Misplaced Pages

#739260

137-464: There are two types of waves that are most commonly studied in classical physics : mechanical waves and electromagnetic waves . In a mechanical wave, stress and strain fields oscillate about a mechanical equilibrium. A mechanical wave is a local deformation (strain) in some physical medium that propagates from particle to particle by creating local stresses that cause strain in neighboring particles too. For example, sound waves are variations of

274-475: A 1 and a 2 denote the amplitude of the wave in the two components of the electric field vector, while θ 1 and θ 2 represent the phases. The product of a Jones vector with a complex number of unit modulus gives a different Jones vector representing the same ellipse, and thus the same state of polarization. The physical electric field, as the real part of the Jones vector, would be altered but

411-421: A birefringent substance, electromagnetic waves of different polarizations travel at different speeds ( phase velocities ). As a result, when unpolarized waves travel through a plate of birefringent material, one polarization component has a shorter wavelength than the other, resulting in a phase difference between the components which increases the further the waves travel through the material. The Jones matrix

548-401: A node . Halfway between two nodes there is an antinode , where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time. A soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in

685-470: A standing wave . Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example, when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut , where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing

822-428: A transmission medium . The propagation and reflection of plane waves—e.g. Pressure waves ( P wave ) or Shear waves (SH or SV-waves) are phenomena that were first characterized within the field of classical seismology, and are now considered fundamental concepts in modern seismic tomography . The analytical solution to this problem exists and is well known. The frequency domain solution can be obtained by first finding

959-634: A vacuum and through some dielectric media (at wavelengths where they are considered transparent ). Electromagnetic waves, as determined by their frequencies (or wavelengths ), have more specific designations including radio waves , infrared radiation , terahertz waves , visible light , ultraviolet radiation , X-rays and gamma rays . Other types of waves include gravitational waves , which are disturbances in spacetime that propagate according to general relativity ; heat diffusion waves ; plasma waves that combine mechanical deformations and electromagnetic fields; reaction–diffusion waves , such as in

1096-413: A container of gas by a function F ( x , t ) {\displaystyle F(x,t)} that gives the pressure at a point x {\displaystyle x} and time t {\displaystyle t} within that container. If the gas was initially at uniform temperature and composition, the evolution of F {\displaystyle F} is constrained by

1233-400: A coordinate system and viewing the polarization ellipse in terms of the x and y polarization components, corresponds to the definition of the Jones vector (below) in terms of those basis polarizations. Axes are selected to suit a particular problem, such as x being in the plane of incidence. Since there are separate reflection coefficients for the linear polarizations in and orthogonal to

1370-406: A coupled oscillating electric field and magnetic field which are always perpendicular to each other; by convention, the "polarization" of electromagnetic waves refers to the direction of the electric field. In linear polarization , the fields oscillate in a single direction. In circular or elliptical polarization , the fields rotate at a constant rate in a plane as the wave travels, either in

1507-1253: A dielectric, η is real and has the value η 0 / n , where n is the refractive index and η 0 is the impedance of free space . The impedance will be complex in a conducting medium. Note that given that relationship, the dot product of E and H must be zero: E → ( r → , t ) ⋅ H → ( r → , t ) = e x h x + e y h y + e z h z = e x ( − e y η ) + e y ( e x η ) + 0 ⋅ 0 = 0 , {\displaystyle {\begin{aligned}{\vec {E}}\left({\vec {r}},t\right)\cdot {\vec {H}}\left({\vec {r}},t\right)&=e_{x}h_{x}+e_{y}h_{y}+e_{z}h_{z}\\&=e_{x}\left(-{\frac {e_{y}}{\eta }}\right)+e_{y}\left({\frac {e_{x}}{\eta }}\right)+0\cdot 0\\&=0,\end{aligned}}} indicating that these vectors are orthogonal (at right angles to each other), as expected. Knowing

SECTION 10

#1732772964740

1644-618: A family of waves by a function F ( A , B , … ; x , t ) {\displaystyle F(A,B,\ldots ;x,t)} that depends on certain parameters A , B , … {\displaystyle A,B,\ldots } , besides x {\displaystyle x} and t {\displaystyle t} . Then one can obtain different waves – that is, different functions of x {\displaystyle x} and t {\displaystyle t} – by choosing different values for those parameters. For example,

