Input parameters

Illustrative geological media

Approximate effective values for exploring the model; real properties depend on composition, porosity, pressure, temperature, and frequency.

Medium properties

Constant effective elastic moduli and density along the entire path.

Density (ρ)

Medium density. Continental crust ≈ 2700 and upper mantle ≈ 3300 kg/m³.

5006000

Bulk modulus (K)

Resistance to uniform compression; it must be greater than zero.

0.1300

Shear modulus (μ)

Medium rigidity. For an ideal fluid μ = 0 and S waves do not propagate.

0250

P-wave velocity (Vₚ)

5.709km/s

S-wave velocity (Vₛ)

3.162km/s

S−P epicentral estimate

The interval uses straight paths and a shallow focus: h ≪ D, so Δ ≈ D.

S−P interval (Δt)

Difference between the S and P arrivals observed at the same station.

060

Approximate epicentral distance (Δ)

212.7km

Valid as an approximation when focal depth h is much smaller than path distance D.

Results

Medium velocities

P-wave velocity (Vₚ)

5.709

km/s

S-wave velocity (Vₛ)

3.162

km/s

Elastic relationships

Vₚ/Vₛ ratio

1.805

Poisson's ratio (ν)

0.2787

S−P result

Approximate epicentral distance (Δ)

212.7

Valid as an approximation when focal depth h is much smaller than path distance D.

Homogeneous travel-time curve — P and S arrivals

Vp/Vs = 1.805

Fundamentals & Explanation

P and S waves in an elastic medium

In an elastic, isotropic, homogeneous solid, a disturbance propagates as two independent modes obtained from the Navier-Cauchy equation. The P wave (primary, compressional) makes particles oscillate along the propagation direction; the S wave(secondary, shear) makes them oscillate perpendicularly. S waves cannot propagate through ideal fluids: that is why Earth's outer core casts an S-wave shadow at teleseismic distances.

Their speeds depend on density $\rho$ and on effective elastic moduli: bulk modulus $K$ and shear modulus $\mu$ . The calculator treats them as constant along the entire path.

Fundamental velocities

P wave (compressional)

V_p = \sqrt{\dfrac{K + \tfrac{4}{3}\mu}{\rho}}

S wave (shear)

V_s = \sqrt{\dfrac{\mu}{\rho}}

Because $K + \tfrac{4}{3}\mu > \mu$ always, P waves are faster than S waves — hence their name, they reach the station first. The ratio $V_p/V_s$ is a lithological diagnostic: continental crust ≈ 1.73, saturated basalts may exceed 1.9, dry sediments drop to 1.5.

Poisson's ratio

Poisson's ratio $\nu$ measures the transverse contraction of a solid under longitudinal extension. For isotropic solids with $V_s > 0$ it is directly tied to the seismic velocities:

\nu = \dfrac{V_p^2 - 2 V_s^2}{2(V_p^2 - V_s^2)}

Stability of an isotropic elastic solid requires $-1 < \nu < 0.5$ ; common rocks usually lie between 0.20 and 0.35. The limit $\nu \to 0.5$ describes a nearly incompressible solid. An ideal fluid with $\mu = 0$ has no S wave or equivalent Poisson test, so the calculator leaves $\nu$ undefined.

Path distance and the S−P epicentral approximation

If P and S follow the same straight path of length $D$ through a homogeneous medium, one station can estimate that distance from their delay. Defining $t_p = D/V_p$ and $t_s = D/V_s$ , their difference is:

\Delta t = t_s - t_p = D \cdot \dfrac{V_p - V_s}{V_p \, V_s} \quad \Longrightarrow \quad D = \Delta t \cdot \dfrac{V_p \, V_s}{V_p - V_s}

Geometrically, for focal depth $h$ , horizontal epicentral distance is $\Delta = \sqrt{D^2-h^2}$ . This calculator adopts $h \ll D$ and therefore reports $\Delta \approx D$ . Homogeneity alone does not remove the depth effect.

The relationship also underpins the Wadati diagram: plotting $t_s-t_p$ against $t_p$ at several stations yields slope $V_p/V_s-1$ , while the intercept can be used to estimate origin time. Real applications use regional or global travel-time curves that include depth, layering, refraction, and Earth curvature.

Typical velocities in Earth

Air: Vₚ ≈ 0.34 km/s

Sea water: Vₚ ≈ 1.5 km/s · Vₛ = 0

Unconsolidated sediments: Vₚ ≈ 1–2.5 km/s

Granite (upper crust): Vₚ ≈ 5.5–6 km/s · Vₛ ≈ 3.0–3.5 km/s

Basalt (oceanic crust): Vₚ ≈ 6–7 km/s · Vₛ ≈ 3.4–3.8 km/s

Upper mantle (peridotite): Vₚ ≈ 7.8–8.4 km/s · Vₛ ≈ 4.4–4.8 km/s

Outer core: Vₚ ≈ 8–10 km/s · Vₛ = 0 (fluid)

Inner core: Vₚ ≈ 11 km/s · Vₛ ≈ 3.5 km/s

Vₚ/Vₛ and Poisson identity

A useful identity links both ratios:

\left(\dfrac{V_p}{V_s}\right)^{\!2} = \dfrac{2(1-\nu)}{1-2\nu}

For $\nu = 0.25$ we get $V_p/V_s = \sqrt{3} \approx 1.732$ (the standard Poisson medium, common in continental crust).

Model limitations

Assumes elastic, isotropic and homogeneous medium. Real Earth is layered and shows azimuthal anisotropy.
The calculated travel-time curve is a pair of straight lines. Real curves form nonlinear branches because of layering, refraction, discontinuities, and source depth.
Ignores intrinsic attenuation $(Q^{-1})$ , dispersion and P↔S conversions at interfaces.
One station gives distance, not direction. Three or more stations can constrain the epicenter; determining hypocenter and origin time requires an inversion with suitable phases, geometry, and a velocity model.

Academic References

[1] Aki, K., & Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books. (Canonical text on elastic-wave theory and seismology.)
[2] Shearer, P. M. (2019). Introduction to Seismology (3rd ed.). Cambridge University Press. (Modern, accessible treatment of travel-time curves and S−P methods.)
[3] Wadati, K. (1933). On the travel time of earthquake waves. Part II. Geophysical Magazine, 7, 101–111. (Origin of the Wadati diagram and the S−P method.)
[4] Dziewonski, A. M., & Anderson, D. L. (1981). Preliminary Reference Earth Model (PREM). Physics of the Earth and Planetary Interiors, 25(4), 297–356.

P- and S-Wave Velocities

Input parameters

Results

Homogeneous travel-time curve — P and S arrivals

Fundamentals & Explanation

P and S waves in an elastic medium

Fundamental velocities

Poisson's ratio

Path distance and the S−P epicentral approximation

Typical velocities in Earth

Vₚ/Vₛ and Poisson identity

Model limitations

Academic References

P and S waves in an elastic medium

Fundamental velocities

Poisson's ratio

Path distance and the S−P epicentral approximation

Typical velocities in Earth

Vₚ/Vₛ and Poisson identity

Model limitations

Academic References