Introduction
Shear or S waves are the transverse elastic body waves that carry information about the Earth’s internal shear response. Particle motion in an S wave is perpendicular to propagation and is restored by shear stresses, in contrast to compressional (P) waves whose motion is parallel to propagation and whose restoring force is volumetric compression. Because they require shear rigidity, S waves do not traverse ideal fluids or media with negligible shear strength, although they can propagate in viscous or otherwise shear‑supporting materials.
S waves travel more slowly than P waves and therefore arrive at seismic stations after the initial compressional phase, a sequencing that gives them the appellation “secondary.” Their inability to pass through a liquid outer core produces an S‑wave shadow zone opposite an earthquake source; the existence and size of that zone are primary constraints on the outer core’s liquid state. At the liquid–solid boundary of the inner core, oblique incidence produces complex P↔S mode conversions: incident P waves can generate S waves in the solid inner core, and those inner‑core S phases can reconvert to P in the outer core. Analyses of these conversions — amplitudes, travel times and ray geometries — provide sensitive probes of inner‑core solidity, elastic moduli and anisotropy.
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S waves originate at the hypocenter and propagate to the surface epicenter and beyond; epicentral distance is a basic coordinate in locating events. Seismometers record S‑wave arrival times and amplitudes, which are essential inputs for earthquake location, focal‑mechanism inversion, magnitude/intensity assessment and ground‑motion prediction in seismic hazard studies. S‑wave behavior also underpins many specialized seismological analyses (e.g., shear‑wave splitting, density–velocity relations via the Adams–Williamson framework, regionalization schemes, and seismite interpretation) and is relevant across the full range of earthquake types and triggering mechanisms.
In 1830 Siméon Denis Poisson presented to the French Academy of Sciences a theoretical treatment of elastic-wave propagation in solids with direct application to earthquake vibrations. His analysis predicted two distinct elastic wave types traveling at different speeds—one at speed a and a second at a/√3—and treated the far-field disturbance locally as plane waves so that particle motions could be described relative to the advancing wavefront. The faster mode produces volumetric compression and dilation with particle displacement aligned with the direction of propagation (longitudinal motion), whereas the slower mode produces shear deformation with particle displacement orthogonal to the propagation direction (transverse motion). Poisson’s two-wave description therefore establishes that seismic disturbances in solid Earth materials consist of two orthogonal deformation modes, each with its own propagation speed, which control how seismic energy is transmitted through and interacts with geological structures at regional and greater distances.
In an isotropic elastic medium the motion is described by the displacement vector u(x,t) = (u1,u2,u3), with spatial derivatives ∂i ≡ ∂/∂xi and temporal derivative ∂t ≡ ∂/∂t. Infinitesimal deformation is measured by the symmetric strain tensor e with components eij = ½(∂i uj + ∂j ui). For a linear, isotropic solid the stress tensor τ is related to strain by the Lamé parameters λ and μ (the latter being the shear modulus): τij = λ δij ∑k ekk + 2μ eij, or equivalently τij = λ δij ∑k ∂k uk + μ(∂i uj + ∂j ui).
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Balance of momentum (Newton’s second law for continua) gives ρ ∂t^2 ui = ∑j ∂j τij, which, after substituting the constitutive relation, yields the seismic wave equation in a homogeneous medium. In vector form this reads ρ ∂t^2 u = (λ + 2μ) ∇(∇·u) − μ ∇×(∇×u), separating volumetric (compressional) effects from shear deformations. Taking the curl produces a homogeneous wave equation for the shear strain field ∇×u: ∂t^2(∇×u) = (μ/ρ) ∇^2(∇×u). Plane-wave solutions of this equation are S (shear) waves that propagate with speed β = √(μ/ρ). Thus μ can be expressed as μ = ρ β^2 (or, in harmonic form, μ = ρ ω^2/k^2), giving the nondispersive relation β = ω/k in an ideal linear, homogeneous, isotropic medium.
Taking the divergence of the vector wave equation isolates compressional motion: the scalar field ∇·u satisfies a wave equation whose solutions are P (primary) waves with speed α = √((λ + 2μ)/ρ), reflecting both Lamé parameters and density. For steady-state, horizontally polarized shear (SH) motion the time-harmonic shear displacement satisfies the Helmholtz equation (∇^2 + k^2)u = 0, the frequency-domain analogue of the time-dependent shear-wave equation. In Earth applications, the shear-wave speed becomes effectively zero in a fluid because μ → 0, which explains the near-absence of S-wave propagation in the liquid outer core; by contrast, the solid inner core supports a nonzero S-wave speed, a contrast that underlies seismic inferences of a liquid outer core and a solid inner core.
In a linear viscoelastic medium shear-wave propagation is governed by the frequency-dependent dispersion relation c(ω) = ω / k(ω) = sqrt(μ(ω)/ρ), where c(ω) is the phase velocity, k(ω) the (generally complex) wavenumber, μ(ω) the complex shear modulus and ρ the mass density. The complex nature of μ(ω) simultaneously encodes stored elastic energy (the real part) and viscous dissipation (the imaginary part); because μ(ω) is complex, k(ω) is also complex so wave motion exhibits both phase propagation and exponential amplitude decay. In practice the measurable phase velocity is set by the real part of the dispersion relation (or equivalently ω divided by Re{k}), while attenuation per unit distance is governed by Im{k}, giving an amplitude factor exp[−Im{k} x].
A common constitutive description is the Kelvin–Voigt (Voigt) model, μ(ω) = μ0 + i ω η, in which μ0 denotes the purely elastic shear stiffness and η the viscosity. In this representation the dissipative contribution grows linearly with angular frequency, so the balance between elastic and viscous behavior depends on ω. At low frequencies (ω → 0) the viscous term becomes negligible and μ(ω) ≈ μ0, yielding a shear speed close to sqrt(μ0/ρ) with vanishing attenuation. At higher frequencies the i ω η term can become comparable to or exceed μ0, producing increased attenuation and frequency-dependent deviations of phase velocity (dispersion) from the low-frequency elastic value.
Therefore, the relation c(ω) = sqrt(μ(ω)/ρ) together with a chosen rheological model like Voigt provides a quantitative framework for predicting how shear-wave phase velocity and amplitude vary with frequency in media exhibiting both elasticity and viscosity. Given μ0, η, ρ and ω one can compute the frequency-dependent phase speed, estimate attenuation rates, and hence model dispersion and dissipative effects relevant to geophysical and material-wave problems.
Magnetic resonance elastography
Magnetic resonance elastography (MRE) is an in vivo imaging modality that probes the mechanical behavior of biological tissues by introducing controlled shear waves at selected frequencies and imaging the resulting motion. A mechanical actuator couples shear waves into the body, creating reproducible wave fields whose propagation is recorded by MRI sequences that encode small tissue displacements. From the imaged wave patterns—principally measured wave speeds and wavelengths—quantitative mechanical parameters such as the shear modulus can be inferred, yielding local estimates of tissue stiffness. Because MRI provides volumetric, high-resolution coverage, MRE produces spatially resolved elastograms that permit localized assessment of elastic properties within organs. This approach has been applied clinically and in research to characterize the mechanical properties of the liver, brain, bone, and other tissues in vivo.