1781-402: A family of waves is to give a mathematical equation that, instead of explicitly giving the value of F ( x , t ) {\displaystyle F(x,t)} , only constrains how those values can change with time. Then the family of waves in question consists of all functions F {\displaystyle F} that satisfy those constraints – that is, all solutions of

1918-399: A frequency of f = c/λ where c is the speed of light), let us take the direction of propagation as the z axis. Being a transverse wave the E and H fields must then contain components only in the x and y directions whereas E z = H z = 0 . Using complex (or phasor ) notation, the instantaneous physical electric and magnetic fields are given by the real parts of

2055-403: A given path on those two components is most easily characterized in the form of a complex 2 × 2 transformation matrix J known as a Jones matrix : e ′ = J e . {\displaystyle \mathbf {e'} =\mathbf {J} \mathbf {e} .} The Jones matrix due to passage through a transparent material is dependent on the propagation distance as well as

2192-537: A homogeneous isotropic non-conducting solid. Note that this equation differs from that of heat flow only in that the left-hand side is ∂ 2 F / ∂ t 2 {\displaystyle \partial ^{2}F/\partial t^{2}} , the second derivative of F {\displaystyle F} with respect to time, rather than the first derivative ∂ F / ∂ t {\displaystyle \partial F/\partial t} . Yet this small change makes

2329-442: A huge difference on the set of solutions F {\displaystyle F} . This differential equation is called "the" wave equation in mathematics, even though it describes only one very special kind of waves. Consider a traveling transverse wave (which may be a pulse ) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling This wave can then be described by

2466-483: A more general formulation with propagation not restricted to the +z direction, then the spatial dependence kz is replaced by k → ∙ r → where k → is called the wave vector , the magnitude of which is the wavenumber. Thus the leading vectors e and h each contain up to two nonzero (complex) components describing the amplitude and phase of the wave's x and y polarization components (again, there can be no z polarization component for

2603-622: A plane wave with those given parameters can then be used to predict its response to a more general case, since a wave with any specified spatial structure can be decomposed into a combination of plane waves (its so-called angular spectrum ). Incoherent states can be modeled stochastically as a weighted combination of such uncorrelated waves with some distribution of frequencies (its spectrum ), phases, and polarizations. Electromagnetic waves (such as light), traveling in free space or another homogeneous isotropic non-attenuating medium, are properly described as transverse waves , meaning that

2740-434: A plane wave's electric field vector E and magnetic field H are each in some direction perpendicular to (or "transverse" to) the direction of wave propagation; E and H are also perpendicular to each other. By convention, the "polarization" direction of an electromagnetic wave is given by its electric field vector. Considering a monochromatic plane wave of optical frequency f (light of vacuum wavelength λ has

2877-419: A purely polarized monochromatic wave the electric field vector over one cycle of oscillation traces out an ellipse. A polarization state can then be described in relation to the geometrical parameters of the ellipse, and its "handedness", that is, whether the rotation around the ellipse is clockwise or counter clockwise. One parameterization of the elliptical figure specifies the orientation angle ψ , defined as

SECTION 20

#1732772964740

3014-426: A random mixture of waves having different spatial characteristics, frequencies (wavelengths), phases, and polarization states. However, for understanding electromagnetic waves and polarization in particular, it is easier to just consider coherent plane waves ; these are sinusoidal waves of one particular direction (or wavevector ), frequency, phase, and polarization state. Characterizing an optical system in relation to

3151-431: A significant imaginary part (or " extinction coefficient ") such as metals; these fields are also not strictly transverse. Surface waves or waves propagating in a waveguide (such as an optical fiber ) are generally not transverse waves, but might be described as an electric or magnetic transverse mode , or a hybrid mode. Even in free space, longitudinal field components can be generated in focal regions, where

3288-410: A sum of sine waves of various frequencies, relative phases, and magnitudes. A plane wave is a kind of wave whose value varies only in one spatial direction. That is, its value is constant on a plane that is perpendicular to that direction. Plane waves can be specified by a vector of unit length n ^ {\displaystyle {\hat {n}}} indicating the direction that

3425-536: A superposition of right and left circularly polarized states, with equal amplitude and phases synchronized to give oscillation in a plane. Polarization is an important parameter in areas of science dealing with transverse waves, such as optics , seismology , radio , and microwaves . Especially impacted are technologies such as lasers , wireless and optical fiber telecommunications , and radar . Most sources of light are classified as incoherent and unpolarized (or only "partially polarized") because they consist of

