time difference $\Delta t_{PS}$ is the key to locating the earthquake focus.

S waves do not propagate in fluids because liquids and gases lack shear rigidity. This property was key to discovering that Earth's outer core is liquid: S waves disappear when crossing it. The combination of both wave types allows both locating earthquakes and diagnosing the physical state of terrestrial materials. A seismogram therefore carries two independent pieces of information in a single recording: the arrival-time gap pinpoints distance, while the mere presence or absence of the S pulse reveals whether the material it crossed was solid or molten.

🔵 Formal level

Basic formulas

The P-wave velocity in an isotropic, homogeneous medium is given by:

P-wave velocity

Where $K_{bulk}$ is the bulk modulus, $G_{shear}$ the shear modulus, and $\rho_{\\text{medio}}$ the density. The presence of both moduli reflects that P waves involve both volume change and shape change.

The S-wave velocity depends only on shear rigidity:

S-wave velocity

The ratio between both velocities is directly related to the Poisson ratio of the medium:

Velocity ratio and Poisson's ratio

For a solid with $\nu$ of 0.25, the $v_p/v_s$ equals exactly the square root of 3 (approximately 1.73). Values above 2 indicate fluid saturation in the pores.

Real student questions

• Why is P always faster than S? Because the $v_p$ formula includes $K_{bulk}$ added to $G_{shear}$, while $v_s$ only includes $G_{shear}$. Adding a positive term in the numerator makes $v_p$ necessarily larger.

• How do we know the outer core is liquid? Because S waves do not traverse it: $G_{shear}$ is zero in a liquid, which makes $v_s$ zero. Only P waves continue (with finite $K_{bulk}$).

• Does the time difference only work for homogeneous media? In theory yes, but in practice corrected travel-time curves are used based on Earth velocity models (such as PREM or IASP91) that account for property variations with depth.

• Can $v_p/v_s$ be less than the square root of 2? In principle yes: it would imply negative $\nu$, corresponding to an auxetic material. They are rare in geology but exist in synthetic materials.

• What about surface waves? Surface waves (Rayleigh and Love) derive from P and S waves interacting with the free surface. They are slower but with larger amplitude at long distances, and cause most seismic damage.

Final synthesis

P and S seismic waves are the two fundamental forms of elastic propagation inside the Earth. Their velocity depends on elastic moduli and medium density, and their ratio reveals the Poisson coefficient and possible fluid presence. The arrival time difference is the basic tool for seismic location, and the extinction of S waves in fluids is the fundamental geophysical evidence for the existence of Earth's liquid core.