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S wave

An S wave, also known as a secondary wave or shear wave, is a type of seismic body wave that propagates through the Earth's interior by inducing oscillatory motion in the medium perpendicular to the direction of wave travel.^[1] These transverse waves shear the material they pass through without altering its volume, distinguishing them from compressional primary (P) waves.^[2] S waves are the second type of body wave to arrive at a seismograph station following an earthquake, typically traveling at speeds of about 3 to 4 km/s in the Earth's crust, which is roughly 60% the velocity of P waves.^[3]^[1] A key property of S waves is their inability to transmit through fluids or gases, as these media cannot sustain the shear stresses required for propagation.^[2] This limitation has been instrumental in revealing the liquid nature of Earth's outer core, as S waves do not pass through it, creating an S-wave shadow zone observable on the planet's opposite side from an earthquake epicenter.^[1] In solid rock, S waves generate both vertical and horizontal ground motions, often producing more intense shaking than P waves and contributing significantly to structural damage near the earthquake source.^[2] Their speeds vary with the medium: slower in unconsolidated sediments (around 1 km/s) and faster in denser mantle materials (up to 8 km/s near the core-mantle boundary).^[2] S waves play a vital role in seismology for mapping Earth's heterogeneous interior, as their travel times and paths recorded by global seismic networks help delineate layer boundaries and material properties.^[4] Unlike surface waves, which are slower and more destructive over large areas, S waves are confined to the planet's volume and diminish in amplitude with distance due to geometric spreading and attenuation.^[5] Detection of S waves on seismograms, appearing as sharper deflections after the initial P-wave arrival, aids in locating earthquake foci and estimating magnitudes.^[6]

Fundamentals

Definition and Characteristics

S waves, also known as secondary waves, are a type of elastic body wave that propagates through solid materials by inducing particle motion perpendicular to the direction of wave travel, thereby transmitting shear stress across the medium.^[1]^[2] These waves are characterized by their transverse polarization, where the oscillations occur in a plane orthogonal to the propagation direction, resulting in shear deformation without altering the volume of the material.^[3]^[7] A key attribute of S waves is their inability to propagate through fluids or gases, as these media lack the shear strength required to support transverse motion; this contrasts with P waves, which are compressional and can travel through any material.^[8]^[9] In the Earth's crust, S wave speeds typically range from 30% to 60% of P wave speeds, for example, approximately 3 to 4 km/s in granite.^[10] Particle motion in S waves involves displacement in the plane perpendicular to propagation, often decomposed into two components: SV waves, where motion is in the vertical plane containing the propagation direction, and SH waves, where motion is horizontal and perpendicular to that plane.^[11]^[12] This shear deformation generates energy dependent on the material's shear modulus μ, with shear stress τ expressed as τ = μ ∂u/∂z, where u is the displacement and z is the direction perpendicular to the motion.^[13]^[14]

Comparison to P Waves

S waves, or shear waves, differ fundamentally from P waves, or primary waves, in their particle motion. While P waves propagate through longitudinal oscillations, where particles move back and forth in the direction of wave travel, causing compression and dilation of the medium, S waves exhibit transverse motion perpendicular to the propagation direction.^[1] In three dimensions, S wave particle paths can trace elliptical trajectories due to the superposition of vertically polarized (SV) and horizontally polarized (SH) components in the plane orthogonal to propagation.^[15] The propagation speeds of S waves are typically slower than those of P waves, with V_s approximately 60% of V_p in typical crustal rocks, resulting in S waves arriving later than P waves on seismograms and allowing for epicenter location via the time difference.^[16] This speed ratio derives from the elastic properties of the medium and is given by V_s / V_p ≈ √(μ / (λ + 2μ)), where λ and μ represent the Lamé parameters characterizing the material's resistance to volumetric and shear deformation, respectively.^[17] S waves require a solid medium to propagate, as they depend on shear rigidity, whereas P waves can travel through solids, liquids, and gases.^[2] Consequently, S waves do not pass through Earth's liquid outer core, creating a shadow zone spanning about 105° to 180° from the earthquake epicenter where no direct S waves are detected, which provided key evidence for the core's fluid nature.^[18] In terms of detection, the transverse shaking induced by S waves generates more intense ground motions than the compressional effects of P waves, making S waves a primary contributor to structural damage during earthquakes.^[19]

