Sinusoidal plane wave
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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 sinusoidal (or monochromatic) plane wave is a special case of
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...
: a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
whose value varies as a sinusoidal function of time and of the distance from some fixed plane. For any position \vec x in space and any time t, the value of such a field can be written as :F(\vec x, t)=A \cos\left(2\pi \nu (\vec x \cdot \vec n - c t) + \varphi\right)\, where \vec n is a unit-length vector, the direction of propagation of the wave, and "\cdot" denotes the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of two vectors. The parameter A, which may be a scalar or a vector, is called the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
of the wave; the coefficient \nu, a positive scalar, its spatial frequency; and the adimensional scalar \varphi, an angle in radians, is its initial phase or
phase shift In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it v ...
. The scalar quantity d = \vec x \cdot \vec n gives the (signed) displacement of the point \vec x from the plane that is perpendicular to \vec n and goes through the origin of the coordinate system. This quantity is constant over each plane perpendicular to \vec n. At time t = 0, the field F varies with the displacement d as a sinusoidal function :F(\vec x, 0)=A \cos\left(2\pi \nu (\vec x \cdot \vec n) + \varphi\right)\, The spatial frequency \nu is the number of full cycles per unit of length along the direction \vec n. For any other value of t, the field values are displaced by the distance c t in the direction \vec n. That is, the whole field seems to travel in that direction with velocity c. For each displacement d, the moving plane perpendicular to \vec n at distance d + c t from the origin is called a
wavefront In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
. This plane lies at distance d from the origin when t=0, and travels in the direction \vec n also with speed c; and the value of the field is then the same, and constant in time, at every one of its points. A sinusoidal plane wave could be a suitable model for 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 ...
within a volume of air that is small compared to the distance of the source (provided that there are no echos from nearly objects). In that case, F(\vec x,t)\, would be a scalar field, the deviation of
air pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The Standard atmosphere (unit), standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equival ...
at point \vec x and time t, away from its normal level. At any fixed point \vec x, the field will also vary sinusoidally with time; it will be a scalar multiple of the amplitude A, between +A and -A When the amplitude A is a vector orthogonal to \vec n, the wave is said to be
transverse Transverse may refer to: *Transverse engine, an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle *Transverse flute, a flute that is held horizontally * Transverse force (or ''Euler force''), the tangen ...
. Such waves may exhibit polarization, if A can be oriented along two non-
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
directions. When A is a vector collinear with \vec n, the wave is said to be
longitudinal Longitudinal is a geometric term of location which may refer to: * Longitude ** Line of longitude, also called a meridian * Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long axis of the vehicle, ...
. These two possibilities are exemplified by the S (shear) waves and P (pressure) waves studied 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 ...
. The formula above gives a purely "kinematic" description of the wave, without reference to whatever physical process may be causing its motion. In a mechanical or electromagnetic wave that is propagating through an
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
medium, the vector \vec n of the apparent propagation of the wave is also the direction in which energy or momentum is actually flowing. However, the two directions may be different in an anisotropic medium.This Wikipedia section has references. Wave vector#Direction of the wave vector


Alternative representations

The same sinusoidal plane wave F above can also be expressed in terms of
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
instead of cosine using the elementary identity \cos a = \sin(a + \pi/2) :F(\vec x, t)=A \sin\left(2\pi \nu (\vec x \cdot \vec n - c t) + \varphi'\right)\, where \varphi' = \varphi + \pi/2. Thus the value and meaning of the phase shift depends on whether the wave is defined in terms of sine or co-sine. Adding any integer multiple of 2\pi to the initial phase \varphi has no effect on the field. Adding an odd multiple of \pi has the same effect as negating the amplitude A. Assigning a negative value for the spatial frequency \nu has the effect of reversing the direction of propagation, with a suitable adjustment of the initial phase. The formula of a sinusoidal plane wave can be written in several other ways: *: F(\vec x,t)=A \cos (2\pi \vec x \cdot \vec n)/\lambda - t/T+ \varphi) :Here \lambda = 1/\nu is the wavelength, the distance between two wavefronts where the field is equal to the amplitude A; and T = \lambda/c is the
period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...
of the field's variation over time, seen at any fixed point in space. Its reciprocal f = 1/T is the temporal frequency of the wave measured in full cycles per unit of time. *: F(\vec x,t)=A \cos (k (\vec x \cdot \vec n) - \omega t + \varphi) :Here k = 2\pi \nu = 2\pi/\lambda is a parameter called the angular wave number (measured in radians per unit of length), and \omega = 2\pi/T is
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the variation at a fixed point (in radians per unit of time). *: F(\vec x,t)=A \cos (2\pi(\vec x \cdot \vec v) - \omega t + \varphi) :where \vec v = \nu \vec n = \vec n/\lambda is the spatial frequency vector or
wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
, a three-dimensional vector \vec v = (v_1,v_2,v_3) where v_i is the number of full cycles that occur per unit of length, at any fixed time, along any straight line parallel to coordinate axis i.


