8.2 Seismic Wave Types
Seismic waves are elastic disturbances that propagate through and along the surface of the Earth, carrying energy released by earthquakes, explosions, and other sources. The study of seismic wave propagation is the primary tool for probing Earth's internal structure, from the thin crust to the inner core. Seismic waves divide into two broad categories: body waves that travel through the Earth's interior, and surface waves that are confined to the near-surface region and decay exponentially with depth.
Body Waves
Body waves propagate through the three-dimensional volume of the Earth, following ray paths governed by Snell's law as they traverse regions of varying elastic properties. Two distinct types exist, corresponding to the two independent elastic deformation modes of a solid medium:
P-Waves (Primary / Compressional)
P-waves are longitudinal compressional waves in which particle motion is parallel to the direction of wave propagation. They alternately compress and dilate the medium, analogous to sound waves in air. P-waves are the fastest seismic waves, arriving first at a seismometer (hence “primary”). They can propagate through solids, liquids, and gases because all materials resist volumetric compression.
P-Wave Velocity
\[ v_P = \sqrt{\frac{K + \frac{4}{3}\mu}{\rho}} \]
where K is the bulk modulus (incompressibility), μ is the shear modulus (rigidity), and ρ is the density. Typical crustal values: vP ≈ 5–7 km/s; upper mantle: 7.8–8.2 km/s; outer core (liquid): 8–10 km/s; inner core: ~11 km/s. Maximum vP ≈ 13.7 km/s at the base of the mantle.
In fluids where μ = 0, the P-wave velocity reduces to \( v_P = \sqrt{K/\rho} \), which is the speed of sound. The sharp drop in vP at the core–mantle boundary (from ~13.7 to ~8.0 km/s) produces the P-wave shadow zone between 104° and 140° epicentral distance, providing direct evidence for Earth's liquid outer core.
S-Waves (Secondary / Shear)
S-waves are transverse shear waves in which particle motion is perpendicular to the propagation direction. They deform the medium by shearing without changing its volume. S-waves are slower than P-waves and arrive second (“secondary”). Crucially, S-waves cannot propagate through fluids because fluids have zero shear modulus (μ = 0).
S-Wave Velocity
\[ v_S = \sqrt{\frac{\mu}{\rho}} \]
Typical crustal values: vS ≈ 3–4 km/s; upper mantle: 4.3–4.7 km/s; lower mantle: up to ~7.3 km/s. S-waves are completely blocked by the liquid outer core, creating an S-wave shadow zone beyond 104° epicentral distance. This observation was the definitive proof (Oldham, 1906; Gutenberg, 1914) that Earth's outer core is liquid.
S-waves can be polarized into two components: SV (motion in the vertical plane containing the ray) and SH (motion in the horizontal plane). These components behave differently at interfaces: SV converts to P at boundaries, while SH does not. The vP/vS ratio in typical crustal rocks is approximately 1.73 (Poisson solid), a useful diagnostic for lithology and fluid saturation.
| Region | vP (km/s) | vS (km/s) | vP/vS | ρ (kg/m³) |
|---|---|---|---|---|
| Upper crust | 5.8–6.3 | 3.2–3.6 | 1.73 | 2700 |
| Lower crust | 6.5–7.2 | 3.6–4.0 | 1.78 | 2900 |
| Uppermost mantle | 7.9–8.2 | 4.4–4.7 | 1.80 | 3300 |
| Lower mantle | 10.5–13.7 | 5.9–7.3 | 1.82 | 4400–5500 |
| Outer core (liquid) | 8.0–10.4 | 0 | — | 9900–12200 |
| Inner core (solid) | 11.0–11.3 | 3.5–3.7 | 3.1 | 12800–13100 |
Values from PREM (Dziewonski & Anderson, 1981) and AK135 (Kennett, Engdahl & Buland, 1995).
Surface Waves
Surface waves are guided by Earth's free surface and decay exponentially with depth. They travel more slowly than body waves but carry the largest amplitudes at teleseismic distances, often dominating seismograms of shallow earthquakes and causing the most structural damage. Two fundamental types exist:
Love Waves (L or LQ)
Named after A.E.H. Love, who predicted their existence mathematically in 1911. Love waves involve purely horizontal, transverse (SH) motion perpendicular to the propagation direction. They require a velocity increase with depth (layered structure) to exist; they cannot propagate in a uniform half-space. Love waves are formed by constructive interference of multiply reflected SH waves trapped in the low-velocity surface layer.
The velocity of Love waves lies between the S-wave velocity of the surface layer and that of the underlying half-space: vS,layer < vLove < vS,halfspace. Love waves are dispersive: longer-period waves sample deeper (faster) structure and therefore travel faster. This dispersion makes Love waves powerful probes of crustal and upper-mantle shear-velocity structure.
Rayleigh Waves (R or LR)
Predicted by Lord Rayleigh in 1885 for a homogeneous elastic half-space. Rayleigh waves involve coupled P-SV motion in the vertical plane of propagation, producing a retrograde elliptical particle motion at the surface (particles move backward at the top of the ellipse, in the direction opposite to wave propagation). The amplitude decays exponentially with depth, with significant motion confined to within approximately one wavelength of the surface.
In a homogeneous half-space, the Rayleigh wave velocity is approximately:
\[ v_R \approx 0.92 \, v_S \]
For a Poisson solid (ν = 0.25), the exact ratio is 0.9194. In a layered Earth, Rayleigh waves are dispersive, with longer periods sensing deeper structure. Fundamental-mode Rayleigh waves at periods of 20–200 s are the primary data for global tomographic models of mantle shear-velocity structure.