3562-409: A system becomes larger or more massive the classical dynamics tends to emerge, with some exceptions, such as superfluidity . This is why we can usually ignore quantum mechanics when dealing with everyday objects and the classical description will suffice. However, one of the most vigorous ongoing fields of research in physics is classical-quantum correspondence . This field of research is concerned with

3699-457: A transverse wave in the + z direction). For a given medium with a characteristic impedance η , h is related to e by: h y = e x η h x = − e y η . {\displaystyle {\begin{aligned}h_{y}&={\frac {e_{x}}{\eta }}\\h_{x}&=-{\frac {e_{y}}{\eta }}.\end{aligned}}} In

3836-420: A wave is mainly a movement of energy through a medium. Most often, the group velocity is the velocity at which the energy moves through this medium. Waves exhibit common behaviors under a number of standard situations, for example: Classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory

3973-632: A wave may be constant (in which case the wave is a c.w. or continuous wave ), or may be modulated so as to vary with time and/or position. The outline of the variation in amplitude is called the envelope of the wave. Mathematically, the modulated wave can be written in the form: u ( x , t ) = A ( x , t ) sin ⁡ ( k x − ω t + ϕ ) , {\displaystyle u(x,t)=A(x,t)\sin \left(kx-\omega t+\phi \right),} where A ( x ,   t ) {\displaystyle A(x,\ t)}

4110-435: A wave's phase and speed concerning energy (and information) propagation. The phase velocity is given as: v p = ω k , {\displaystyle v_{\rm {p}}={\frac {\omega }{k}},} where: The phase speed gives you the speed at which a point of constant phase of the wave will travel for a discrete frequency. The angular frequency ω cannot be chosen independently from

4247-534: Is a unitary matrix : | g 1 | = | g 2 | = 1 . Media termed diattenuating (or dichroic in the sense of polarization), in which only the amplitudes of the two polarizations are affected differentially, may be described using a Hermitian matrix (generally multiplied by a common phase factor). In fact, since any matrix may be written as the product of unitary and positive Hermitian matrices, light propagation through any sequence of polarization-dependent optical components can be written as

Wave - Misplaced Pages Continue

4384-582: Is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a vector in the Cartesian three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} . However, in many cases one can ignore one dimension, and let x {\displaystyle x} be a point of the Cartesian plane R 2 {\displaystyle \mathbb {R} ^{2}} . This

4521-407: Is a property of transverse waves which specifies the geometrical orientation of the oscillations . In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image) , for example, in a musical instrument like a guitar string . Depending on how

4658-440: Is almost always confined to some finite region of space, called its domain . For example, the seismic waves generated by earthquakes are significant only in the interior and surface of the planet, so they can be ignored outside it. However, waves with infinite domain, that extend over the whole space, are commonly studied in mathematics, and are very valuable tools for understanding physical waves in finite domains. A plane wave

4795-494: Is an important mathematical idealization where the disturbance is identical along any (infinite) plane normal to a specific direction of travel. Mathematically, the simplest wave is a sinusoidal plane wave in which at any point the field experiences simple harmonic motion at one frequency. In linear media, complicated waves can generally be decomposed as the sum of many sinusoidal plane waves having different directions of propagation and/or different frequencies . A plane wave

4932-538: Is classified as a transverse wave if the field disturbance at each point is described by a vector perpendicular to the direction of propagation (also the direction of energy transfer); or longitudinal wave if those vectors are aligned with the propagation direction. Mechanical waves include both transverse and longitudinal waves; on the other hand electromagnetic plane waves are strictly transverse while sound waves in fluids (such as air) can only be longitudinal. That physical direction of an oscillating field relative to

5069-443: Is commonly referred to as transverse-magnetic (TM), and has also been termed pi-polarized or π -polarized , or tangential plane polarized . S -polarization is also called transverse-electric (TE), as well as sigma-polarized or σ-polarized , or sagittal plane polarized . Degree of polarization ( DOP ) is a quantity used to describe the portion of an electromagnetic wave which is polarized. DOP can be calculated from

