Editing Transverse wave

{{short description|Moving wave that has oscillations perpendicular to the direction of the wave}}
{{More citations needed|date=May 2019}}
[[File:Onde cisaillement impulsion 1d 30 petit.gif|thumb|305px|Illustration of a simple (plane) transverse wave propagating through an elastic medium in the horizontal direction, with particles being displaced in the vertical direction.  Only one layer of the material is shown]]
[[File:Electromagneticwave3D.gif|thumb|305px|Illustration of the electric (red) and magnetic (blue) fields along a ray in a simple light wave.  For any plane perpendicular to the ray, each field has always the same value at all points of the plane.]]
[[Image:Ondes cisaillement 2d 20 petit.gif|thumb|305px|Propagation of a transverse spherical wave in a 2d grid (empirical model)]]

In [[physics]], a '''transverse wave''' is a [[wave]] that oscillates perpendicularly to the direction of the wave's advance. In contrast, a [[longitudinal wave]] travels in the direction of its oscillations. All waves move energy from place to place without transporting the matter in the [[transmission medium]] if there is one.<ref>{{Cite web |title=Transverse Waves |url=https://www.physics.wisc.edu/ingersollmuseum/exhibits/waves/transverse/ |access-date=2024-03-06 |website=L.R. Ingersoll Physics Museum |language=en-US}}</ref><ref>{{Cite web |date=2020-03-05 |title=Explainer: Understanding waves and wavelengths |url=https://www.snexplores.org/article/explainer-understanding-waves-and-wavelengths |access-date=2024-03-06 |language=en-US}}</ref> [[Electromagnetic waves]] are transverse without requiring a medium.<ref>{{Cite web |title=Transverse Waves |url=https://www.memphis.edu/physics/demonstrations/transverse_waves.php |access-date=2024-03-06 |website=www.memphis.edu |language=en}}</ref> The designation “transverse” indicates the direction of the wave is perpendicular to the displacement of the particles of the medium through which it passes, or in the case of EM waves, the oscillation is perpendicular to the direction of the wave.<ref>{{Cite web |title=Physics Tutorial: The Anatomy of a Wave |url=https://www.physicsclassroom.com/class/waves/Lesson-2/The-Anatomy-of-a-Wave |access-date=2024-03-06 |website=www.physicsclassroom.com}}</ref>

A simple example is given by the waves that can be created on a horizontal length of string by anchoring one end and moving the other end up and down. Another example is the waves that are created on the membrane of a [[drum]]. The waves propagate in directions that are parallel to the membrane plane, but each point in the membrane itself gets displaced up and down, perpendicular to that plane. [[Light]] is another example of a transverse wave, where the oscillations are the [[electric field|electric]] and [[magnetic field]]s, which point at right angles to the ideal light rays that describe the direction of propagation.

Transverse waves commonly occur in [[elasticity (physics)|elastic]] solids due to the [[shear stress]] generated; the oscillations in this case are the displacement of the solid particles away from their relaxed position, in directions perpendicular to the propagation of the wave.  These displacements correspond to a local [[shear deformation]] of the material. Hence a transverse wave of this nature is called a '''shear wave'''. Since fluids cannot resist shear forces while at rest, propagation of transverse waves inside the bulk of fluids is not possible.<ref>{{Cite web|title=Fluid Mechanics II: Viscosity and Shear stresses|url=http://www.homepages.ucl.ac.uk/~uceseug/Fluids2/Notes_Viscosity.pdf#page=1|access-date=|website=}}</ref> In [[seismology]], shear waves are also called '''secondary waves''' or '''S-waves'''.

Transverse waves are contrasted with [[longitudinal waves]], where the oscillations occur in the direction of the wave.  The standard example of a longitudinal wave is a [[sound wave]] or "pressure wave" in gases, liquids, or solids, whose oscillations cause compression and expansion of the material through which the wave is propagating.  Pressure waves are called "primary waves", or "P-waves" in geophysics.

[[Gravity wave|Water wave]]s involve both longitudinal and transverse motions.<ref>{{Cite web|title=Longitudinal and Transverse Wave Motion|url=https://www.acs.psu.edu/drussell/demos/waves/wavemotion.html}}</ref>

==Mathematical formulation==
Mathematically, the simplest kind of transverse wave is a '''plane linearly polarized sinusoidal''' one.  "Plane" here means that the direction of propagation is unchanging and the same over the whole medium; "[[polarization (waves)|linearly polarized]]" means that the direction of displacement too is unchanging and the same over the whole medium; and the magnitude of the displacement is a [[sinusoidal]] function only of time and of position along the direction of propagation.