Propagation Theory

In Isotropic Media

In isotropic media, S waves propagate as transverse shear waves in homogeneous elastic solids where material properties do not vary with direction. The velocity of an S wave, denoted V_s, is given by V_s = \sqrt{\mu / \rho}, where \mu is the shear modulus and \rho is the material density.^[17] This formula arises from the linear elastic constitutive relation, known as Hooke's law in its generalized form for isotropic solids, which connects the shear stress tensor components \tau_{ij} to the strain tensor e_{ij} via \tau_{ij} = \lambda \delta_{ij} e_{kk} + 2\mu e_{ij}, with \lambda as the first Lamé parameter.^[20] For pure shear deformation, where volumetric changes are absent (e_{kk} = 0), the relation simplifies to \tau_{ij} = 2\mu e_{ij}, emphasizing the shear modulus \mu as the key stiffness parameter governing transverse motion.^[20] Combining this with Newton's second law of motion and the kinematic relation between strain and displacement leads to the S wave speed as the square root of the ratio of shear rigidity to inertial mass density.^[17] The governing wave equation for S waves in an isotropic elastic solid derives from the Navier-Cauchy equations of elastodynamics. For the displacement vector \mathbf{u} satisfying the condition of zero divergence (\nabla \cdot \mathbf{u} = 0) to ensure incompressibility characteristic of shear waves, the equation takes the form

\frac{\partial^2 \mathbf{u}}{\partial t^2} = \frac{\mu}{\rho} \nabla^2 \mathbf{u}.

^[17] This scalar Helmholtz-like equation describes plane or spherical wave solutions propagating at speed V_s, with particle motion perpendicular to the direction of propagation.^[20] In one dimension, for a shear displacement u_y(x, t) along the y-direction varying with x, it reduces to the standard wave equation \partial^2 u_y / \partial t^2 = V_s^2 \partial^2 u_y / \partial x^2.^[17] At interfaces between two isotropic media, S waves undergo reflection and refraction governed by Snell's law, adapted for shear velocities: \sin i_1 / V_{s1} = \sin i_2 / V_{s2}, where i_1 and i_2 are the incidence and refraction angles, and V_{s1}, V_{s2} are the S wave speeds in the respective media.^[21] Unlike in fluids, where only compressional waves transmit, solid interfaces permit mode conversions, such that an incident S wave (specifically SV-polarized) can produce reflected and transmitted P and SV waves, or vice versa for incident P waves generating S components.^[22] The reflection and transmission coefficients depend on the velocity contrast, density, and angles, often computed via the Zoeppritz equations for precise amplitudes. Critical angles arise when the incident wave approaches from the slower medium (V_{s1} < V_{s2}), defined by \sin i_c = V_{s1} / V_{s2}, beyond which total internal reflection occurs for the shear mode, with evanescent waves in the faster medium.^[21] In ideal isotropic elastic solids, S waves experience no intrinsic attenuation from material damping, as the medium is assumed perfectly elastic without energy dissipation mechanisms. The primary amplitude reduction stems from geometric spreading, where wavefront expansion in three dimensions causes the displacement amplitude to decay inversely with distance r as $1/r for spherical waves, conserving energy flux across expanding surfaces.^[23] This effect is purely kinematic, arising from the ray theory approximation in homogeneous media, and scales with the square root of the ray Jacobian for more general wavefront geometries.^[23]