5206-673: Is considered to be modern, and its introduction represented a major paradigm shift , then the previous theories, or new theories based on the older paradigm, will often be referred to as belonging to the area of "classical physics". As such, the definition of a classical theory depends on context. Classical physical concepts are often used when modern theories are unnecessarily complex for a particular situation. Most often, classical physics refers to pre-1900 physics, while modern physics refers to post-1900 physics, which incorporates elements of quantum mechanics and relativity . Classical theory has at least two distinct meanings in physics. In

5343-434: Is dependent on the polarization state of a wave, properties known as birefringence and polarization dichroism (or diattenuation ) respectively, then the polarization state of a wave will generally be altered. In such media, an electromagnetic wave with any given state of polarization may be decomposed into two orthogonally polarized components that encounter different propagation constants . The effect of propagation over

5480-416: Is depicted in the animation on the right. Note that circular or elliptical polarization can involve either a clockwise or counterclockwise rotation of the field, depending on the relative phases of the components. These correspond to distinct polarization states, such as the two circular polarizations shown above. The orientation of the x and y axes used in this description is arbitrary. The choice of such

5617-401: Is generally characterized by the principle of complete determinism , although deterministic interpretations of quantum mechanics do exist. From the point of view of classical physics as being non-relativistic physics, the predictions of general and special relativity are significantly different from those of classical theories, particularly concerning the passage of time, the geometry of space,

Wave - Misplaced Pages Continue

5754-413: Is identical to one of those basis polarizations. Since the phase shift, and thus the change in polarization state, is usually wavelength-dependent, such objects viewed under white light in between two polarizers may give rise to colorful effects, as seen in the accompanying photograph. Circular birefringence is also termed optical activity , especially in chiral fluids, or Faraday rotation , when due to

5891-448: Is meant to signify that, in the derivative with respect to some variable, all other variables must be considered fixed.) This equation can be derived from the laws of physics that govern the diffusion of heat in solid media. For that reason, it is called the heat equation in mathematics, even though it applies to many other physical quantities besides temperatures. For another example, we can describe all possible sounds echoing within

6028-439: Is more commonly called in astronomy to avoid confusion with the horizontal coordinate system ) corresponding to due north. Another coordinate system frequently used relates to the plane of incidence . This is the plane made by the incoming propagation direction and the vector perpendicular to the plane of an interface, in other words, the plane in which the ray travels before and after reflection or refraction. The component of

6165-402: Is normally not even mentioned. On the other hand, sound waves in a bulk solid can be transverse as well as longitudinal, for a total of three polarization components. In this case, the transverse polarization is associated with the direction of the shear stress and displacement in directions perpendicular to the propagation direction, while the longitudinal polarization describes compression of

6302-532: Is passed through an ideal polarizer (where g 1 = 1 and g 2 = 0 ) exactly half of its initial power is retained. Practical polarizers, especially inexpensive sheet polarizers, have additional loss so that g 1 < 1 . However, in many instances the more relevant figure of merit is the polarizer's degree of polarization or extinction ratio , which involve a comparison of g 1 to g 2 . Since Jones vectors refer to waves' amplitudes (rather than intensity ), when illuminated by unpolarized light

6439-627: Is polarized. In a vacuum , the components of the electric field propagate at the speed of light , so that the phase of the wave varies in space and time while the polarization state does not. That is, the electric field vector e of a plane wave in the + z direction follows: e ( z + Δ z , t + Δ t ) = e ( z , t ) e i k ( c Δ t − Δ z ) , {\displaystyle \mathbf {e} (z+\Delta z,t+\Delta t)=\mathbf {e} (z,t)e^{ik(c\Delta t-\Delta z)},} where k

6576-458: Is the wavelength of the emitted note, and f = c / λ {\displaystyle f=c/\lambda } is its frequency .) Many general properties of these waves can be inferred from this general equation, without choosing specific values for the parameters. As another example, it may be that the vibrations of a drum skin after a single strike depend only on the distance r {\displaystyle r} from

6713-446: Is the wavenumber . As noted above, the instantaneous electric field is the real part of the product of the Jones vector times the phase factor e − i ω t {\displaystyle e^{-i\omega t}} . When an electromagnetic wave interacts with matter, its propagation is altered according to the material's (complex) index of refraction . When the real or imaginary part of that refractive index

