Key Geophysical Equations to Know for Intro to Geophysics

These key geophysical equations provide essential insights into Earth's physical processes. They explain gravity, seismic wave behavior, heat transfer, and fluid dynamics, helping us understand the planet's structure and the forces shaping it. Each equation plays a vital role in geophysics.

1. Gravitational acceleration (g = GM/r^2)

   • Describes the force of gravity acting on an object at a distance r from the center of a mass M.
   • G is the gravitational constant, approximately 6.674 × 10^-11 N(m/kg)^2.
   • The equation shows that gravitational acceleration decreases with the square of the distance from the mass.

2. Seismic wave velocity equations (P-waves and S-waves)

   • P-waves (primary waves) are compressional waves that travel fastest through solids, liquids, and gases.
   • S-waves (secondary waves) are shear waves that only travel through solids and are slower than P-waves.
   • The velocity of seismic waves is influenced by the material properties, such as density and elasticity.

3. Snell's Law for seismic wave refraction

   • Describes how seismic waves change direction when they pass through different media.
   • The law is expressed as n1sin(θ1) = n2sin(θ2), where n is the refractive index and θ is the angle of incidence/refraction.
   • Important for understanding wave propagation in the Earth's layered structure.

4. Heat flow equation (Fourier's Law)

   • States that the heat transfer rate through a material is proportional to the negative gradient of temperature.
   • Mathematically expressed as q = -k*(dT/dx), where q is heat flow, k is thermal conductivity, and dT/dx is the temperature gradient.
   • Essential for studying geothermal gradients and heat transfer in the Earth's crust.

5. Magnetic dipole equation

   • Describes the magnetic field generated by a magnetic dipole moment (m).
   • The magnetic field (B) at a distance r from the dipole is given by B = (μ0/4π) * (2m cos(θ)/r^3), where μ0 is the permeability of free space.
   • Important for understanding the Earth's magnetic field and magnetic anomalies.

6. Bouguer gravity anomaly equation

   • Used to correct gravity measurements for the effects of topography and density variations in the Earth's crust.
   • The equation is Δg = g_obs - g_ref - Δg_topo, where g_obs is observed gravity, g_ref is reference gravity, and Δg_topo accounts for topographic effects.
   • Helps in identifying subsurface geological structures.

7. Radioactive decay equation

   • Describes the process by which unstable isotopes lose energy by emitting radiation.
   • The decay is characterized by the equation N(t) = N0 * e^(-λt), where N0 is the initial quantity, λ is the decay constant, and t is time.
   • Fundamental for dating geological materials and understanding nuclear processes in the Earth.

8. Darcy's Law for fluid flow in porous media

   • Describes the flow of fluid through a porous medium, stating that flow rate is proportional to the pressure gradient.
   • Mathematically expressed as Q = -kA(dP/dx), where Q is the flow rate, k is permeability, A is cross-sectional area, and dP/dx is the pressure gradient.
   • Key for understanding groundwater movement and hydrocarbon reservoirs.

9. Navier-Stokes equations for fluid dynamics

   • A set of nonlinear partial differential equations that describe the motion of fluid substances.
   • They account for viscosity, pressure, and external forces acting on the fluid.
   • Essential for modeling complex fluid flows in geophysical processes, such as ocean currents and atmospheric dynamics.

10. Elastic wave equation

• Governs the propagation of elastic waves in a medium, relating stress and strain.
• The equation is expressed as ∇²u = (1/c²)(∂²u/∂t²), where u is the displacement, c is the wave speed, and ∇² is the Laplacian operator.
• Important for understanding seismic wave propagation and material response to stress.