# S-wave

In seismology, S-waves, secondary waves, or shear waves (sometimes called an elastic S-wave) 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 surface waves.[1]

The S-wave is a transverse wave, meaning that, in the simplest situation, the oscillations of the particles of the medium is perpendicular to the direction of wave propagation, and the main restoring force comes from shear stress.[2]

The shadow zone of a P-wave. S-waves don't penetrate the outer core, so they're shadowed everywhere more than 104° away from the epicenter (from USGS)

Its name, S for secondary, comes from the fact that it is the second direct arrival on an earthquake seismogram, after the compressional primary wave, or P-wave, because S-waves travel slower in rock. Unlike the P-wave, the S-wave cannot travel through the molten outer core of the Earth, and this causes a shadow zone for S-waves opposite to where they originate. They can still appear in 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 seismologists to determine some physical properties of the inner core.[3]

Plane shear wave
Propagation of a spherical S-wave in a 2d grid (empirical model)

## 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 a/3. At a sufficient distance from the source, when they can be considered plane waves 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).[4]

## 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 ${\displaystyle {\boldsymbol {u}}=(u_{1},u_{2},u_{3})}$ be the displacement vector of a particle of such a medium from its "resting" position ${\displaystyle {\boldsymbol {x}}=(x_{1},x_{2},x_{3})}$ due elastic vibrations, understood to be a function of the rest position ${\displaystyle {\boldsymbol {x}}}$ and time ${\displaystyle t}$. The deformation of the medium at that point can be described by the strain tensor ${\displaystyle {\boldsymbol {e}}}$, the 3×3 matrix whose elements are

${\displaystyle e_{ij}={\frac {1}{2}}(\partial _{i}u_{j}+\partial _{j}u_{i})}$

where ${\displaystyle \partial _{i}}$ denotes partial derivative with respect to position coordinate ${\displaystyle x_{i}}$. The strain tensor is related to the 3×3 stress tensor ${\displaystyle {\boldsymbol {\tau }}}$ by the equation

${\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}e_{kk}+2\mu e_{ij}}$

Here ${\displaystyle \delta _{ij}}$ is the Kronecker delta (1 if ${\displaystyle i=j}$, 0 otherwise) and ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are the Lamé parameters (${\displaystyle \mu }$ being the material's shear modulus). It follows that

${\displaystyle \tau _{ij}=\lambda \delta _{ij}{\bigl (}\sum _{k}\partial _{k}u_{k}{\bigr )}+\mu (\partial _{i}u_{j}+\partial _{j}u_{i})}$

From Newton's law of inertia, one also gets

${\displaystyle \rho \partial _{t}^{2}u_{i}=\sum _{j}\partial _{j}\tau _{ij}}$

where ${\displaystyle \rho }$ is the density (mass per unit volume) of the medium at that point, and ${\displaystyle \partial _{t}}$ denotes partial derivative with respect to time. Combining the last two equations one gets the seismic wave equation in homogeneous media

${\displaystyle \rho \partial _{t}^{2}u_{i}=\lambda \partial _{i}{\bigl (}\sum _{k}\partial _{k}u_{k}{\bigr )}+\mu {\bigl (}\sum _{j}\partial _{i}\partial _{j}u_{j}+\mu \partial _{j}\partial _{j}u_{i}{\bigr )}}$

Using the nabla operator notation of vector calculus, ${\displaystyle \nabla =(\partial _{1},\partial _{2},\partial _{3})}$, with some approximations, this equation can be written as

${\displaystyle \rho \partial _{t}^{2}{\boldsymbol {u}}=\left(\lambda +2\mu \right)\nabla (\nabla \cdot {\boldsymbol {u}})-\mu \nabla \times (\nabla \times {\boldsymbol {u}})}$

Taking the curl of this equation and applying vector identities, one gets

${\displaystyle \partial _{t}^{2}(\nabla \times {\boldsymbol {u}})={\frac {\mu }{\rho }}\nabla ^{2}(\nabla \times {\boldsymbol {u}})}$

This formula is the wave equation applied to the vector quantity ${\displaystyle \nabla \times {\boldsymbol {u}}}$, which is the material's shear strain. Its solutions, the S-waves, are linear combinations of sinusoidal plane waves of various wavelengths and directions of propagation, but all with the same speed ${\displaystyle \beta =\textstyle {\sqrt {\mu /\rho }}}$

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

The steady-state SH waves are defined by the Helmholtz equation[5]

${\displaystyle (\nabla ^{2}+k^{2}){\boldsymbol {u}}=0}$

where k is the wave number.

