The P-wave, or Primary wave, is the fastest compressional seismic wave that travels through the Earth’s interior. They are the first waves detected by seismographs after an earthquake. P-waves cause particles in the medium to vibrate back and forth in the direction the wave is moving, similar to sound. Calculating their speed and predicting their arrival time is fundamental to seismology, providing data necessary to understand Earth’s structure and locate seismic events.
Material Inputs for Velocity
The speed at which a P-wave travels depends entirely on the physical properties of the material it passes through. Velocity changes dramatically as the wave moves between layers of the Earth, such as the crust, mantle, or outer core. P-wave velocity is determined by three material properties: density, bulk modulus, and shear modulus.
Density (\(\rho\)) represents the mass per unit volume of the rock or fluid and is inversely related to wave speed. Although seismic velocity often increases with depth, the elastic properties change even faster. The bulk modulus (\(K\)) measures a material’s resistance to compression, which is important because P-waves are compressional waves.
The shear modulus (\(\mu\)) represents the material’s stiffness or resistance to distortion of shape. P-waves depend on the shear modulus as their propagation involves the elastic response of the solid material. For fluids, like the Earth’s outer core, the shear modulus is zero because fluids cannot resist a change in shape. This causes P-waves to slow down significantly when they enter the liquid outer core.
Deriving P-Wave Velocity
The P-wave velocity (\(V_p\)) is calculated using a foundational equation from elasticity theory that incorporates the three material properties of the medium. The formula combines the bulk modulus (\(K\)), the shear modulus (\(\mu\)), and the density (\(\rho\)) into the expression \(V_p = \sqrt{((K + (4/3)\mu) / \rho)}\). The term \((K + (4/3)\mu)\) is sometimes referred to as the P-wave modulus.
The numerator, which includes both the bulk and shear moduli, represents the material’s total elastic resistance to the P-wave’s compressional motion. This value is divided by the density, and the square root is taken of the ratio to yield the final velocity. For example, if a wave moves from solid rock to water, the shear modulus (\(\mu\)) instantaneously becomes zero.
For P-waves in water, the formula simplifies to \(V_p = \sqrt{(K / \rho)}\), since the \((4/3)\mu\) term vanishes. This loss of the shear term causes the P-wave velocity to decrease significantly in liquids. In typical crustal rock, \(V_p\) ranges between 4,800 and 6,700 meters per second, compared to only about 1,500 meters per second in water.
Calculating P-Wave Arrival Time
Once the P-wave velocity is determined for a given path, calculating the wave’s travel time to a seismograph station is the next step. The simplest calculation for a homogeneous medium uses the basic relationship: Time equals Distance divided by Velocity. This calculation is effective for small, localized studies where rock properties are relatively uniform.
For tracking P-waves over large distances, the simple formula is insufficient because the Earth’s interior has many layers with varying densities and moduli. Seismologists instead rely on global travel-time curves, which are standardized charts developed from observing thousands of earthquakes. These curves account for the complex effects of refraction and reflection as the waves pass through the Earth’s curved layers.
Using these curves, a seismologist matches the distance from the earthquake source to the recorded P-wave arrival time. The measured arrival time is compared to the travel-time curve to determine the travel duration. Subtracting this travel time from the P-wave’s arrival time at the station yields the precise moment the earthquake began, known as the origin time.
Locating the Earthquake Source (Triangulation)
The calculated P-wave arrival time is the first piece of data used to locate an earthquake’s source, or epicenter. To determine the distance from a single seismic station, seismologists use the time difference between the arrival of the P-wave and the slower-traveling S-wave. This difference in arrival time is known as the S-P interval.
Since the P-wave travels faster than the S-wave, the time gap between their arrivals increases the farther the station is from the earthquake. The measured S-P interval is matched against standardized travel-time curves to convert the time difference directly into a distance from the station. This distance defines a circle on a map, with the seismic station at the center and the calculated distance as the radius.
The final location of the earthquake’s epicenter is found through triangulation. By calculating the distance from at least three different seismic stations, three separate circles are drawn on a map. The single point where all three distance circles intersect marks the precise location of the epicenter.