P-Wave and S-Wave Velocity Calculator
Find seismic P-wave and S-wave velocities, their ratio, and Poisson's ratio from a material's bulk modulus, shear modulus, and density.
🌍 What are P-Wave and S-Wave Velocity?
P-waves (primary, compressional) and S-waves (secondary, shear) are the two body-wave types that travel through Earth's interior, radiating outward from an earthquake or any other seismic source. Their velocities depend on the material's elastic properties, Vp = sqrt((K + 4/3*G) / rho) and Vs = sqrt(G / rho), where K is bulk modulus, G is shear modulus, and rho is density.
Seismologists and exploration geophysicists use these calculations constantly, to interpret seismic refraction and reflection survey data, estimate rock type and fluid content from measured Vp/Vs ratios, model how earthquake waves propagate through Earth's layered interior, and understand why S-waves vanish through Earth's liquid outer core while P-waves pass straight through.
A common point of confusion is expecting P-waves and S-waves to travel at similar speeds. They don't, P-waves are always faster (typically 1.6 to 1.8 times the S-wave velocity in most crustal rocks), which is exactly why seismographs record a clear P-wave arrival first, followed by the S-wave, the time gap between them is the standard method for estimating distance to an earthquake's source.
This calculator computes P-wave velocity, S-wave velocity, their ratio, and the derived Poisson's ratio from your material's bulk modulus, shear modulus, and density, and plots how both wave velocities change with density at fixed elastic moduli.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Granite (Classic Poisson Solid)
K = 50 GPa, G = 30 GPa, ρ = 2700 kg/m³
Example 2 — Sandstone
K = 15 GPa, G = 8 GPa, ρ = 2200 kg/m³
Example 3 — Upper Mantle Peridotite
K = 130 GPa, G = 80 GPa, ρ = 3300 kg/m³
❓ Frequently Asked Questions
🔗 Related Calculators
What is the difference between P-waves and S-waves?
P-waves (primary, or compressional waves) vibrate the medium in the direction of travel and can pass through solids, liquids, and gases. S-waves (secondary, or shear waves) vibrate perpendicular to the direction of travel and can only pass through solids, since liquids and gases have no shear strength.
What is the formula for P-wave and S-wave velocity?
Vp = sqrt((K + 4/3*G) / rho) and Vs = sqrt(G / rho), where K is the bulk modulus, G is the shear modulus, and rho is density, all in consistent SI units (pascals and kg/m3), giving velocity in meters per second.
Why do P-waves always travel faster than S-waves?
Vp always exceeds Vs because the P-wave formula includes the full bulk modulus term (K + 4/3*G) under the square root, while the S-wave formula only has G. Since K and G are both positive, the P-wave's numerator is always larger, mathematically guaranteeing Vp > Vs in any elastic solid.
Why can't S-waves travel through Earth's outer core?
S-waves require shear rigidity (a nonzero shear modulus G) to propagate, and liquids have essentially zero shear modulus, they can't resist shear deformation. Since S-waves from earthquakes are observed to disappear behind Earth's outer core, this was the key historical evidence that the outer core is liquid.
What is the Vp/Vs ratio used for?
The Vp/Vs ratio (and the Poisson's ratio derived from it) is a key diagnostic in seismology and reservoir geophysics, since it's sensitive to rock type, porosity, and pore fluid content in ways that Vp or Vs alone are not, higher Vp/Vs often indicates fluid saturation or softer, more ductile material.
How is Poisson's ratio related to Vp/Vs?
Poisson's ratio sigma = ((Vp/Vs)^2 - 2) / (2*((Vp/Vs)^2 - 1)). A Vp/Vs ratio of exactly sqrt(3) (about 1.732) corresponds to sigma = 0.25, the widely cited 'Poisson solid' reference value used as a rough benchmark for many common crustal rocks.
What are typical Vp and Vs values for common rocks?
Approximate ranges: sedimentary rocks (sandstone, shale) Vp 2,000 to 4,500 m/s; crystalline rocks (granite, basalt) Vp 5,000 to 6,500 m/s; the upper mantle Vp is typically 8,000 to 8,500 m/s. S-wave velocities are typically 50 to 60% of the corresponding P-wave velocity.
How does porosity affect P-wave and S-wave velocity?
Higher porosity generally lowers both bulk and shear modulus (more void space, less rigid rock matrix), reducing both Vp and Vs. Fluid saturation in those pores raises the effective bulk modulus (fluids resist compression) while leaving shear modulus nearly unchanged, which is why Vp is more fluid-sensitive than Vs.
Does this formula apply to soil as well as rock?
Yes, the same isotropic elastic-medium equations apply to any material describable by bulk and shear moduli, including soils, though soil moduli are typically far lower than rock (often megapascals rather than gigapascals) and highly sensitive to confining pressure and saturation.
What units does this calculator use?
Bulk modulus K and shear modulus G are entered in gigapascals (GPa), density in kilograms per cubic meter (kg/m3). Velocities are returned in meters per second, with the Vp/Vs ratio and Poisson's ratio reported as dimensionless numbers.