6850-520: Is the (first) derivative of F {\displaystyle F} with respect to t {\displaystyle t} ; and ∂ 2 F / ∂ x i 2 {\displaystyle \partial ^{2}F/\partial x_{i}^{2}} is the second derivative of F {\displaystyle F} relative to x i {\displaystyle x_{i}} . (The symbol " ∂ {\displaystyle \partial } "

6987-647: Is the amplitude envelope of the wave, k {\displaystyle k} is the wavenumber and ϕ {\displaystyle \phi } is the phase . If the group velocity v g {\displaystyle v_{g}} (see below) is wavelength-independent, this equation can be simplified as: u ( x , t ) = A ( x − v g t ) sin ⁡ ( k x − ω t + ϕ ) , {\displaystyle u(x,t)=A(x-v_{g}t)\sin \left(kx-\omega t+\phi \right),} showing that

SECTION 50

#1732772964740

7124-417: Is the case, for example, when studying vibrations of a drum skin. One may even restrict x {\displaystyle x} to a point of the Cartesian line R {\displaystyle \mathbb {R} } – that is, the set of real numbers . This is the case, for example, when studying vibrations in a violin string or recorder . The time t {\displaystyle t} , on

7261-549: Is the heat that is being generated per unit of volume and time in the neighborhood of x {\displaystyle x} at time t {\displaystyle t} (for example, by chemical reactions happening there); x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} are the Cartesian coordinates of the point x {\displaystyle x} ; ∂ F / ∂ t {\displaystyle \partial F/\partial t}

7398-536: Is the length of the bore; and n {\displaystyle n} is a positive integer (1,2,3,...) that specifies the number of nodes in the standing wave. (The position x {\displaystyle x} should be measured from the mouthpiece , and the time t {\displaystyle t} from any moment at which the pressure at the mouthpiece is maximum. The quantity λ = 4 L / ( 2 n − 1 ) {\displaystyle \lambda =4L/(2n-1)}

7535-466: Is the velocity of the object and c is the speed of light. For velocities much smaller than that of light, one can neglect the terms with c and higher that appear. These formulas then reduce to the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities. Computer modeling has to be as real as possible. Classical physics would introduce an error as in

7672-402: Is the wavelength in the medium (whose refractive index is n ) and T = 1/ f is the period of the wave. Here e x , e y , h x , and h y are complex numbers. In the second more compact form, as these equations are customarily expressed, these factors are described using the wavenumber k = 2π n / λ 0 and angular frequency (or "radian frequency") ω = 2π f . In

7809-493: The Belousov–Zhabotinsky reaction ; and many more. Mechanical and electromagnetic waves transfer energy , momentum , and information , but they do not transfer particles in the medium. In mathematics and electronics waves are studied as signals . On the other hand, some waves have envelopes which do not move at all such as standing waves (which are fundamental to music) and hydraulic jumps . A physical wave field

7946-461: The Helmholtz decomposition of the displacement field, which is then substituted into the wave equation . From here, the plane wave eigenmodes can be calculated. The analytical solution of SV-wave in a half-space indicates that the plane SV wave reflects back to the domain as a P and SV waves, leaving out special cases. The angle of the reflected SV wave is identical to the incidence wave, while

8083-463: The Stokes parameters . A perfectly polarized wave has a DOP of 100%, whereas an unpolarized wave has a DOP of 0%. A wave which is partially polarized, and therefore can be represented by a superposition of a polarized and unpolarized component, will have a DOP somewhere in between 0 and 100%. DOP is calculated as the fraction of the total power that is carried by the polarized component of

8220-405: The electric displacement D and magnetic flux density B still obey the above geometry but due to anisotropy in the electric susceptibility (or in the magnetic permeability ), now given by a tensor , the direction of E (or H ) may differ from that of D (or B ). Even in isotropic media, so-called inhomogeneous waves can be launched into a medium whose refractive index has

8357-454: The electric field vector E {\displaystyle E} , or the magnetic field vector H {\displaystyle H} , or any related quantity, such as the Poynting vector E × H {\displaystyle E\times H} . In fluid dynamics , the value of F ( x , t ) {\displaystyle F(x,t)} could be

SECTION 60

#1732772964740

8494-405: The ellipticity angle , χ = arctan ⁡ b / a {\textstyle \chi =\arctan b/a} = arctan ⁡ 1 / ε {\textstyle =\arctan 1/\varepsilon } as is shown in the figure. The angle χ is also significant in that the latitude (angle from the equator) of the polarization state as represented on

