P-Wave Velocity (Primary Waves) Equation, Derivation & Rearrangements

Core idea

Overview

Primary waves (P-waves) are compressional seismic waves that travel through the Earth's interior by pushing and pulling material in the direction of propagation. This equation mathematically links their velocity to the elastic properties—bulk and shear moduli—and the mass density of the medium through which they move.

When to use: This equation is used to calculate the speed of the fastest seismic waves through Earth's layers, assuming the material behaves as an isotropic elastic solid. It is fundamental in earthquake seismology and reflection seismology for interpreting subsurface structures based on seismic travel times.

Why it matters: Since P-waves are the first to arrive at seismic monitoring stations, their velocity is critical for early warning systems that alert cities before more damaging waves arrive. Measuring these velocities allows geophysicists to determine the physical state of the Earth's interior, such as identifying the liquid nature of the outer core where the shear modulus drops to zero.

Symbols

Variables

v = P-Wave Velocity, K = Bulk Modulus, G = Shear Modulus, $ρ$ = Density

v

P-Wave Velocity

m/s

K

Bulk Modulus

Pa

G

Shear Modulus

Pa

ρ

Density

k g / m^{3}

Walkthrough

Derivation

Formula: P-Wave Velocity

P-wave velocity depends on elastic stiffness (bulk and shear moduli) and density of the medium.

Medium is homogeneous and isotropic (simple elastic model).
Small-amplitude linear elastic waves.

1

State the elastic-wave relation:

Higher stiffness increases wave speed; higher density reduces speed because more mass must be accelerated.

v = \frac{K + \frac{4}{3} G}{ρ}

Result

v = \frac{K + \frac{4}{3} G}{ρ}

Source: Geophysics — Elastic Waves (intro)

Free formulas

Rearrangements

Solve for $v$

Make v the subject

v = \frac{K + \frac{4}{3} G}{ρ}

v is already the subject of the formula.

Difficulty: 1/5

Solve for $ρ$

Make rho the subject

ρ = \frac{K + \frac{4}{3} G}{v ^{2}}

To make $ρ$ (density) the subject, first square both sides to remove the square root, then multiply by $ρ$ to clear the denominator, and finally divide by $v^{2}$ .

Difficulty: 2/5

Solve for $K$

Make K the subject

K = ρ v^{2} - \frac{4}{3} G

To make K (Bulk Modulus) the subject, first square both sides to remove the square root, then multiply by density, and finally subtract the shear modulus term.

Difficulty: 2/5

Solve for $G$

P-Wave Velocity (Primary Waves) - Make G the subject

G = \frac{3}{4} (ρ v^{2} - K)

To make G (Shear Modulus) the subject, first square both sides to remove the square root, then clear the denominator $ρ$ (density). Isolate the term containing G by subtracting K (Bulk Modulus), and finally multiply by $\frac{3}{4}$ .

Difficulty: 4/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows an inverse square root relationship where velocity decreases as density increases, causing the curve to approach the horizontal axis as density grows and rise toward infinity as density nears zero. For a geology student, this means that materials with lower density allow for significantly faster compressional wave travel, while high-density materials act to slow these waves down. The most important feature of this curve is that the velocity never reaches zero, which indicates that even in extremely dense media, compressional waves will always maintain some measurable speed.

Graph type: power_law

Why it behaves this way

Intuition

A pulse traveling through a line of stiff springs and heavy beads, where stiffer springs (moduli) snap back faster and heavier beads (density) slow the signal down.

K

Bulk modulus

Measures the material's resistance to uniform compression; a higher bulk modulus means the material is less compressible and transmits pressure pulses more rapidly.

G

Shear modulus (rigidity)

Measures the resistance to shape deformation; because P-waves involve longitudinal strain, the material's rigidity contributes to the total restoring force.

ρ

Mass density

Represents the inertia of the medium; a higher density means more mass must be moved by the wave, which tends to slow the propagation speed.

Signs and relationships

sqrt(...): The square root arises from the wave equation where the velocity is proportional to the square root of the ratio of the elastic modulus to the density.
1/ρ: Density is in the denominator because inertia opposes the acceleration of the medium; for a constant stiffness, a heavier material responds more slowly to the passing wave.
K + 4/3 G: The addition indicates that both volumetric and shear stiffness contribute to the total longitudinal stiffness (the P-wave modulus) of the solid.

Free study cues

Insight

Canonical usage

This equation is typically used with SI units for all quantities to ensure dimensional consistency and obtain velocity in meters per second.

Common confusion

A common mistake is mixing units from different systems (e.g., density in g/cm3 with moduli in Pa) without proper conversion, leading to incorrect velocity calculations.

Unit systems

$v$ m/s · Velocity of the P-wave.

$K$ Pa (Pascals) or N/m^2 · Bulk modulus, representing resistance to compression.

$G$ Pa (Pascals) or N/m^2 · Shear modulus, representing resistance to shear. For fluids, G is zero.

$ρ$ kg/m^3 · Mass density of the medium.

Ballpark figures

Quantity:
Quantity:

One free problem

Practice Problem

A granite sample has a bulk modulus (K) of 35 GPa and a shear modulus (G) of 25 GPa. Given a density (rho) of 2700 kg/m³, calculate the velocity (v) of the primary wave passing through this rock.

Bulk Modulus35000000000 Pa

Shear Modulus25000000000 Pa

Density2700 kg/m^3

Solve for: $v$

Hint: Calculate the sum of K and 4/3 times G first, then divide by density before taking the square root.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In a physics application involving P-Wave Velocity (Primary Waves), P-Wave Velocity (Primary Waves) is used to calculate P-Wave Velocity from Bulk Modulus, Shear Modulus, and Density. 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 all elastic moduli are converted from GPa or MPa to Pascals for unit consistency.
Recognize that in liquids, the shear modulus (G) is zero, simplifying the calculation to the root of K/rho.
Note that while higher density theoretically decreases velocity, it is often accompanied by even larger increases in stiffness in Earth's deep interior.

Avoid these traps

Common Mistakes

Using only one modulus in the formula.
Incorrect units for density (kg/ $m^{3}$ is standard).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

P-wave velocity depends on elastic stiffness (bulk and shear moduli) and density of the medium.

This equation is used to calculate the speed of the fastest seismic waves through Earth's layers, assuming the material behaves as an isotropic elastic solid. It is fundamental in earthquake seismology and reflection seismology for interpreting subsurface structures based on seismic travel times.

Since P-waves are the first to arrive at seismic monitoring stations, their velocity is critical for early warning systems that alert cities before more damaging waves arrive. Measuring these velocities allows geophysicists to determine the physical state of the Earth's interior, such as identifying the liquid nature of the outer core where the shear modulus drops to zero.

Using only one modulus in the formula. Incorrect units for density (kg/m^3 is standard).