The motion of such a wave can be expressed mathematically as follows.  Let <math>\widehat{d}</math> be the direction of propagation (a [[vector (mathematics)|vector]] with unit length), and <math>\vec{o}</math> any reference point in the medium.  Let <math>\widehat{u}</math> be the direction of the oscillations (another unit-length vector perpendicular to ''d'').  The displacement of a particle at any point <math>\vec{p}</math> of the medium and any time ''t'' (seconds) will be
<math display="block">S(\vec{p},t) = A \sin\left( (2\pi)\frac{t-\frac{(\vec{p}-\vec{o})}{v}\cdot\widehat{d}}{T} + \phi\right) \widehat{u}</math>
where ''A'' is the wave's '''amplitude''' or '''strength''', ''T'' is its '''period''', ''v'' is the '''speed''' of propagation, and <math>\phi</math> is its '''phase''' at t = 0 seconds at <math>\vec{o}</math>.  All these parameters are [[real number]]s. The symbol "•" denotes the [[inner product]] of two vectors.

By this equation, the wave travels in the direction <math>\widehat{d}</math> and the oscillations occur back and forth along the direction <math>\widehat{u}</math>.   The wave is said to be linearly polarized in the direction <math>\widehat{u}</math>.

An observer that looks at a fixed point <math>\vec{p}</math> will see the particle there move in a simple [[Simple harmonic motion|harmonic]] (sinusoidal) motion with period ''T'' seconds, with maximum particle displacement ''A'' in each sense; that is, with a '''frequency''' of ''f'' = 1/''T'' full oscillation cycles every second.  A snapshot of all particles at a fixed time ''t'' will show the same displacement for all particles on each plane perpendicular to <math>\widehat{d}</math>, with the displacements in successive planes forming a sinusoidal pattern, with each full cycle extending along <math>\widehat{d}</math> by the '''wavelength''' ''λ'' = ''v'' ''T'' = ''v''/''f''.  The whole pattern moves in the direction <math>\widehat{d}</math>with speed ''V''.

The same equation describes a plane linearly polarized sinusoidal light wave, except that the "displacement" ''S''(<math>\vec{p}</math>, ''t'') is the electric field at point <math>\vec{p}</math> and time ''t''.  (The magnetic field will be described by the same equation, but with a "displacement" direction that is perpendicular to both <math>\widehat{d}</math> and <math>\widehat{u}</math>, and a different amplitude.)

===Superposition principle===
In a [[homogeneous]] [[Linear combination|linear]] medium, complex oscillations (vibrations in a material or light flows) can be described as the [[wave superposition|superposition]] of many simple sinusoidal waves, either transverse or longitudinal.

The vibrations of a violin string create [[standing waves]],<ref>University Physics, Vol. 1, Chapter 16.6, “Standing Waves and Resonance” University of Central Florida, https://pressbooks.online.ucf.edu/osuniversityphysics/chapter/16-6-standing-waves-and-resonance/.</ref> for example, which can be analyzed as the sum of many transverse waves of different frequencies moving in opposite directions to each other, that displace the string either up or down or left to right. The [[antinodes]] of the waves align in a superposition .

=== Circular polarization === 
If the medium is linear and allows multiple independent displacement directions for the same travel direction <math>\widehat{d}</math>, we can choose two mutually perpendicular directions of polarization, and express any wave linearly polarized in any other direction as a linear combination (mixing) of those two waves.

By combining two waves with same frequency, velocity, and direction of travel, but with different phases and independent displacement directions, one obtains a [[circular polarization|circularly]] or [[elliptical polarization|elliptically polarized]] wave.  In such a wave the particles describe circular or elliptical trajectories, instead of moving back and forth.

It may help understanding to revisit the thought experiment with a taut string mentioned above. Notice that you can also launch waves on the string by moving your hand to the right and left instead of up and down. This is an important point. There are two independent (orthogonal) directions that the waves can move. (This is true for any two directions at right angles, up and down and right and left are chosen for clarity.) Any waves launched by moving your hand in a straight line are linearly polarized waves.

But now imagine moving your hand in a circle. Your motion will launch a spiral wave on the string. You are moving your hand simultaneously both up and down and side to side. The maxima of the side to side motion occur a quarter wavelength (or a quarter of a way around the circle, that is 90 degrees or π/2 radians) from the maxima of the up and down motion. At any point along the string, the displacement of the string will describe the same circle as your hand, but delayed by the propagation speed of the wave. Notice also that you can choose to move your hand in a clockwise circle or a counter-clockwise circle. These alternate circular motions produce right and left circularly polarized waves.