8631-462: The right-hand or in the left-hand direction. Light or other electromagnetic radiation from many sources, such as the sun, flames, and incandescent lamps , consists of short wave trains with an equal mixture of polarizations; this is called unpolarized light . Polarized light can be produced by passing unpolarized light through a polarizer , which allows waves of only one polarization to pass through. The most common optical materials do not affect

8768-469: The superfluidity case. In order to produce reliable models of the world, one can not use classical physics. It is true that quantum theories consume time and computer resources, and the equations of classical physics could be resorted to in order to provide a quick solution, but such a solution would lack reliability. Computer modeling would use only the energy criteria to determine which theory to use: relativity or quantum theory, when attempting to describe

8905-427: The xy -plane, along the page, with the wave propagating in the z direction, perpendicular to the page. The first two diagrams below trace the electric field vector over a complete cycle for linear polarization at two different orientations; these are each considered a distinct state of polarization (SOP). The linear polarization at 45° can also be viewed as the addition of a horizontally linearly polarized wave (as in

9042-421: The Jones matrix can be written as J = T [ g 1 0 0 g 2 ] T − 1 , {\displaystyle \mathbf {J} =\mathbf {T} {\begin{bmatrix}g_{1}&0\\0&g_{2}\end{bmatrix}}\mathbf {T} ^{-1},} where g 1 and g 2 are complex numbers describing the phase delay and possibly

9179-463: The Poincaré sphere (see below) is equal to ±2 χ . The special cases of linear and circular polarization correspond to an ellipticity ε of infinity and unity (or χ of zero and 45°) respectively. Full information on a completely polarized state is also provided by the amplitude and phase of oscillations in two components of the electric field vector in the plane of polarization. This representation

9316-427: The amplitude attenuation due to propagation in each of the two polarization eigenmodes . T is a unitary matrix representing a change of basis from these propagation modes to the linear system used for the Jones vectors; in the case of linear birefringence or diattenuation the modes are themselves linear polarization states so T and T can be omitted if the coordinate axes have been chosen appropriately. In

9453-439: The angle between the major axis of the ellipse and the x -axis along with the ellipticity ε = a/b , the ratio of the ellipse's major to minor axis. (also known as the axial ratio ). The ellipticity parameter is an alternative parameterization of an ellipse's eccentricity e = 1 − b 2 / a 2 , {\textstyle e={\sqrt {1-b^{2}/a^{2}}},} or

9590-470: The angle of the reflected P wave is greater than the SV wave. For the same wave frequency, the SV wavelength is smaller than the P wavelength. This fact has been depicted in this animated picture. Similar to the SV wave, the P incidence, in general, reflects as the P and SV wave. There are some special cases where the regime is different. Wave velocity is a general concept, of various kinds of wave velocities, for

9727-467: The argument x − vt . Constant values of this argument correspond to constant values of F , and these constant values occur if x increases at the same rate that vt increases. That is, the wave shaped like the function F will move in the positive x -direction at velocity v (and G will propagate at the same speed in the negative x -direction). In the case of a periodic function F with period λ , that is, F ( x + λ − vt ) = F ( x − vt ),

9864-400: The bar. Then the temperatures at later times can be expressed by a function F {\displaystyle F} that depends on the function h {\displaystyle h} (that is, a functional operator ), so that the temperature at a later time is F ( h ; x , t ) {\displaystyle F(h;x,t)} Another way to describe and study

10001-454: The behavior of an object. A physicist would use a classical model to provide an approximation before more exacting models are applied and those calculations proceed. In a computer model, there is no need to use the speed of the object if classical physics is excluded. Low-energy objects would be handled by quantum theory and high-energy objects by relativity theory. Polarization (waves) Polarization ( also polarisation )

10138-451: The birefringence. The birefringence (as well as the average refractive index) will generally be dispersive , that is, it will vary as a function of optical frequency (wavelength). In the case of non-birefringent materials, however, the 2 × 2 Jones matrix is the identity matrix (multiplied by a scalar phase factor and attenuation factor), implying no change in polarization during propagation. For propagation effects in two orthogonal modes,

10275-420: The branches of theory sometimes included in classical physics are variably: In contrast to classical physics, " modern physics " is a slightly looser term that may refer to just quantum physics or to 20th- and 21st-century physics in general. Modern physics includes quantum theory and relativity, when applicable. A physical system can be described by classical physics when it satisfies conditions such that