Complex exponential form

A plane sinusoidal wave may also be expressed in terms of the
complex exponential The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...
function :e^ = \exp(\boldsymbolz) = \cos z + \boldsymbol\sin z where e is the base of the
natural exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, al ...
, and \boldsymbol\, is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, defined by the equation \boldsymbol^2 = -1. With those tools, one defines the complex exponential plane wave as :U(\vec x,t)\;=\; A \exp boldsymbol(2\pi\nu(\vec x\cdot\vec n - c t) +\varphi);=\; A \exp boldsymbol(2\pi\vec x \cdot \vec v - \omega t + \varphi)/math> where A,\nu,\vec n,c,\vec v,\omega, \varphi are as defined for the (real) sinusoidal plane wave. This equation gives a field U(\vec x,t) whose value is a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, or a vector with complex coordinates. The original wave expression is now simply the real part, :F(\vec x,t) = \text (\vec x,t)/math> To appreciate this equation's relationship to the earlier ones, below is this same equation expressed using sines and cosines. Observe that the first term equals the real form of the plane wave just discussed. :U (\vec x, t ) = A \cos (2\pi\nu \vec n \cdot \vec x - \omega t + \varphi ) + \boldsymbol A \sin (2\pi\nu \vec n \cdot \vec x - \omega t + \varphi ) :U (\vec x, t ) = \qquad \ \ F (\vec x, t ) \qquad \qquad + \boldsymbol A \sin (2\pi\nu \vec n \cdot \vec x - \omega t + \varphi ) The introduced complex form of the plane wave can be simplified by using a
complex-valued amplitude In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to a ...
C\, substitute the real valued amplitude A\,.
Specifically, since the complex form :\exp boldsymbol(2\pi\vec x \cdot \vec v - \omega t +\varphi)\;=\; \exp boldsymbol(2\pi\nu \vec n\cdot\vec x - \omega t ),e^ one can absorb the
phase factor For any complex number written in polar form (such as ), the phase factor is the complex exponential factor (). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e. not necessarily on the ...
e^ into a
complex amplitude Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
by letting C=A e^, resulting in the more compact equation :U(\vec x,t)= C \exp boldsymbol(2\pi\vec x\cdot\vec v - \omega t)/math> While the complex form has an imaginary component, after the necessary calculations are performed in the complex plane, its real value (which corresponds to the wave one would actually physically observe or measure) can be extracted giving a real valued equation representing an actual plane wave. :\operatorname (\vec x,t) F (\vec x, t ) = A \cos (2\pi\nu \vec n \cdot \vec x - \omega t + \varphi ) The main reason one would choose to work with complex exponential form of plane waves is that complex exponentials are often algebraically easier to handle than the trigonometric sines and cosines. Specifically, the angle-addition rules are extremely simple for exponentials. Additionally, when using
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
techniques for waves in a
lossy medium In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in r ...
, the resulting
attenuation In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at variable att ...
is easier to deal with using complex Fourier
coefficients In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
. If a wave is traveling through a lossy medium, the amplitude of the wave is no longer constant, and therefore the wave is strictly speaking no longer a true plane wave. In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
the solutions of the Schrödinger wave equation are by their very nature complex-valued and in the simplest instance take a form identical to the complex plane wave representation above. The imaginary component in that instance however has not been introduced for the purpose of mathematical expediency but is in fact an inherent part of the “wave”. In
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
, one can utilize an even more compact expression by using
four-vectors In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
. :The
four-position In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a r ...
\vec x = (ct,\vec x) :The four-wavevector 2\pi\nu \vec n = \left(\frac,2\pi\nu \vec n\right) :The scalar product 2\pi\nu \vec n\cdot\vec x = \omega t - 2\pi\nu \vec n\cdot\vec x Thus, :U(\vec x,t)= C \exp boldsymbol(2\pi\nu \vec n\cdot\vec x - \omega t )/math> becomes :U(\vec x)= C \exp \boldsymbol(2\pi\nu \vec n\cdot\vec x)/math>


