S wave

 ( Waves ) 

In seismology and other areas involving elastic waves, S waves, secondary waves, or shear waves (sometimes called elastic S waves) are a type of elastic wave and are one of the two main types of elastic body waves, so named because they move through the body of an object, unlike .

S waves are , meaning that the direction of particle movement of an S wave is perpendicular to the direction of wave propagation, and the main restoring force comes from shear stress. Therefore, S waves cannot propagate in liquids with zero (or very low) viscosity; however, they may propagate in liquids with high viscosity. Similarly, S waves cannot travel through gases.

The name secondary wave comes from the fact that they are the second type of wave to be detected by an earthquake seismograph, after the compressional primary wave, or P wave, because S waves travel more slowly in solids. Unlike P waves, S waves cannot travel through the molten outer core of the Earth, and this causes a shadow zone for S waves opposite to their origin. They can still propagate through the solid inner core: when a P wave strikes the boundary of molten and solid cores at an oblique angle, S waves will form and propagate in the solid medium. When these S waves hit the boundary again at an oblique angle, they will in turn create P waves that propagate through the liquid medium. This property allows seismology to determine some physical properties of the Earth's inner core.

---

History In 1830, the mathematician Siméon Denis Poisson presented to the French Academy of Sciences an essay ("memoir") with a theory of the propagation of elastic waves in solids. In his memoir, he states that an earthquake would produce two different waves: one having a certain speed $a$ and the other having a speed $\backslash frac\{a\}\{\backslash sqrt\; 3\}$. At a sufficient distance from the source, when they can be considered in the region of interest, the first kind consists of expansions and compressions in the direction perpendicular to the wavefront (that is, parallel to the wave's direction of motion); while the second consists of stretching motions occurring in directions parallel to the front (perpendicular to the direction of motion). From p.595: " On verra aisément que cet ébranlement donnera naissance à deux ondes sphériques qui se propageront uniformément, l'une avec une vitesse a , l'autre avec une vitesse b ou a / " ... (One will easily see that this quake will give birth to two spherical waves that will be propagated uniformly, one with a speed a, the other with a speed b or a /√3 ... ) From p.602: ... " à une grande distance de l'ébranlement primitif, et lorsque les ondes mobiles sont devenues sensiblement planes dans chaque partie très-petite par rapport à leurs surfaces entières, il ne subsiste plus que des vitesses propres des molécules, normales ou parallèles à ces surfaces; les vitesses normal ayant lieu dans les ondes de la première espèce, où elles sont accompagnées de dilations qui leur sont proportionnelles, et les vitesses parallèles appartenant aux ondes de la seconde espèce, où elles ne sont accompagnées d'aucune dilatation ou condensation de volume, mais seulement de dilatations et de condensations linéaires." ( ... at a great distance from the original quake, and when the moving waves have become roughly planes in every tiny part in relation to their entire surface, there remain in only the molecules' own speeds, normal or parallel to these surfaces; the normal speeds occur in waves of the first type, where they are accompanied by expansions that are proportional to them, and the parallel speeds belonging to waves of the second type, where they are not accompanied by any expansion or contraction of volume, but only by linear stretchings and squeezings.)

---

Theory

---

Isotropic medium For the purpose of this explanation, a solid medium is considered isotropic if its strain (deformation) in response to stress is the same in all directions. Let $\backslash boldsymbol\{u\}\; =\; (u\_1,u\_2,u\_3)$ be the displacement vector of a particle of such a medium from its "resting" position $\backslash boldsymbol\{x\}=(x\_1,x\_2,x\_3)$ due elastic vibrations, understood to be a function of the rest position $\backslash boldsymbol\{x\}$ and time $t$. The deformation of the medium at that point can be described by the strain tensor $\backslash boldsymbol\{e\}$, the 3×3 matrix whose elements are $$e\_\{i\; j\}\; =\; \backslash tfrac\{1\}\{2\}\; \backslash left(\; \backslash partial\_i\; u\_j\; +\; \backslash partial\_j\; u\_i\; \backslash right)$$

where $\backslash partial\_i$ denotes partial derivative with respect to position coordinate $x\_i$. The strain tensor is related to the 3×3 stress tensor $\backslash boldsymbol\{\backslash tau\}$ by the equation $$\backslash tau\_\{i\; j\}\; =\; \backslash lambda\backslash delta\_\{i\; j\}\backslash sum\_\{k\}\; e\_\{k\; k\}\; +\; 2\backslash mu\; e\_\{i\; j\}$$

Here $\backslash delta\_\{ij\}$ is the Kronecker delta (1 if $i\; =\; j$, 0 otherwise) and $\backslash lambda$ and $\backslash mu$ are the Lamé parameters ($\backslash mu$ being the material's shear modulus). It follows that $$\backslash tau\_\{i\; j\}\; =\; \backslash lambda\backslash delta\_\{i\; j\}\; \backslash sum\_\{k\}\; \backslash partial\_k\; u\_k\; +\; \backslash mu\; \backslash left(\; \backslash partial\_i\; u\_j\; +\; \backslash partial\_j\; u\_i\; \backslash right)$$

