2.1 Seismic Wave Propagation
Elastic Waves in the Earth
Seismic waves are elastic disturbances that propagate through the Earth following earthquakes, explosions, or controlled sources. They provide the primary tool for imaging Earth's interior, as their velocities depend on the elastic moduli and density of the material they traverse. The two fundamental body wave types -- compressional (P) and shear (S) waves -- respond differently to the physical state of the medium, enabling us to distinguish solid from liquid regions.
The mathematical framework governing seismic wave propagation is rooted in continuum mechanics and the theory of elasticity. By solving the elastic wave equation under appropriate boundary conditions, we can predict wave travel times, amplitudes, and waveforms observed at seismograph stations around the globe.
The Elastic Wave Equation
For a homogeneous, isotropic, linearly elastic medium, the equation of motion relates the displacement field u to the elastic properties via the Lamé parameters λ and μ (shear modulus). Starting from Newton's second law applied to a continuous medium and using Hooke's law for the stress-strain relation:
\(\rho \frac{\partial^2 \mathbf{u}}{\partial t^2} = (\lambda + 2\mu)\,\nabla(\nabla \cdot \mathbf{u}) - \mu\,\nabla \times (\nabla \times \mathbf{u})\)
where ρ is density, u is the displacement vector, λ is the first Lamé parameter, and μ is the shear modulus (rigidity). Using the Helmholtz decomposition, the displacement field can be separated into irrotational (compressional) and solenoidal (shear) components:
\(\mathbf{u} = \nabla\phi + \nabla \times \boldsymbol{\psi}\)
Substituting this decomposition into the wave equation yields two independent scalar wave equations: one for the scalar potential φ (P-waves) and one for the vector potential ψ (S-waves), each propagating at different characteristic velocities.
P-wave and S-wave Velocities
The compressional (P) wave velocity and the shear (S) wave velocity are determined by the elastic moduli and density of the medium:
\(v_P = \sqrt{\frac{\lambda + 2\mu}{\rho}} = \sqrt{\frac{K + \frac{4}{3}\mu}{\rho}}\)
\(v_S = \sqrt{\frac{\mu}{\rho}}\)
where K is the bulk modulus (incompressibility). Since K and μ are always positive for real materials, P-waves are always faster than S-waves. The ratio of P-wave to S-wave velocity is governed by Poisson's ratio ν:
\(\frac{v_P}{v_S} = \sqrt{\frac{2(1-\nu)}{1-2\nu}}\)
For a Poisson solid (ν = 0.25), vP/vS = √3 ≈ 1.73. Typical values in the Earth:
| Material | vP (km/s) | vS (km/s) | vP/vS | ν |
|---|---|---|---|---|
| Granite | 5.5 - 6.0 | 3.0 - 3.5 | 1.71 | 0.24 |
| Basalt | 5.5 - 6.5 | 3.0 - 3.5 | 1.80 | 0.28 |
| Upper Mantle | 8.0 - 8.5 | 4.5 - 4.7 | 1.78 | 0.27 |
| Outer Core (liquid) | 8.0 - 10.3 | 0 | — | 0.50 |
| Inner Core | 11.0 - 11.3 | 3.5 - 3.7 | 3.1 | 0.44 |
Snell's Law & Ray Theory
In a medium with velocity that varies with depth, seismic rays bend according to Snell's law. At an interface between two layers with different velocities, the ray parameter p is conserved:
\(p = \frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2} = \frac{\sin\theta(r)}{v(r)} = \text{constant}\)
The ray parameter p has dimensions of slowness (s/km) and is constant along the entire ray path. For a spherically symmetric Earth, the modified form uses:
\(p = \frac{r \sin\theta(r)}{v(r)} = \frac{r_0}{v(r_0)}\)
where r is the radial distance from Earth's center, θ(r) is the angle of incidence, and r0 is the turning-point radius where the ray becomes horizontal (θ = 90°).
Reflection
When a P or S wave encounters a discontinuity, part of the energy reflects. At a solid-liquid boundary (e.g., CMB), P converts entirely to reflected P and transmitted P, but transmitted S vanishes since μ = 0 in liquids.
