Transversal Wave
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In
physics Physics is the natural science that studies matter, its 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 which r ...
, a transverse wave is a
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a
longitudinal wave Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves ...
which travels in the direction of its oscillations.
Water wave In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of t ...
s are an example of transverse 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 she ...
. 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 or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
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 that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic field A magnetic field is a vector 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 to its own velocity and to ...
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 used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...
solids due to the
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
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 and the propagation of elastic waves through the Earth or through other ...
, shear waves are also called secondary waves or S-waves. Transverse waves are contrasted with
longitudinal waves Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical wave, Mechanical l ...
, 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.


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 just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
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 ''d'' be the direction of propagation (a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
with unit length), and ''o'' any reference point in the medium. Let ''u'' be the direction of the oscillations (another unit-length vector perpendicular to ''d''). The displacement of a particle at any point ''p'' of the medium and any time ''t'' (seconds) will be S(p,t) = A u \sin\left(\frac + \phi\right) where ''A'' is the wave's amplitude or strength, ''T'' is its period, ''v'' is the speed of propagation, and ''φ'' is its phase at ''o''. 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. The symbol "•" denotes the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of two vectors. By this equation, the wave travels in the direction ''d'' and the oscillations occur back and forth along the direction ''u''. The wave is said to be linearly polarized in the direction ''u''. An observer that looks at a fixed point ''p'' will see the particle there move in a simple
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
(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 ''d'', with the displacements in successive planes forming a sinusoidal pattern, with each full cycle extending along ''d'' by the wavelength ''λ'' = ''v'' ''T'' = ''v''/''f''. The whole pattern moves in the direction ''d'' with speed ''V''. The same equation describes a plane linearly polarized sinusoidal light wave, except that the "displacement" ''S''(''p'', ''t'') is the electric field at point ''p'' and time ''t''. (The magnetic field will be described by the same equation, but with a "displacement" direction that is perpendicular to both ''d'' and ''u'', and a different amplitude.)


Superposition principle

In a
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
medium, complex oscillations (vibrations in a material or light flows) can be described as the superposition of many simple sinusoidal waves, either transverse or longitudinal. The vibrations of a violin string, for example, can be analyzed as the sum of many transverse waves of different frequencies, that displace the string either up or down or left to right. The ripples in a pond can be analyzed as a combination of transverse and longitudinal waves (
gravity wave In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
s) that propagate together.


Circular polarization

If the medium is linear and allows multiple independent displacement directions for the same travel direction ''d'', 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 elli ...
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 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 in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves ...
*
Luminiferous aether Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), so ...
– 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 Polarization (waves), polarized shear wave enters an anisotropic medium (Fig. 1). The incident shear wave splits into two polarized shear waves (Fig. 2). ...
*
Sinusoidal plane-wave solutions of the electromagnetic wave equation Sinusoidal plane-wave solutions are particular solutions to the electromagnetic wave equation. The general solution of the electromagnetic wave equation in homogeneous, linear, time-independent media can be written as a linear superposition 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 microwav ...
*
Elastography Elastography is any of a class of medical imaging modalities that map the elastic properties and stiffness of soft tissue.Sarvazyan A, Hall TJ, Urban MW, Fatemi M, Aglyamov SR, Garra BSOverview of elastography–an emerging branch of medical im ...
*
Shear-wave elasticity imaging Elastography is any of a class of medical imaging modalities that map the elasticity (physics), elastic properties and stiffness of soft tissue.Sarvazyan A, Hall TJ, Urban MW, Fatemi M, Aglyamov SR, Garra BSOverview of elastography–an emerging ...


References


External links


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