In Anisotropic and Viscoelastic Media

In anisotropic media, such as crystals or layered rock formations, the propagation velocity of S waves varies with direction due to the directional dependence of elastic properties. Unlike in isotropic media, where S waves have a single velocity, in anisotropic media the S wave splits into two quasi-shear modes: the SH wave (horizontal polarization perpendicular to the propagation plane) and the qSV wave (polarization in the propagation plane), each exhibiting direction-dependent velocities due to the medium's elastic anisotropy.^[24] This velocity anisotropy leads to non-uniform wave speeds, where in weakly anisotropic transversely isotropic (VTI) media common in sedimentary rocks, the SH shear wave velocity can be approximated using Thomsen's parameters as V_{SH}(\theta) \approx V_{s0} (1 + \gamma \sin^2 \theta), with \theta denoting the angle from the symmetry axis, V_{s0} the reference vertical shear velocity, and \gamma quantifying the shear anisotropy strength as the normalized difference between horizontal and vertical shear moduli. The qSV velocity approximation is more complex, involving parameters δ and ε in addition to γ and the P-to-S velocity ratio. These parameters enable corrections for imaging distortions in seismic surveys.^[24] Viscoelastic media introduce frequency-dependent damping to S wave propagation, modeled through a complex shear modulus \mu^* = \mu' + i \mu'', where \mu' represents the storage modulus and \mu'' the loss modulus associated with energy dissipation. The quality factor Q, defined as Q = 2\pi times the ratio of energy stored to energy lost per cycle, quantifies this attenuation.^[25] In such media, S wave amplitude decays exponentially as A = A_0 e^{-\pi f t / Q}, with f as frequency and t as travel time, reflecting the material's anelastic behavior under oscillatory stress.^[26] The adaptation of Huygens' principle in anisotropic media results in wavefronts that deviate from spherical shapes, often forming ellipses for shear waves in transversely isotropic conditions, as secondary wavelets construct the propagating front based on directionally varying velocities. This contrasts with isotropic cases and affects ray paths and interference patterns in complex geological structures.^[27] Laboratory experiments using ultrasonic techniques measure these effects in rock samples, revealing anisotropy-induced velocity variations and viscoelastic damping through transmitted shear wave analysis in controlled setups. For instance, ultrasonic pulses applied to granitic rocks demonstrate directional shear speed differences up to 10-20% aligned with layering, validating theoretical models without requiring field-scale deployments.

Historical Development

Early Observations

The origins of S wave observations date to the mid-19th century, when Irish engineer Robert Mallet conducted artificial explosions to study seismic propagation during his investigations of earthquakes in Italy, laying foundational empirical groundwork, though Mallet's instruments—simple seismoscopes—lacked the precision to clearly differentiate wave types.^[28] Key seismic events in the late 19th century further highlighted these secondary arrivals. During the 1889 Tokyo earthquake, Japanese seismologist Fusakichi Omori utilized early mechanical seismographs to document delayed transverse wave phases following the initial compressional signals, providing some of the first instrumental records of these patterns in a major urban event.^[29] This observation was bolstered by distant recordings in Europe, such as at Potsdam and Wilhelmshaven, where horizontal pendulums captured the secondary phases traveling across continents.^[29] The role of advancing instrumentation was pivotal in enabling consistent S wave detection. Evolving from rudimentary seismoscopes, which only indicated motion direction without timing, to mechanical seismographs in the 1890s—exemplified by John Milne's horizontal pendulum designs—these devices incorporated transverse components to measure shear motions perpendicular to wave propagation, allowing separation of S wave signals from primary ones.^[28] Milne's instruments, deployed in a nascent global network of about 40 stations by 1900, facilitated routine capture of these transverse arrivals.^[30] In 1900, Richard Dixon Oldham analyzed records from the 1897 Assam earthquake and identified the secondary phase as a transverse shear wave, distinct from the primary compressional wave and surface waves.^[29] A significant confirmation of S-P time delays occurred in 1909, when Croatian geophysicist Andrija Mohorovičić analyzed records from the Kulpa Valley earthquake near Zagreb, observing consistent intervals between primary (P) and secondary (S) wave arrivals that indicated varying propagation speeds through layered Earth structures.^[31] Early interpretations often conflated these secondary transverse phases with surface waves due to overlapping arrival times and instrumental limitations in distinguishing body from surface motions.^[30] This confusion was progressively resolved in the 1910s through data from expanding global seismic networks, which revealed the distinct deeper propagation paths of S waves compared to the slower, crust-bound surface waves.^[30]