## References

1. ^ What are seismic waves? UPSeis at Michigan Tech
2. ^ S wave US Geological Survey
3. ^ University of Illinois at Chicago (17 July 1997). "Lecture 16 Seismographs and the earth's interior". Archived from the original on 7 May 2002. Retrieved 8 June 2010.
4. ^ Poisson, S. D. (1831), "Mémoire sur la propagation du mouvement dans les milieux élastiques" (Memoir on the propagation of motion in elastic media). Mémoires de l'Académie des Sciences de l'Institut de France, volume 10, pages 549–605. 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/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."
5. ^ Sheikhhassani, Ramtin (2013). "Scattering of a plane harmonic SH wave by multiple layered inclusions". Wave Motion. 51 (3): 517–532. doi:10.1016/j.wavemoti.2013.12.002.

The Adams–Williamson equation, named after L. H. Adams and E. D. Williamson, is an equation used to determine density as a function of radius, more commonly used to determine the relation between the velocities of seismic waves and the density of the Earth's interior. Given the average density of rocks at the Earth's surface and profiles of the P-wave and S-wave speeds as function of depth, it can predict how density increases with depth. It assumes that the compression is adiabatic and that the Earth is spherically symmetric, homogeneous, and in hydrostatic equilibrium. It can also be applied to spherical shells with that property. It is an important part of models of the Earth's interior such as the Preliminary reference Earth model (PREM).

Earthquake

An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth, resulting from the sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in size from those that are so weak that they cannot be felt to those violent enough to toss people around and destroy whole cities. The seismicity, or seismic activity, of an area is the frequency, type and size of earthquakes experienced over a period of time. The word tremor is also used for non-earthquake seismic rumbling.

At the Earth's surface, earthquakes manifest themselves by shaking and displacing or disrupting the ground. When the epicenter of a large earthquake is located offshore, the seabed may be displaced sufficiently to cause a tsunami. Earthquakes can also trigger landslides and occasionally, volcanic activity.

In its most general sense, the word earthquake is used to describe any seismic event—whether natural or caused by humans—that generates seismic waves. Earthquakes are caused mostly by rupture of geological faults but also by other events such as volcanic activity, landslides, mine blasts, and nuclear tests. An earthquake's point of initial rupture is called its focus or hypocenter. The epicenter is the point at ground level directly above the hypocenter.

Epicenter

The epicenter, epicentre or epicentrum in seismology is the point on the Earth's surface directly above a hypocenter or focus, the point where an earthquake or an underground explosion originates.

Hypocenter

A hypocenter (or hypocentre) (from Ancient Greek: ὑπόκεντρον [hypόkentron] for 'below the center') is the point of origin of an earthquake or a subsurface nuclear explosion. In seismology, it is a synonym of the focus. The term hypocenter is also used as a synonym for ground zero, the surface point directly beneath a nuclear airburst.

Kim Hyung-jun discography

South Korean singer and the youngest member of SS501, Kim Hyung-jun has released two EPs, 10 singles, ten soundtrack contribution songs, four collaboration songs, and three DVDs.

During 2005-2010, Kim has had three solo songs from SS501 albums: "Sayonara Ga Dekinai" from Kokoro, "I Am" from U R Man album, and "Hey G" from SS501 Solo Collection. He also released two Korean singles entitled Men from Mars, Women from Venus and S-Wave 1st Present, and contributed "Lonely Girl" with Kim Kyu-jong from Bad Girl Diary OST.

On March 8, 2011, Kim released his debut solo mini album My Girl with music videos for the two lead tracks "oH! aH!" and "Girl". A Japanese version was released on April 6 with two bonus tracks of Japanese versions of the two lead tracks.In July 2012, he released his second mini album entitled Escape with its single "Sorry, I'm Sorry". The album includes five songs, and a 25-minute drama MV starring Kang Ji-hwan, Lee Ki-woo, and himself. He, then, held his first solo live tour ‘2012 1st Story in Japan' in April and '2012 2nd Story in Japan' in August consecutively.In February 2013, it was announced that Kim will be having his very first solo concert in Korea to commemorate his 2nd anniversary since his solo debut in 2011. Ticket sales were pre-released on February 12 Kim Hyung Jun "The First", the name of the concert, was held on March 9 at the Woori Art Hall at the Olympic Park, organized by SPLUS Entertainment and managed by SH Creative Works.

