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In
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, a transverse wave is a
wave In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium ...
that oscillates perpendicularly to the direction of the wave's advance. In contrast, a
longitudinal wave Longitudinal waves are waves which oscillate in the direction which is parallel to the direction in which the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal ...
travels in the direction of its oscillations. All waves move energy from place to place without transporting the matter in the
transmission medium A transmission medium is a system or substance that can mediate the propagation of signals for the purposes of telecommunication. Signals are typically imposed on a wave of some kind suitable for the chosen medium. For example, data can modula ...
if there is one.
Electromagnetic waves In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength, ran ...
are transverse without requiring a medium. 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. 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 drum is a member of the percussion group of musical instruments. In the Hornbostel–Sachs classification system, it is a membranophone. Drums consist of at least one membrane, called a drumhead or drum skin, that is stretched over a ...
. 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 Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
is another example of a transverse wave, where the oscillations are the
electric Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
and
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
s, which point at right angles to the ideal light rays that describe the direction of propagation. Transverse waves commonly occur in
elastic Elastic is a word often used to describe or identify certain types of elastomer, Elastic (notion), elastic used in garments or stretch fabric, stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rub ...
solids due to the
shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
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. In
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes (or generally, quakes) and the generation and propagation of elastic ...
, 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 In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
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.
Water wave In fluid dynamics, a wind wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result of the wind blowing over the water's surface. The contact distance in the direction of the wind is ...
s involve both longitudinal and transverse motions.


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; " 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 A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is '' simple harmonic motion''; as rotation, it correspond ...
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 \widehat be the direction of propagation (a
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
with unit length), and \vec any reference point in the medium. Let \widehat be the direction of the oscillations (another unit-length vector perpendicular to ''d''). The displacement of a particle at any point \vec of the medium and any time ''t'' (seconds) will be S(\vec,t) = A \sin\left( (2\pi)\frac + \phi\right) \widehat where ''A'' is the wave's amplitude or strength, ''T'' is its period, ''v'' is the speed of propagation, and \phi is its phase at t = 0 seconds at \vec. All these parameters are
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. The symbol "•" denotes the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
of two vectors. By this equation, the wave travels in the direction \widehat and the oscillations occur back and forth along the direction \widehat. The wave is said to be linearly polarized in the direction \widehat. An observer that looks at a fixed point \vec will see the particle there move in a simple
harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
(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 \widehat, with the displacements in successive planes forming a sinusoidal pattern, with each full cycle extending along \widehat by the wavelength ''λ'' = ''v'' ''T'' = ''v''/''f''. The whole pattern moves in the direction \widehatwith speed ''V''. The same equation describes a plane linearly polarized sinusoidal light wave, except that the "displacement" ''S''(\vec, ''t'') is the electric field at point \vec and time ''t''. (The magnetic field will be described by the same equation, but with a "displacement" direction that is perpendicular to both \widehat and \widehat, and a different amplitude.)


Superposition principle

In a
homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
medium, complex oscillations (vibrations in a material or light flows) can be described as the
superposition In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ...
of many simple sinusoidal waves, either transverse or longitudinal. The vibrations of a violin string create
standing waves In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect t ...
, 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 A node is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effec ...
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 \widehat, 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 circularly or
elliptically polarized In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An el ...
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.


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: dK = \frac 1 2 \ dm \ v_y^2 = \frac12 \ \mu dx \ A^2 \omega^2 \cos^2 \left(\frac - \omega t\right) In one wavelength, kinetic energy K = \frac 1 2 \mu A ^2 \omega^2 \int ^\lambda _0 \cos^2 \left(\frac - \omega t\right) dx = \frac14 \mu A^2 \omega^2 \lambda Using
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
the potential energy in mass element dU = \frac 1 2 \ dm \omega ^ 2 \ y ^ 2 = \frac 1 2 \ \mu dx \omega ^ 2 \ A^2 \sin^2 \left(\frac - \omega t\right) And the potential energy for one wavelength U = \frac 1 2 \mu A ^2 \omega^2 \int ^\lambda _0 \sin^2 \left(\frac - \omega t\right) dx = \frac 1 4 \mu A^2 \omega^2 \lambda So, total energy in one wavelength K + U = \frac 1 2 \mu A^2 \omega^2 \lambda Therefore average power is \frac 1 2 \mu A^2 \omega^2 v_x


See also

*
Longitudinal wave Longitudinal waves are waves which oscillate in the direction which is parallel to the direction in which the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal ...
*
Luminiferous aether Luminiferous aether or ether (''luminiferous'' meaning 'light-bearing') was the postulated Transmission medium, medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empt ...
– 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 Shear wave splitting, also called seismic birefringence, is the phenomenon that occurs when a polarized shear wave enters an anisotropic medium. The incident shear wave splits into two polarized shear waves. Shear wave splitting is typically ...
*
Sinusoidal plane-wave solutions of the electromagnetic wave equation Sinusoidal plane-wave solutions are particular solutions to the wave equation. The general solution of the electromagnetic wave equation in homogeneous, linear, time-independent media can be written as a linear superposition of plane-waves of ...
*
Transverse mode A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular (i.e., transverse) to the radiation's propagation direction. Transverse modes occur in radio waves and micr ...
* Elastography * Shear-wave elasticity imaging


References


External links


Interactive simulation of transverse wave
*
Transverse and Longitudinal Waves
Introductory module on these waves at Connexions {{Authority control Wave mechanics Acoustics Waves Polarization (waves)