To the extent your circle is imperfect, a regular motion will describe an ellipse, and produce elliptically polarized waves. At the extreme of eccentricity your ellipse will become a straight line, producing linear polarization along the major axis of the ellipse. An elliptical motion can always be decomposed into two orthogonal linear motions of unequal amplitude and 90 degrees out of phase, with circular polarization being the special case where the two linear motions have the same amplitude.

[[File:Polarizacio.jpg|thumb|upright=1.25|Circular polarization mechanically generated on a rubber thread, converted to linear polarization by a mechanical polarizing filter.]]
<!--Raleigh waves, gravity waves, wave bending, P-S wave conversion, ... -->
<!--TO CLEAN UP
==Mechanical transverse waves==
The compressional wave speed is related to the bulk elasticity modulus of the medium while the shear wave speed is related to the shear elasticity modulus.

===Strings===

===Membranes===

===Solids===

==Electromagnetic waves== 
Electromagnetic waves behave in this same way. Electromagnetic waves are also two-dimensional transverse waves. 
Transverse waves are waves that travel perpendicular to the direction of the vibration.
Ray theory does not describe phenomena such as interference and diffraction, which require wave theory (involving the phase of the wave).
You can think of a [[Ray (optics)|ray of light]], in [[optics]], as an idealized narrow beam of electromagnetic radiation. Rays are used to model the propagation of [[light]] through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of [[Ray tracing (physics)|ray tracing]]. <ref name=zmx1>{{cite web  |url=http://kb-en.radiantzemax.com/Knowledgebase/What-is-a-ray  |title=What is a ray?  |first=Ken  |last=Moore  |date=2005-07-25  |work=ZEMAX Users' Knowledge Base  |access-date=2008-05-30  |url-status=dead  |archive-url=https://archive.today/20130624192746/http://kb-en.radiantzemax.com/Knowledgebase/What-is-a-ray  |archive-date=2013-06-24  }}</ref>            
A light ray is a line or curve that is perpendicular to the light's wavefronts (and is therefore collinear with the wave vector). Light rays bend at the interface between two dissimilar media and may be curved in a medium in which the [[refractive index]] changes. Geometric optics describes how rays propagate through an optical system.<ref name=zmx1>{{cite web  |url=http://kb-en.radiantzemax.com/Knowledgebase/What-is-a-ray  |title=What is a ray?  |first=Ken  |last=Moore  |date=2005-07-25  |work=ZEMAX Users' Knowledge Base  |access-date=2008-05-30  |url-status=dead  |archive-url=https://archive.today/20130624192746/http://kb-en.radiantzemax.com/Knowledgebase/What-is-a-ray  |archive-date=2013-06-24  }}</ref>

This two-dimensional nature should not be confused with the two components of an electromagnetic wave, the electric and magnetic field components, which are shown in the light wave diagram here. Each of these fields, the electric and the magnetic, exhibits two-dimensional transverse wave behavior, just like the waves on a string.-->
<!--MATERIAL FROM [[shear waves]] before redirection -- probably already all in [[elastography]]:
In soft biological tissues the bulk modulus varies no more than 10% while variation of the shear elasticity modulus may be several orders of magnitude depending on the structure and state of tissue.<ref>Sarvazyan AP, Skovoroda AR, Emelianov SY, Fowlkes JB, Pipe JG, Adler RS, Buxton RB, Carson PL. Biophysical bases of elasticity imaging. In: Acoustical Imaging. Ed. Jones JP, Plenum Press, New York and London, 1995; 21: 223-240.</ref><ref>Sarvazyan AP, Urban MW, Greenleaf JF. Acoustic waves in medical imaging and diagnostics. Ultrasound Med Biol. 2013 Jul;39(7):1133-46. {{doi|10.1016/j.ultrasmedbio.2013.02.006}}. Epub 2013 Apr 30. Review. {{PMID|23643056}}</ref>