Left ventricular hypertrophy

Left ventricular hypertrophy (LVH) is thickening of the heart muscle of the left ventricle of the heart, that is, left-sided ventricular hypertrophy.

Lehmann discontinuity

The Lehmann discontinuity is an abrupt increase of P-wave and S-wave velocities at the depth of 220 km (140 mi), discovered by seismologist Inge Lehmann. The thickness is 220 km. It appears beneath continents, but not usually beneath oceans, and does not readily appear in globally averaged studies. Several explanations have been proposed: a lower limit to the pliable asthenosphere, a phase transition, and most plausibly, depth variation in the shear wave anisotropy. Further discussion of the Lehmann discontinuity can be found in the book Deformation of Earth Materials by Shun-ichirō Karato.

P-wave

A P-wave is one of the two main types of elastic body waves, called seismic waves in seismology. P-waves travel faster than other seismic waves and hence are the first signal from an earthquake to arrive at any affected location or at a seismograph. P-waves may be transmitted through gases, liquids, or solids.

QRS complex

The QRS complex is the combination of three of the graphical deflections seen on a typical electrocardiogram (EKG or ECG). It is usually the central and most visually obvious part of the tracing; in other words, it's the main spike seen on an ECG line. It corresponds to the depolarization of the right and left ventricles of the human heart and contraction of the large ventricular muscles.

In adults, the QRS complex normally lasts 0.06–0.10 s; in children and during physical activity, it may be shorter. The Q, R, and S waves occur in rapid succession, do not all appear in all leads, and reflect a single event and thus are usually considered together. A Q wave is any downward deflection immediately following the P wave. An R wave follows as an upward deflection, and the S wave is any downward deflection after the R wave. The T wave follows the S wave, and in some cases, an additional U wave follows the T wave.

Right bundle branch block

A right bundle branch block (RBBB) is a heart block in the right bundle branch of the electrical conduction system.During a right bundle branch block, the right ventricle is not directly activated by impulses travelling through the right bundle branch. The left ventricle however, is still normally activated by the left bundle branch. These impulses are then able to travel through the myocardium of the left ventricle to the right ventricle and depolarize the right ventricle this way. As conduction through the myocardium is slower than conduction through the Bundle of His-Purkinje fibres, the QRS complex is seen to be widened. The QRS complex often shows an extra deflection that reflects the rapid depolarisation of the left ventricle followed by the slower depolarisation of the right ventricle.

It is seen in healthy individuals in about 1.5-3%.

Scattering length

The scattering length in quantum mechanics describes low-energy scattering. It is defined as the following low-energy limit:

${\displaystyle \lim _{k\to 0}k\cot \delta (k)=-{\frac {1}{a}}\;,}$

where ${\displaystyle a}$ is the scattering length, ${\displaystyle k}$ is the wave number, and ${\displaystyle \delta (k)}$ is the phase shift of the outgoing spherical wave. The elastic cross section, ${\displaystyle \sigma _{e}}$, at low energies is determined solely by the scattering length:

${\displaystyle \lim _{k\to 0}\sigma _{e}=4\pi a^{2}\;.}$
Seismic refraction

Seismic refraction is a geophysical principle (see refraction) governed by Snell's Law. Used in the fields of engineering geology, geotechnical engineering and exploration geophysics, seismic refraction traverses (seismic lines) are performed using a seismograph(s) and/or geophone(s), in an array and an energy source. The seismic refraction method utilizes the refraction of seismic waves on geologic layers and rock/soil units in order to characterize the subsurface geologic conditions and geologic structure.

The methods depend on the fact that seismic waves have differing velocities in different types of soil (or rock): in addition, the waves are refracted when they cross the boundary between different types (or conditions) of soil or rock. The methods enable the general soil types and the approximate depth to strata boundaries, or to bedrock, to be determined.

