S-P Travel Time

Core idea

Overview

The S-P travel time equation relates the distance from a seismic source to the time interval between the arrival of P-waves and S-waves at a recording station. This calculation exploits the velocity difference between the faster primary (P) compressional waves and the slower secondary (S) shear waves to determine the earthquake's epicentral distance.

When to use: This equation is used when a seismogram clearly displays the distinct onset times of both P-waves and S-waves. It assumes a homogeneous crustal model where seismic velocities remain relatively constant along the travel path, making it most effective for local or regional seismic events.

Why it matters: Calculating the S-P gap is the foundational step in earthquake triangulation. By determining the distance from at least three different stations, seismologists can pinpoint the exact epicenter and depth of an earthquake, which is vital for emergency response and tsunami forecasting.

Symbols

Variables

D = Distance to Epicentre, Gap = S-P Time Diff, $V_{p}$ = P-Wave Velocity, $V_{s}$ = S-Wave Velocity

D

Distance to Epicentre

km

Gap

S-P Time Diff

s

V_{p}

P-Wave Velocity

km/s

V_{s}

S-Wave Velocity

km/s

Walkthrough

Derivation

Understanding S-P Time for Epicentre Distance

The difference in arrival times of P-waves and S-waves at a seismometer allows the distance to the earthquake epicentre to be calculated.

P and S waves travel at constant velocities through the crust.
The earthquake source is a point.

1

Write travel times for each wave:

Both waves travel the same distance D, but S-waves are slower, so they arrive later.

t_{P} = \frac{D}{V _{P}}, t_{S} = \frac{D}{V _{S}}

2

Calculate the time gap:

The measured S-P time gap Δt depends on the distance and the difference in reciprocal velocities.

Δ t = t_{S} - t_{P} = D (\frac{1}{V _{S}} - \frac{1}{V _{P}})

3

Solve for distance:

Rearranging gives the epicentre distance D from the S-P time gap.

D = Δ t \cdot \frac{V _{P} V _{S}}{V _{P} - V _{S}}

Note: Three seismometers at different locations are needed to pinpoint the epicentre by triangulation.

Result

D = Δ t \cdot \frac{V _{P} V _{S}}{V _{P} - V _{S}}

Source: A-Level Geology — Seismology

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a straight line passing through the origin, where the epicentre distance (D) is plotted against the time difference between S and P waves. This linear relationship occurs because the term involving wave velocities acts as a constant gradient, meaning distance increases proportionally with the time interval.

Graph type: linear

Why it behaves this way

Intuition

Visualize two waves (P and S) originating from a single point (earthquake focus) and propagating outwards. The P-wave forms an expanding sphere ahead of the slower S-wave's expanding sphere.

D

The epicentral distance, which is the horizontal distance from the seismic station to the earthquake's epicentre on the Earth's surface.

This is the unknown distance we aim to calculate. A larger time gap between P and S wave arrivals indicates a greater distance to the earthquake.

T_{s} - T_{p}

The time interval between the arrival of the S-wave and the P-wave at a seismic recording station.

This time difference directly measures how much longer the slower S-wave took to cover the same distance compared to the faster P-wave. It's the primary observable from a seismogram used in this calculation.

V_{p}

The average velocity of primary (P) compressional waves through the medium.

P-waves are the fastest seismic waves, traveling through solids, liquids, and gases by compression and expansion. A higher

V_{p}

means P-waves reach the station sooner.

V_{s}

The average velocity of secondary (S) shear waves through the medium.

S-waves are slower than P-waves and travel by shearing motion, only propagating through solid materials. A higher

V_{s}

means S-waves reach the station sooner, but always after P-waves.

\frac{V _{p} V _{s}}{V _{p} - V _{s}}

A composite velocity factor that converts the observed S-P time difference into the epicentral distance. It represents the effective speed at which the S-P time gap accumulates per in the relevant context.

This factor accounts for the relative speed difference between P and S waves. If P and S waves travel at very similar speeds (small

V_{p}

-

V_{s}

), then a small time difference corresponds to a large distance, making this

Signs and relationships

V_p - V_s: This term represents the absolute difference in velocities between P-waves and S-waves. Since P-waves are inherently faster than S-waves ( $V_{p}$ > $V_{s}$ ), this difference is always positive, ensuring the calculated distance D
V_p V_s: This product, combined with the denominator, forms the effective velocity factor. It ensures that the scaling from time difference to distance correctly reflects the absolute magnitudes of the wave velocities, not just

Free study cues

Insight

Canonical usage

This equation requires consistent units for time and velocity to yield the epicentral distance in the corresponding length unit.

Common confusion

A common mistake is using inconsistent units, such as velocities in meters per second (m/s) while the time difference is in seconds (s), but expecting the distance in kilometres (km) without proper conversion.

Unit systems

$D$ km - The calculated distance to the earthquake epicentre. Its unit will match the length unit used in the velocity terms.

$T_{s} - T_{p}$ s - The time difference between the arrival of the S-wave and the P-wave at the seismic station. Typically measured in seconds.

$V_{p}$ km/s - The velocity of the P-wave (primary, compressional wave). Must be in consistent length and time units with V_s.

$V_{s}$ km/s - The velocity of the S-wave (secondary, shear wave). Must be in consistent length and time units with V_p.

Ballpark figures

Quantity:
Quantity:
Quantity:

One free problem

Practice Problem

A seismograph at a remote station records a P-wave arrival at 08:12:10 and an S-wave arrival at 08:12:34. If the regional P-wave velocity is 6.0 km/s and the S-wave velocity is 3.5 km/s, calculate the distance from the station to the earthquake epicenter.

S-P Time Diff24 s

P-Wave Velocity6 km/s

S-Wave Velocity3.5 km/s

Solve for: $D$

Hint: First calculate the S-P time gap by finding the difference in arrival times, then apply the formula.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In a physics application involving S-P Travel Time, S-P Travel Time is used to calculate Epicentre Distance from S-P Time Diff, P-Wave Velocity, and S-Wave Velocity. The result matters because it helps predict motion, energy transfer, waves, fields, or circuit behaviour and check whether the answer is plausible.

Study smarter

Tips

Ensure velocity units (km/s) and time units (s) are consistent to output distance in km.
Variable GAP represents the S-P interval (Ts - Tp) in seconds.
Check that the S-wave velocity is lower than the P-wave velocity, as P-waves are always faster in the same medium.

Avoid these traps

Common Mistakes

Using arrival times directly rather than the difference between them (S minus P).
Convert units and scales before substituting, especially when the inputs mix km, s, km/s.
Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The difference in arrival times of P-waves and S-waves at a seismometer allows the distance to the earthquake epicentre to be calculated.

This equation is used when a seismogram clearly displays the distinct onset times of both P-waves and S-waves. It assumes a homogeneous crustal model where seismic velocities remain relatively constant along the travel path, making it most effective for local or regional seismic events.

Calculating the S-P gap is the foundational step in earthquake triangulation. By determining the distance from at least three different stations, seismologists can pinpoint the exact epicenter and depth of an earthquake, which is vital for emergency response and tsunami forecasting.

Using arrival times directly rather than the difference between them (S minus P). Convert units and scales before substituting, especially when the inputs mix km, s, km/s. Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.