Theoretical Advancements

The theoretical framework for S waves, or shear waves, advanced significantly in the early 20th century as elastic wave theory was extended to model Earth's spherical structure. A.E.H. Love's seminal 1911 treatise integrated shear wave propagation into the broader theory of elasticity, emphasizing transverse vibrations in solid media and their role in geodynamical problems, including preliminary considerations of wave paths in a spherically symmetric Earth. This built upon Lord Rayleigh's earlier 1885 analysis of wave motion in elastic solids, which provided foundational equations for transverse waves, later applied to body wave dynamics in planetary interiors. By the 1930s and 1940s, these concepts were formalized in global Earth models; Harold Jeffreys and Keith E. Bullen developed the Jeffreys-Bullen (JB) model, incorporating radial S-wave velocity profiles V_s(r) derived from travel-time data, marking the first comprehensive spherical Earth model distinguishing P and S velocities across mantle layers. Works by Leon Knopoff and collaborators in this era further refined S-wave inclusion by addressing attenuation and source mechanisms, such as the 1964 body force equivalents for seismic dislocations, which modeled how shear waves are excited by fault ruptures in elastic media. Advancements in the 1960s focused on handling Earth's heterogeneity, particularly anisotropy and layering. George E. Backus introduced averaging techniques in 1965 to derive effective isotropic parameters for weakly anisotropic layered media, enabling computation of quasi-S wave velocities and polarizations in the upper mantle from fine-scale variations.^[32] This method, known as Backus averaging, approximated long-wavelength S-wave propagation in transversely isotropic models, providing a bridge between microscopic fabric and macroscopic seismic observations. Building on early empirical travel-time data from global seismicity, these developments quantified how azimuthal anisotropy affects S-wave splitting.^[32] Viscoelastic extensions emerged in the mid-1970s to account for attenuation in S-wave propagation. Hsi-Ping Liu, Don L. Anderson, and Hiroo Kanamori's 1976 model used a generalized standard linear solid with multiple relaxation mechanisms to explain velocity dispersion and nearly constant Q in the mantle, deriving frequency-dependent V_s(r) profiles that reconciled observed anelasticity with elastic theory. This framework extended isotropic elastic equations to include damping, predicting S-wave phase velocities varying by up to 5% across seismic frequencies. The 1980s marked computational breakthroughs, enabling numerical simulation of S waves in complex geometries. Jean Virieux's finite-difference schemes, starting with SH-wave modeling in 1984 and extending to P-SV in 1986, discretized the velocity-stress equations on staggered grids to propagate shear waves through heterogeneous spherical models, accurately predicting shadow zones and scattering without analytical approximations. These methods facilitated validation of theoretical V_s(r) against synthetic seismograms. Culminating in the 1981 Preliminary Reference Earth Model (PREM) by Adam M. Dziewonski and Don L. Anderson, which specified high-resolution S-velocity profiles V_s(r) from surface to core, incorporating anisotropy in the upper mantle (2-4% velocity variation) and integrating free-oscillation, surface-wave, and body-wave data for a spherically symmetric reference.^[33] PREM's V_s(r) decreases from ~4.5 km/s in the upper mantle to ~7 km/s in the lower mantle, establishing a benchmark for subsequent anisotropic and viscoelastic refinements.^[33]