From Newton's law of inertia, one also gets $$\backslash rho\; \backslash partial\_t^2\; u\_i\; =\; \backslash sum\_j\; \backslash partial\_j\backslash tau\_\{i\; j\}$$ where $\backslash rho$ is the density (mass per unit volume) of the medium at that point, and $\backslash partial\_t$ denotes partial derivative with respect to time. Combining the last two equations one gets the seismic wave equation in homogeneous media $$\backslash rho\; \backslash partial\_t^2\; u\_i\; =\; \backslash lambda\backslash partial\_i\; \backslash sum\_k\; \backslash partial\_k\; u\_k\; +\; \backslash mu\backslash sum\_j\; \backslash bigl(\backslash partial\_i\backslash partial\_j\; u\_j\; +\; \backslash partial\_j\backslash partial\_j\; u\_i\backslash bigr)$$

Using the nabla operator notation of vector calculus, $\backslash nabla\; =\; (\backslash partial\_1,\; \backslash partial\_2,\; \backslash partial\_3)$, with some approximations, this equation can be written as $$\backslash rho\; \backslash partial\_t^2\; \backslash boldsymbol\{u\}\; =\; \backslash left(\backslash lambda\; +\; 2\backslash mu\; \backslash right)\; \backslash nabla\backslash left(\backslash nabla\; \backslash cdot\; \backslash boldsymbol\{u\}\backslash right)\; -\; \backslash mu\backslash nabla\; \backslash times\; \backslash left(\backslash nabla\; \backslash times\; \backslash boldsymbol\{u\}\backslash right)$$

Taking the curl of this equation and applying vector identities, one gets $$\backslash partial\_t^2(\backslash nabla\backslash times\backslash boldsymbol\{u\})\; =\; \backslash frac\{\backslash mu\}\{\backslash rho\}\backslash nabla^2\; \backslash left(\backslash nabla\backslash times\backslash boldsymbol\{u\}\backslash right)$$

This formula is the wave equation applied to the vector quantity $\backslash nabla\backslash times\; \backslash boldsymbol\{u\}$, which is the material's shear strain. Its solutions, the S waves, are linear combinations of sinusoidal of various and directions of propagation, but all with the same speed $\backslash beta\; =\; \backslash sqrt\{\backslash mu/\backslash rho\}$. Assuming that the medium of propagation is linear, elastic, isotropic, and homogeneous, this equation can be rewritten as $\backslash mu=\backslash rho\; \backslash beta^2=\backslash rho\; \backslash omega^2\; /\; k^2$ where ω is the angular frequency and is the wavenumber. Thus, $\backslash beta\; =\; \backslash omega\; /\; k$.

Taking the divergence of seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity $\backslash nabla\; \backslash cdot\; \backslash boldsymbol\{u\}$, which is the material's compression strain. The solutions of this equation, the P waves, travel at the faster speed $\backslash alpha\; =\; \backslash sqrt\{(\backslash lambda\; +\; 2\backslash mu)/\backslash rho\}$.

The steady state SH waves are defined by the Helmholtz equation

$$\backslash left(\backslash nabla^2\; +\; k^2\; \backslash right)\; \backslash boldsymbol\{u\}=0$$ where is the wave number.

---

S waves in viscoelastic materials Similar to in an elastic medium, in a Viscoelasticity material, the speed of a shear wave is described by a similar relationship $c(\backslash omega)\; =\; \backslash omega\; /\; k(\backslash omega)=\backslash sqrt\{\backslash mu(\backslash omega)/\backslash rho\}$, however, here, $\backslash mu$ is a complex, frequency-dependent shear modulus and $c(\backslash omega)$ is the frequency dependent phase velocity. One common approach to describing the shear modulus in viscoelastic materials is through the Voigt Model which states: $\backslash mu(\backslash omega)=\backslash mu\_0+i\backslash omega\backslash eta$, where $\backslash mu\_0$ is the stiffness of the material and $\backslash eta$ is the viscosity.

---

S wave technology

---

Magnetic resonance elastography

Magnetic resonance elastography (MRE) is a method for studying the properties of biological materials in living organisms by propagating shear waves at desired frequencies throughout the desired organic tissue. This method uses a vibrator to send the shear waves into the tissue and magnetic resonance imaging to view the response in the tissue. The measured wave speed and wavelengths are then measured to determine elastic properties such as the . MRE has seen use in studies of a variety of human tissues including liver, brain, and bone tissues.

---

See also

• Earthquake Early Warning (Japan)
• Lamb waves
• Longitudinal wave
• Love wave
• Rayleigh wave
• Shear wave splitting

---

Further reading

• Peter Shearer (1999). 9780521660235, Cambridge University Press. ISBN 9780521660235
• (2025). 9780935702965, University Science Books. . ISBN 9780935702965
• C. M. R. Fowler (1990). 9780521385909, Cambridge University Press. . ISBN 9780521385909

Categories: Waves, Seismology

Post Comment