Dispersion & Velocity Concepts
In a layered Earth, surface wave velocity depends on frequency because waves of different periods sample different depth ranges. This phenomenon is called dispersion. Short-period waves are sensitive to shallow structure; long-period waves penetrate deeper and sense faster material. The result is that a surface wave packet spreads out as it propagates, with different frequency components arriving at different times.
Group and Phase Velocity
\[ U = c + k \, \frac{dc}{dk} \]
where U is the group velocity (speed of energy transport), c = ω/k is the phase velocity (speed of individual crests), and k is the wavenumber. When dc/dk < 0 (normal dispersion), the group velocity is less than the phase velocity. Measuring group and phase velocity dispersion curves constrains the depth-dependent velocity structure beneath the propagation path.
Plane Wave Solution
\[ u(\mathbf{x}, t) = A \, e^{\,i(\mathbf{k} \cdot \mathbf{x} - \omega t)} \]
The fundamental plane-wave solution to the elastic wave equation, where A is amplitude, k is the wave vector, x is position, and ω = 2πf is angular frequency. The dispersion relation ω(k) encodes all information about wave propagation in the medium.
Free Oscillations (Normal Modes)
Very large earthquakes (Mw > 7.5) excite the Earth's free oscillations — standing-wave patterns in which the entire planet vibrates as a resonant body. These normal modes were first observed after the 1960 Chile earthquake (Mw 9.5) and provide constraints on Earth's deep radial structure, including density and anelasticity.
Spheroidal Modes (nSl)
Involve radial and horizontal motion, coupled P-SV deformation. They are sensitive to both compressional and shear structure, as well as density. The gravest mode, 0S2, has a period of approximately 54 minutes and involves a “football-shaped” oscillation of the entire planet. The radial mode 0S0 (“breathing mode”) has a period of ~20.5 minutes.
Toroidal Modes (nTl)
Involve purely horizontal, rotational (SH) motion with no radial displacement. They are sensitive only to shear-velocity and density structure. Toroidal modes do not exist in a fluid; their observation in the core would indicate inner-core solidity. The gravest toroidal mode, 0T2, has a period of approximately 44 minutes.
Normal-mode observations provide the most complete constraints on Earth's 1-D radial structure. The Preliminary Reference Earth Model (PREM; Dziewonski & Anderson, 1981) was constructed primarily from normal-mode eigenfrequencies combined with body-wave travel times. Mode splitting due to Earth's rotation and 3-D heterogeneity further constrains lateral structure and inner-core anisotropy.
Attenuation & the Quality Factor Q
As seismic waves propagate, they lose energy to anelastic processes (grain-boundary friction, fluid flow in pore spaces, dislocation motion). This intrinsic attenuation causes amplitude decay beyond the geometric spreading expected for an elastic medium. The attenuation is quantified by the dimensionless quality factor Q:
Amplitude Attenuation
\[ A(r) = A_0 \, r^{-1} \, \exp\!\left(-\frac{\pi f \, r}{Q \, v}\right) \]
where A0 is the source amplitude, r is the propagation distance, f is frequency, v is wave velocity, and Q is the quality factor. The r−1 term accounts for geometric spreading (body waves); the exponential term represents intrinsic attenuation. Low Q means high attenuation; high Q means low attenuation.
| Region | QP | QS | Interpretation |
|---|---|---|---|
| Asthenosphere | ~200 | ~80 | High attenuation; partial melt, high T |
| Upper mantle (lid) | ~600 | ~300 | Moderate attenuation |
| Lower mantle | ~1000 | ~500 | Low attenuation; cold, rigid |
| Outer core | ~10,000 | — | Very low attenuation for P (liquid, no S) |
| Inner core | ~600 | ~100 | Moderate QS; possibly due to partial melt |
Converted Phases & Receiver Functions
When a seismic wave encounters a sharp velocity discontinuity (such as the Moho or the 410-km discontinuity), it partially converts between P and S modes. The most commonly exploited conversion is the P-to-S conversion at the receiver side: an incoming teleseismic P-wave generates a converted S-wave (Ps) at each discontinuity beneath the recording station.
The receiver function technique isolates these converted phases by deconvolving the vertical component (dominated by the incoming P) from the radial component (which contains the Ps conversions). The time delay between the direct P arrival and the Ps conversion is proportional to the depth of the discontinuity and the velocity structure above it:
\( t_{Ps} - t_P = H \left( \frac{1}{v_S} - \frac{1}{v_P} \right) \cos(i) \)
where H is the depth to the discontinuity and i is the incidence angle. This method has mapped crustal thickness globally (e.g., EARS database) and resolved the 410-km, 520-km, and 660-km mantle transition-zone discontinuities beneath thousands of stations worldwide. Receiver functions are a cornerstone of modern passive-source seismology.
Key Takeaways
- P-waves (compressional) travel through all materials; S-waves (shear) are blocked by fluids
- The absence of S-waves beyond 104° proved the liquid outer core
- Love waves (SH) require layering; Rayleigh waves (P-SV) exist even in a half-space
- Surface wave dispersion provides depth-dependent velocity structure: the basis of seismic tomography
- Normal modes (free oscillations) constrain the deepest 1-D structure of the Earth, including density
- Attenuation (Q) reveals temperature, composition, and partial melt in the mantle
- P-to-S converted phases (receiver functions) map discontinuity depths beneath seismic stations