==Shear Waves in Elastography==
The bulk modulus is defined by short range molecular interaction forces and depends mainly on molecular composition of tissue which is typically 75% water with little variation. In contrast to that, the shear modulus is defined by long range interactions and is highly sensitive to structural changes.  The wide range of variability makes the shear elasticity modulus and, respectively, the shear wave speed, highly sensitive to physiological and pathological structural changes of tissue. For this reason, the use of shear waves in new diagnostic methods and devices has been extensively investigated over the last two decades. Numerous new methods were developed most notable of which are [[Shear Wave Elasticity Imaging]] (SWEI), [[Magnetic Resonance Elastography]] (MRE), Supersonic Shear Imaging  (SSI), Shearwave Dispersion Ultrasound Vibrometry (SDUV), Harmonic Motion Imaging (HMI), Comb-push Ultrasound Shear Elastography (CUSE), and Spatially Modulated Ultrasound Radiation Force (SMURF).<ref>Sarvazyan AP, Rudenko OV, Swanson SD, Fowlkes JB, and Emelianov SY, Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics. Ultrasound Med. Biol. 1998; 24: 1419-35.</ref><ref>Muthupillai R, Lomas DJ, Rossman PJ, Greenleaf JF, Manduca A, and Ehman RL, Magnetic resonance elastography by direct visualization of propagating acoustic strain waves. Science 1995; 269: 1854-7.</ref><ref>Bercoff J, Tanter M, and Fink M, Supersonic shear imaging: a new technique for soft tissue elasticity mapping. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 2004; 51: 396-409.</ref><ref>Chen S, Urban MW, Pislaru C, Kinnick R, Zheng Y, Yao A, and Greenleaf JF, Shearwave dispersion ultrasound vibrometry (SDUV) for measuring tissue elasticity and viscosity. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 2009; 56: 55-6.</ref><ref>Vappou J, Maleke C, and Konofagou EE, Quantitative viscoelastic parameters measured by harmonic motion imaging. Phys. Med. Biol. 2009; 54: 3579-3594.</ref><ref>Song P, Zhao H, Manduca A, Urban M W, Greenleaf J F, and Chen S, "Comb-push ultrasound shear elastography (CUSE): a novel method for two-dimensional shear elasticity imaging of soft tissues," IEEE Trans. Med. Imaging, vol. 31, pp. 1821-1832, 2012.</ref><ref>9.	McAleavey S. A., Menon M., and Orszulak J., "Shear-modulus estimation by application of spatially-modulated impulsive acoustic radiation force," Ultrason. Imaging, vol. 29, pp. 87-104, 2007.</ref>
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=== Power in a transverse wave in string ===
(Let the linear mass density of the string be μ.)

The kinetic energy of a mass element in a transverse wave is given by:
<math display="block"> dK = \frac 1 2 \ dm \ v_y^2 = \frac12 \ \mu dx \ A^2 \omega^2 \cos^2 \left(\frac{2 \pi x}{\lambda} - \omega t\right)</math>

In one wavelength, kinetic energy
<math display="block"> K = \frac 1 2 \mu A ^2 \omega^2 \int ^\lambda _0 \cos^2 \left(\frac{2 \pi x}{\lambda} - \omega t\right) dx = \frac14 \mu A^2 \omega^2 \lambda</math>

Using [[Hooke's law]] the potential energy in mass element
<math display="block"> dU = \frac 1 2 \ dm \omega ^ 2 \ y ^ 2 = \frac 1 2 \ \mu dx \omega ^ 2 \ A^2 \sin^2 \left(\frac{2 \pi x}{\lambda} - \omega t\right)</math>

And the potential energy for one wavelength
<math display="block"> U = \frac 1 2 \mu A ^2 \omega^2 \int ^\lambda _0 \sin^2 \left(\frac{2 \pi x}{\lambda} - \omega t\right) dx = \frac 1 4 \mu A^2 \omega^2 \lambda</math>

So, total energy in one wavelength <math display="inline"> K + U = \frac 1 2 \mu A^2 \omega^2 \lambda</math>

Therefore average power is <math display="inline"> \frac 1 2 \mu A^2 \omega^2 v_x</math><ref>{{Cite web|title=16.4 Energy and Power of a Wave - University Physics Volume 1 {{!}} OpenStax|url=https://openstax.org/books/university-physics-volume-1/pages/16-4-energy-and-power-of-a-wave|access-date=2022-01-28|website=openstax.org|date=19 September 2016 |language=en}}</ref>

==See also==
* [[Longitudinal wave]]
* [[Luminiferous aether]] – the postulated medium for light waves; accepting that light was a transverse wave prompted a search for evidence of this physical medium
* [[Shear wave splitting]]
* [[Sinusoidal plane-wave solutions of the electromagnetic wave equation]]
* [[Transverse mode]]
* [[Elastography]]
* [[Shear-wave elasticity imaging]]

== References ==
{{Reflist}}

==External links==
*[http://www.phy.hk/wiki/englishhtm/TwaveA.htm Interactive simulation of transverse wave]
*[http://www.acoustics.salford.ac.uk/feschools/waves/wavetypes.htm Wave types explained with high speed film and animations]
*{{ScienceWorld |title= Transverse Wave |urlname= physics/TransverseWave}}
*[http://cnx.org/content/m12378/latest/ Transverse and Longitudinal Waves] Introductory module on these waves at [[OpenStax CNX|Connexions]]

{{Strings (music)}}
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[[Category:Wave mechanics]]
[[Category:Acoustics]]
[[Category:Waves]]
[[Category:Polarization (waves)]]