Applications

In Seismology and Geophysics

In seismology, S waves play a crucial role in earthquake analysis by enabling the determination of epicenter locations through the difference in arrival times between S and P waves (S-P times). This time differential, combined with travel-time tables such as the Jeffreys-Bullen tables, allows seismologists to triangulate the hypocenter by accounting for the slower propagation speed of S waves (approximately 60% of P wave velocity in the crust).^[34]^[35] Additionally, the surface-wave magnitude scale (M_s) measures the amplitude of Rayleigh surface waves—which involve shear motion similar to S waves—at periods around 20 seconds to assess the energy released by intermediate to large earthquakes.^[36] S wave tomography provides detailed three-dimensional images of Earth's mantle structure, revealing velocity anomalies that map convective flows and subducting slabs, thereby informing models of mantle convection dynamics. High-resolution global models highlight large low-velocity provinces in the lowermost mantle, linked to upwelling plumes.^[37]^[38] Receiver function techniques further exploit S waves by isolating S-to-P conversions at seismic discontinuities, such as the 410 km and 660 km transitions in the mantle, to probe the depth and sharpness of these interfaces without interference from crustal multiples. Stacked S-to-P receiver functions enhance resolution for upper mantle features like the midlithospheric discontinuity.^[39] For seismic hazard assessment, S waves are amplified in soft sediments, leading to intensified ground motions; the 1985 Michoacán earthquake (M_w 8.0) demonstrated this in Mexico City, where lakebed clays caused spectral amplifications of 3 to 7 times at periods of 1-2 seconds, exacerbating building collapses. The VS30 parameter, defined as the time-averaged shear-wave velocity in the top 30 m of soil, serves as a proxy for site classification in seismic zoning, with values below 360 m/s indicating high-risk soft sites prone to amplification.^[40]^[41]^[42] In exploration geophysics, vertical seismic profiling (VSP) employs S wave sources to image subsurface reservoirs with improved resolution, capturing SV-P and S-S reflections that delineate fluid contacts and lithological boundaries in hydrocarbon traps. This method integrates borehole receivers with surface vibrators to calibrate surface seismic data and monitor CO2 storage sites.^[43]^[44]

In Biomedical Imaging

S waves, or shear waves, play a crucial role in biomedical imaging through elastography techniques, which non-invasively assess tissue mechanical properties by measuring wave propagation speeds to quantify stiffness. These methods exploit the fact that pathological changes, such as fibrosis or tumors, alter tissue elasticity, allowing differentiation between healthy and diseased states without invasive biopsies. Elastography primarily focuses on shear modulus, derived from shear wave velocity V_s, as stiffer tissues propagate waves faster.^[45] Magnetic Resonance Elastography (MRE) generates shear waves using external mechanical actuators that vibrate at audible frequencies, typically 50-100 Hz, to induce controlled deformations in the target tissue. These waves are then imaged via phase-contrast MRI sequences that capture micron-level displacements synchronized with the actuation. Post-processing involves inverting the wave data to produce spatial maps of the shear modulus, calculated as \mu(\mathbf{r}) = \rho V_s^2, where \rho is tissue density and V_s is the local shear wave speed; this approach enables quantitative visualization of tissue heterogeneity.^[46]^[47] Ultrasound-based shear wave elastography employs acoustic radiation force impulse (ARFI) imaging, where focused ultrasound pulses create localized pushing forces to generate transient shear waves within the tissue. These waves are tracked in real-time using ultrafast ultrasound imaging at frame rates exceeding 10,000 Hz, allowing precise measurement of propagation speeds. For incompressible soft tissues, the Young's modulus E is estimated from the shear modulus via E \approx 3\mu, providing a clinically relevant metric for elasticity assessment.^[48]^[49] Clinically, these techniques excel in staging liver fibrosis, where shear wave speeds exceeding approximately 1.8 m/s often indicate cirrhosis, enabling non-invasive monitoring of disease progression in chronic liver conditions. In neuro-oncology, S wave elastography aids brain tumor characterization by distinguishing stiff malignant lesions from surrounding parenchyma, improving surgical planning and resection margins. However, applications are limited in highly attenuating tissues, such as fatty liver or scarred regions, where rapid wave damping reduces signal quality and penetration depth.^[50]^[51]^[52]^[53] Advancements in 3D MRE, developed since the early 2000s, extend imaging to volumetric acquisitions, capturing full three-dimensional wave fields for more accurate stiffness mapping in complex organs like the brain or liver. Recent integrations combine MRE with diffusion tensor imaging (DTI) to analyze anisotropic tissue properties, revealing directional variations in shear modulus aligned with microstructural fiber orientations, which enhances diagnostic specificity in conditions like multiple sclerosis or tumors.^[54]^[55]^[56]

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