Seismic tomography

Seismic tomography is a technique for imaging the subsurface of the Earth with seismic waves produced by earthquakes or explosions. P-, S-, and surface waves can be used for tomographic models of different resolutions based on seismic wavelength, wave source distance, and the seismograph array coverage. The data received at seismometers are used to solve an inverse problem, wherein the locations of reflection and refraction of the wave paths are determined. This solution can be used to create 3D images of velocity anomalies which may be interpreted as structural, thermal, or compositional variations. Geoscientists use these images to better understand core, mantle, and plate tectonic processes.

Seismic wave

Seismic waves are waves of energy that travel through the Earth's layers, and are a result of earthquakes, volcanic eruptions, magma movement, large landslides and large man-made explosions that give out low-frequency acoustic energy. Many other natural and anthropogenic sources create low-amplitude waves commonly referred to as ambient vibrations. Seismic waves are studied by geophysicists called seismologists. Seismic wave fields are recorded by a seismometer, hydrophone (in water), or accelerometer.

The propagation velocity of seismic waves depends on density and elasticity of the medium as well as the type of wave. Velocity tends to increase with depth through Earth's crust and mantle, but drops sharply going from the mantle to the outer core.Earthquakes create distinct types of waves with different velocities; when reaching seismic observatories, their different travel times help scientists to locate the source of the hypocenter. In geophysics the refraction or reflection of seismic waves is used for research into the structure of the Earth's interior, and man-made vibrations are often generated to investigate shallow, subsurface structures.

A seismic shadow zone is an area of the Earth's surface where seismographs can only barely detect an earthquake after its seismic waves have passed through the Earth. When an earthquake occurs, seismic waves radiate out spherically from the earthquake's focus. The primary seismic waves are refracted by the liquid outer core of the Earth and are not detected between 104° and 140° (between approximately 11,570 and 15,570 km or 7,190 and 9,670 mi) from the epicenter. The secondary seismic waves cannot pass through the liquid outer core and are not detected more than 104° (approximately 11,570 km or 7,190 mi) from the epicenter. P waves that have been converted to s-waves on leaving the outer core may be detected beyond 140 degrees.

The reason for this is that the velocity for P-waves and S-waves is governed by both the different properties in the material which they travel through and the different mathematical relationships they share in each case. The three properties are: incompressibility (${\displaystyle k}$), density (${\displaystyle p}$) and rigidity (${\displaystyle u}$). P-wave velocity is equal to ${\displaystyle {\sqrt {(k+{\tfrac {4}{3}}u)/p}}}$ whereas S-wave velocity is equal to ${\displaystyle {\sqrt {(u/p)}}}$ and so S-wave velocity is entirely dependent on the rigidity of the material it travels through. Liquids, however, have zero rigidity, hence always making the S-wave velocity overall zero and as such S-waves lose all velocity when travelling through a liquid. P-waves, however, are only partially dependent on rigidity and as such still maintain some velocity (if greatly reduced) when travelling through a liquid. Analysis of the seismology of various recorded earthquakes and their shadow zones led geologist Richard Oldham to deduce in 1906 the liquid nature of the Earth's outer core.

Supershear earthquake

A supershear earthquake is an earthquake in which the propagation of the rupture along the fault surface occurs at speeds in excess of the seismic shear wave (S-wave) velocity. This causes an effect analogous to a sonic boom.

The Gun Club

The Gun Club were an American post-punk/blues band from Los Angeles, California, United States, that existed from 1979 to 1996. Created and led by singer, guitarist and songwriter Jeffrey Lee Pierce, they merged the contemporary genre of punk rock with the more traditional genres of rockabilly and country music along with X, the Flesh Eaters and the Blasters. The Gun Club has been called a "tribal psychobilly blues" band and initiators of the U.S. wave of cowpunk.

Types of earthquake

This is a list of different types of earthquake.

Wave 105

Wave 105 is a British regional commercial radio station broadcasting across East Dorset, South Hampshire, Isle of Wight and parts of West Sussex and Wiltshire. Playing a mix of adult contemporary music, it combines presenter-led shows with local news and information, entertainment guides and competitions. The station forms part of the Bauer City network, although its entire output and playlist is locally produced and takes no network programming.

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