Mode Conversion
At solid-solid interfaces, an incident P-wave generates reflected P and S waves, and transmitted P and S waves. The partition of energy is governed by the Zoeppritz equations, which depend on the impedance contrast.
Travel Times & Shadow Zones
The travel time T of a seismic ray as a function of epicentral distance Δ is given by integrating along the ray path. For a radially symmetric Earth:
\(T(\Delta) = 2\int_{r_0}^{R} \frac{r\,dr}{v(r)\sqrt{r^2/v^2(r) - p^2}} \quad , \quad \Delta(p) = 2\int_{r_0}^{R} \frac{p\,v(r)\,dr}{r\sqrt{r^2/v^2(r) - p^2}}\)
The T(Δ) curve reveals the internal structure of the Earth through several key features:
P-wave Shadow Zone (104° - 140°)
The liquid outer core refracts P-waves downward due to the sharp velocity decrease at the CMB (Gutenberg discontinuity). Waves entering the core are bent so strongly that no direct P arrivals are observed between 104° and 140° epicentral distance. Beyond 140°, the core-refracted wave (PKP) re-emerges.
S-wave Shadow Zone (beyond ~104°)
Because S-waves cannot propagate through liquids (μ = 0), no direct S arrivals are observed beyond ~104°. This was the decisive evidence that the outer core is liquid, first recognized by Oldham (1906) and confirmed by Gutenberg (1914).
Seismic Phase Nomenclature
P = compressional in mantle, K = compressional in outer core, I = compressional in inner core, S = shear in mantle, J = shear in inner core, c = reflection from CMB, i = reflection from ICB. Example: PKiKP = P through mantle, K through outer core, reflected at ICB, K back through outer core, P through mantle.
Body Waves vs Surface Waves
Seismic waves fall into two broad categories: body waves that travel through the interior, and surface waves that propagate along the free surface.
Body Waves
- ▶P-waves (Primary): Longitudinal, particle motion parallel to propagation. Travel through solids, liquids, and gases. Fastest wave type.
- ▶S-waves (Secondary): Transverse, particle motion perpendicular to propagation. Travel only through solids (μ > 0). Split into SV (vertical) and SH (horizontal).
Surface Waves
- ▶Rayleigh waves: Retrograde elliptical particle motion in the sagittal plane. Velocity ≈ 0.92 vS in a homogeneous half-space. Dispersive in layered media.
- ▶Love waves: Horizontally polarized shear waves trapped in a low-velocity surface layer. Require a velocity increase with depth. Always dispersive.
Surface waves have larger amplitudes and longer durations than body waves at teleseismic distances because their energy spreads over a 2D wavefront (amplitude decays as 1/√r) rather than a 3D wavefront (amplitude decays as 1/r). They cause the most destruction in earthquakes.
Attenuation & Quality Factor Q
Real Earth materials are not perfectly elastic. Anelastic processes (grain boundary sliding, dislocation motion, partial melting) cause seismic energy to be dissipated as heat. The dimensionless quality factor Q characterizes this intrinsic attenuation:
\(Q = \frac{2\pi E}{\Delta E} \quad \Rightarrow \quad A(x) = A_0 \, \exp\!\left(-\frac{\omega x}{2 v Q}\right)\)
where E is the peak elastic energy stored per cycle, ΔE is the energy dissipated per cycle, ω is angular frequency, x is distance, and v is wave velocity. High Q means low attenuation (efficient propagation); low Q means high attenuation (energy loss).
| Region | QP | QS (Qμ) | Implication |
|---|---|---|---|
| Upper Mantle (LVZ) | ~150 | ~80 | High attenuation, partial melting |
| Lower Mantle | ~700 | ~300 | Low attenuation, cold & stiff |
| Outer Core | ~10,000 | — | Very low attenuation (liquid, no shear) |
| Inner Core | ~400 | ~100 | Moderate attenuation, near melting point |
Key Concepts Summary
vP > vS
P-waves always faster; √3 ratio for Poisson solid
104° - 140°
P-wave shadow zone from core refraction
Q ∼ 80
Low Qμ in LVZ indicates partial melt