Key Geophysical Equations

Why This Matters

Geophysics is fundamentally a quantitative science—you're using mathematical relationships to probe a planet you can't directly see or touch. These equations aren't just formulas to memorize; they're the tools that let you interpret gravity surveys, predict earthquake wave arrivals, date ancient rocks, and model everything from groundwater flow to mantle convection. On exams, you're being tested on whether you understand why each equation works and when to apply it to real Earth problems.

The equations in this guide fall into distinct physical categories: wave propagation, potential fields, transport phenomena, and time-dependent processes. Recognizing which category a problem falls into will help you select the right approach. Don't just memorize the symbols—know what physical principle each equation captures and how changing one variable affects the system. That's what separates a student who can solve problems from one who just recognizes formulas.

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Wave Propagation Equations

Seismic waves are your primary window into Earth's interior. These equations describe how mechanical energy travels through rock, changes direction at boundaries, and reveals the properties of materials it passes through. The key insight: wave velocity depends on the elastic properties and density of the medium.

Seismic Wave Velocity Equations (P-waves and S-waves)

• P-wave velocity $V_p = \sqrt{\frac{K + \frac{4}{3}\mu}{\rho}}$—where $K$ is bulk modulus, $\mu$ is shear modulus, and $\rho$ is density
• S-wave velocity $V_s = \sqrt{\frac{\mu}{\rho}}$—S-waves require shear strength, which is why they cannot travel through liquids ($\mu = 0$)
• Shadow zones result from these velocity relationships; the liquid outer core blocks S-waves entirely and refracts P-waves, creating predictable gaps in seismic recordings

Snell's Law for Seismic Wave Refraction

• $\frac{\sin\theta_1}{V_1} = \frac{\sin\theta_2}{V_2}$—waves bend toward the normal when entering slower material, away from normal when entering faster material
• Critical angle occurs when $\sin\theta_c = \frac{V_1}{V_2}$, producing head waves that travel along layer boundaries—essential for refraction seismology
• Layered Earth interpretation depends on this equation; velocity generally increases with depth, causing rays to curve back toward the surface

Elastic Wave Equation

• $\nabla^2 u = \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}$—this is the fundamental wave equation relating spatial curvature ($\nabla^2$) to temporal acceleration
• Displacement field $u$ describes how particles move as the wave passes; $c$ is the phase velocity determined by material properties
• Boundary conditions applied to this equation predict reflections, transmissions, and surface wave behavior—the mathematical foundation for all seismic modeling

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Potential Field Equations

Gravity and magnetic fields are "potential fields"—they exist everywhere in space and can be measured remotely at Earth's surface. These equations let you work backward from surface measurements to infer subsurface structure. The unifying principle: field strength decreases with distance according to inverse-square or inverse-cube laws.

Gravitational Acceleration

• $g = \frac{GM}{r^2}$—gravitational acceleration decreases with the square of distance from the center of mass
• $G = 6.674 \times 10^{-11}$ N·m²/kg² is the universal gravitational constant; small variations in $g$ at Earth's surface reveal density anomalies below
• Reference value at Earth's surface is approximately 9.81 m/s², but local variations of a few milligals ($10^{-5}$ m/s²) indicate ore bodies, basins, or other structures

Bouguer Gravity Anomaly Equation

• $\Delta g_{Bouguer} = g_{obs} - g_{ref} - \Delta g_{FA} - \Delta g_{Bouguer\ correction}$—systematically removes known effects to isolate subsurface density variations
• Free-air correction accounts for elevation above the reference ellipsoid; Bouguer correction removes the gravitational effect of rock mass between you and sea level
• Negative anomalies typically indicate low-density material (sedimentary basins, mountain roots); positive anomalies suggest dense material (mafic intrusions, uplifted mantle)

Magnetic Dipole Equation

• $B = \frac{\mu_0}{4\pi} \cdot \frac{2m\cos\theta}{r^3}$—magnetic field strength falls off with the cube of distance, making it more sensitive to shallow sources than gravity
• $\mu_0 = 4\pi \times 10^{-7}$ T·m/A is the permeability of free space; $m$ is the magnetic dipole moment
• Earth's field approximates a geocentric axial dipole, but magnetic anomalies from crustal rocks reveal everything from ore deposits to seafloor spreading history

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Heat and Mass Transport Equations

These equations describe how energy and matter move through Earth materials—whether it's heat conducting through the lithosphere or water flowing through an aquifer. The common thread: transport rate is proportional to a gradient (temperature, pressure, or concentration) and a material property (conductivity, permeability).

Heat Flow Equation (Fourier's Law)

• $q = -k\frac{dT}{dx}$—heat flows from hot to cold, with the rate proportional to the temperature gradient and thermal conductivity $k$
• Negative sign indicates heat flows in the direction of decreasing temperature; $q$ is measured in W/m² (heat flux)
• Geothermal gradient averages about 25-30°C/km in continental crust, but varies dramatically near mid-ocean ridges, subduction zones, and areas with radioactive heat production

Darcy's Law for Fluid Flow in Porous Media

• $Q = -\frac{kA}{\mu}\frac{dP}{dx}$—volumetric flow rate depends on permeability $k$, cross-sectional area $A$, fluid viscosity $\mu$, and pressure gradient
• Permeability is an intrinsic rock property (measured in m² or darcys); don't confuse it with porosity, which is the fraction of void space
• Applications include groundwater hydrology, petroleum reservoir engineering, and modeling fluid circulation in geothermal systems and mid-ocean ridge hydrothermal vents

Navier-Stokes Equations for Fluid Dynamics

• $\rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla\mathbf{v}\right) = -\nabla P + \mu\nabla^2\mathbf{v} + \rho\mathbf{g}$—balances inertia, pressure gradients, viscous forces, and gravity
• Nonlinear terms ($\mathbf{v} \cdot \nabla\mathbf{v}$) make these equations notoriously difficult to solve; most geophysical applications use simplified versions
• Mantle convection models adapt these equations for highly viscous flow ($Re \ll 1$), while atmospheric and ocean models must handle turbulence and rotation (Coriolis effects)

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Time-Dependent Processes

Some geophysical phenomena unfold over time in predictable ways. These equations capture exponential decay and other temporal evolution—essential for geochronology and understanding long-term Earth processes.

Radioactive Decay Equation

• $N(t) = N_0 e^{-\lambda t}$—the number of parent atoms decreases exponentially with time; $\lambda$ is the decay constant
• Half-life $t_{1/2} = \frac{\ln 2}{\lambda}$—the time for half the parent atoms to decay; ranges from fractions of a second to billions of years depending on the isotope
• Geochronology applications include $^{14}$C dating (up to ~50,000 years), K-Ar dating (millions to billions of years), and U-Pb dating (the gold standard for ancient rocks and zircons)

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Quick Reference Table

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Self-Check Questions

1. Both Fourier's Law and Darcy's Law describe transport proportional to a gradient. What is the key physical difference between what is being transported, and how does this affect their applications in geothermal systems?

2. Why do S-waves produce a shadow zone at the outer core while P-waves are only refracted? Which term in the seismic velocity equations explains this?

3. Compare the distance dependence of gravity ($1/r^2$) and magnetic ($1/r^3$) fields. If you're surveying for a shallow ore body versus a deep sedimentary basin, which method would be more sensitive to each target?

4. The radioactive decay equation uses the decay constant $\lambda$, but geologists often report half-life instead. Derive the relationship between them, and explain why $^{14}$C dating can't be used for rocks older than about 50,000 years.

5. An FRQ asks you to explain how seismologists determined that Earth's outer core is liquid. Which equations from this guide would you use, and what specific observations support this conclusion?