P and S Seismic Waves

Analysis of body wave propagation in elastic, homogeneous, and isotropic media. Calculates compressional and shear velocities, Poisson's ratio, and estimates local epicentral distances using the S-P method.

Input parameters

Reference geological media

Density (ρ)

Density of the isotropic elastic medium. Crust ≈ 2700, mantle ≈ 3300 kg/m³.

5006000

Bulk Modulus (K)

Resistance of the medium to volumetric compression.

0300

Shear Modulus (μ)

Rigidity of the medium. In fluids μ = 0 and S waves cannot propagate.

0250

S − P interval (Δt)

Time difference between the arrival of the S wave and the P wave at a station.

0300

Results

P-wave velocity (Vₚ)

5.709

km/s

S-wave velocity (Vₛ)

3.162

km/s

Vₚ/Vₛ ratio

1.805

Poisson's ratio (ν)

0.2787

Epicentral distance (Δ)

212.7

Travel-time curve — P and S arrival vs distance

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 the elastic moduli: the bulk modulus $K$ (incompressibility) and the shear modulus $\mu$ (rigidity).

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 how much a material contracts laterally when stretched. For isotropic media it is directly tied to the seismic velocities:

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

The physical range is $-1 \le \nu \le 0.5$ , but real materials almost always satisfy $0 \le \nu \le 0.5$ . $\nu = 0.5$ corresponds to an incompressible fluid; typical rocks lie between 0.20 and 0.35.

Epicentral distance via the S − P method

If the medium is homogeneous, a single station can estimate the epicentral distance from the wave 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 \Delta = \Delta t \cdot \dfrac{V_p \, V_s}{V_p - V_s}

This is the basis of the Wadati rule: when $t_s - t_p$ is plotted against $t_p$ for several stations, the slope is $V_p/V_s - 1$ and the intercept allows estimating the earthquake's origin time. The approximation assumes a flat Earth and constant velocities; for distances beyond ~1500 km a radial model is needed (PREM, IASP91).

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 resulting hodochrone is a pair of straight lines; in the real Earth those curves bend slightly toward the origin because velocity increases with depth.
Ignores intrinsic attenuation $(Q^{-1})$ , dispersion and P↔S conversions at interfaces.
The single-station S−P method gives distance, not direction: triangulation with ≥ 3 stations is required to locate the hypocenter.

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 Seismic Waves

Input parameters

Results

Travel-time curve — P and S arrival vs distance

Fundamentals & Explanation

P and S waves in an elastic medium

Fundamental velocities

Poisson's ratio

Epicentral distance via the S − P method

Typical velocities in Earth

Vₚ/Vₛ and Poisson identity

Model limitations

Academic References

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P and S waves in an elastic medium

Fundamental velocities

Poisson's ratio

Epicentral distance via the S − P method

Typical velocities in Earth

Vₚ/Vₛ and Poisson identity

Model limitations

Academic References