Applications

The equations describing
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
in a homogeneous
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
medium admit as special solutions that are sinusoidal plane waves. In
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
, the field F is typically the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
,
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 ...
, or
vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field v, a ''vecto ...
, which in an isotropic medium is perpendicular to the direction of propagation \vec n. The amplitude A is then a vector of the same nature, equal to the maximum-strength field. The propagation speed c will be the speed of light in the medium. The equations that describe vibrations in a homogeneous elastic solid also admit solutions that are sinusoidal plane waves, both transverse and longitudinal. These two types have different propagation speeds, that depend on the density and the
Lamé parameters In continuum mechanics, Lamé parameters (also called the Lamé coefficients, Lamé constants or Lamé moduli) are two material-dependent quantities denoted by λ and μ that arise in strain-stress relationships. In general, λ and μ are indi ...
of the medium. The fact that the medium imposes a propagation speed means that the parameters \omega and k must satisfy a
dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
characteristic of the medium. The dispersion relation is often expressed as a function, \omega(k). The ratio \omega/, k, gives the magnitude of the
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
, and the derivative \partial\omega/\partial k gives the
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
. For electromagnetism in an isotropic medium with index of refraction r, the phase velocity is c/r, which equals the group velocity if the index is not frequency-dependent. In linear uniform media, a general solution to the
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
can be expressed as a superposition of sinusoidal plane waves. This approach is known as the
angular spectrum method The angular spectrum method is a technique for modeling the propagation of a wave field. This technique involves expanding a complex wave field into a summation of infinite number of plane waves of the same frequency and different directions. Its ...
. The form of the planewave solution is actually a general consequence of
translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equatio ...
. More generally, for periodic structures having discrete translational symmetry, the solutions take the form of
Bloch wave In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who di ...
s, most famously in
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
line atomic materials but also in
photonic crystal A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of Crystal structure, natural crystals gives rise to X-ray crystallograp ...
s and other periodic wave equations. As another generalization, for structures that are only uniform along one direction x (such as a
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
along the x direction), the solutions (waveguide modes) are of the form exp (kx-\omega t)/math> multiplied by some amplitude function a(y,z). This is a special case of a
separable partial differential equation A separable partial differential equation is one that can be broken into a set of separate equations of lower dimensionality (fewer independent variables) by a method of separation of variables. This generally relies upon the problem having some s ...
.


Polarized electromagnetic plane waves

Represented in the first illustration toward the right is a linearly polarized,
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
. Because this is a plane wave, each blue
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 ...
, indicating the perpendicular displacement from a point on the axis out to the sine wave, represents the magnitude and direction of the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
for an entire plane that is perpendicular to the axis. Represented in the second illustration is a
circularly polarized In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to t ...
, electromagnetic plane wave. Each blue vector indicating the perpendicular displacement from a point on the axis out to the helix, also represents the magnitude and direction of the electric field for an entire plane perpendicular to the axis. In both illustrations, along the axes is a series of shorter blue vectors which are scaled down versions of the longer blue vectors. These shorter blue vectors are extrapolated out into the block of black vectors which fill a volume of space. Notice that for a given plane, the black vectors are identical, indicating that the magnitude and direction of the electric field is constant along that plane. In the case of the linearly polarized light, the field strength from plane to plane varies from a maximum in one direction, down to zero, and then back up to a maximum in the opposite direction. In the case of the circularly polarized light, the field strength remains constant from plane to plane but its direction steadily changes in a rotary type manner. Not indicated in either illustration is the electric field’s corresponding
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 ...
which is proportional in strength to the electric field at each point in space but is at a right angle to it. Illustrations of the magnetic field vectors would be virtually identical to these except all the vectors would be rotated 90 degrees about the axis of propagation so that they were perpendicular to both the direction of propagation and the electric field vector. The ratio of the amplitudes of the electric and magnetic field components of a plane wave in free space is known as the free-space wave-impedance, equal to 376.730313 ohms.


See also

*
Angular spectrum method The angular spectrum method is a technique for modeling the propagation of a wave field. This technique involves expanding a complex wave field into a summation of infinite number of plane waves of the same frequency and different directions. Its ...
*
Collimated beam A collimated beam of light or other electromagnetic radiation has parallel rays, and therefore will spread minimally as it propagates. A perfectly collimated light beam, with no divergence, would not disperse with distance. However, diffraction pr ...
* Plane waves in a vacuum *
Plane wave expansion In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves: e^ = \sum_^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\hat \cdot \hat), where * is the imaginary unit, * is a wave vector of length , * ...
*
Rectilinear propagation Rectilinear propagation describes the tendency of electromagnetic waves (light) to travel in a straight line. Light does not deviate when travelling through a homogeneous medium, which has the same refractive index throughout; otherwise, light suf ...
*
Wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...


References

{{Reflist * J. D. Jackson, ''Classical Electrodynamics'' (Wiley: New York, 1998). * L. M. Brekhovskikh, "Waves in Layered Media, Series:Applied Mathematics and Mechanics, Vol. 16, (Academic Press, 1980). Wave mechanics