10412-480: The case of linear birefringence (with two orthogonal linear propagation modes) with an incoming wave linearly polarized at a 45° angle to those modes. As a differential phase starts to accrue, the polarization becomes elliptical, eventually changing to purely circular polarization (90° phase difference), then to elliptical and eventually linear polarization (180° phase) perpendicular to the original polarization, then through circular again (270° phase), then elliptical with

10549-430: The center of the skin to the strike point, and on the strength s {\displaystyle s} of the strike. Then the vibration for all possible strikes can be described by a function F ( r , s ; x , t ) {\displaystyle F(r,s;x,t)} . Sometimes the family of waves of interest has infinitely many parameters. For example, one may want to describe what happens to

10686-435: The combination n ^ ⋅ x → {\displaystyle {\hat {n}}\cdot {\vec {x}}} , any displacement in directions perpendicular to n ^ {\displaystyle {\hat {n}}} cannot affect the value of the field. Plane waves are often used to model electromagnetic waves far from a source. For electromagnetic plane waves,

10823-1575: The complex quantities occurring in the following equations. As a function of time t and spatial position z (since for a plane wave in the + z direction the fields have no dependence on x or y ) these complex fields can be written as: E → ( z , t ) = [ e x e y 0 ] e i 2 π ( z λ − t T ) = [ e x e y 0 ] e i ( k z − ω t ) {\displaystyle {\vec {E}}(z,t)={\begin{bmatrix}e_{x}\\e_{y}\\0\end{bmatrix}}\;e^{i2\pi \left({\frac {z}{\lambda }}-{\frac {t}{T}}\right)}={\begin{bmatrix}e_{x}\\e_{y}\\0\end{bmatrix}}\;e^{i(kz-\omega t)}} and H → ( z , t ) = [ h x h y 0 ] e i 2 π ( z λ − t T ) = [ h x h y 0 ] e i ( k z − ω t ) , {\displaystyle {\vec {H}}(z,t)={\begin{bmatrix}h_{x}\\h_{y}\\0\end{bmatrix}}\;e^{i2\pi \left({\frac {z}{\lambda }}-{\frac {t}{T}}\right)}={\begin{bmatrix}h_{x}\\h_{y}\\0\end{bmatrix}}\;e^{i(kz-\omega t)},} where λ = λ 0 / n

10960-483: The context of quantum mechanics , classical theory refers to theories of physics that do not use the quantisation paradigm , which includes classical mechanics and relativity . Likewise, classical field theories , such as general relativity and classical electromagnetism , are those that do not use quantum mechanics. In the context of general and special relativity, classical theories are those that obey Galilean relativity . Depending on point of view, among

11097-428: The crystal. It was this effect that provided the first discovery of polarization, by Erasmus Bartholinus in 1669. Media in which transmission of one polarization mode is preferentially reduced are called dichroic or diattenuating . Like birefringence, diattenuation can be with respect to linear polarization modes (in a crystal) or circular polarization modes (usually in a liquid). Devices that block nearly all of

11234-412: The different propagation of waves in two such components in circularly birefringent media (see below) or signal paths of coherent detectors sensitive to circular polarization. Regardless of whether polarization state is represented using geometric parameters or Jones vectors, implicit in the parameterization is the orientation of the coordinate frame. This permits a degree of freedom, namely rotation about

11371-399: The discoverer of that particular equation. Computer modeling is essential for quantum and relativistic physics. Classical physics is considered the limit of quantum mechanics for a large number of particles. On the other hand, classic mechanics is derived from relativistic mechanics . For example, in many formulations from special relativity, a correction factor ( v / c ) appears, where v

11508-408: The discovery of how the laws of quantum physics give rise to classical physics found at the limit of the large scales of the classical level. Today, a computer performs millions of arithmetic operations in seconds to solve a classical differential equation , while Newton (one of the fathers of the differential calculus) would take hours to solve the same equation by manual calculation, even if he were

11645-611: The dispersion relation, we have dispersive waves. The dispersion relationship depends on the medium through which the waves propagate and on the type of waves (for instance electromagnetic , sound or water waves). The speed at which a resultant wave packet from a narrow range of frequencies will travel is called the group velocity and is determined from the gradient of the dispersion relation : v g = ∂ ω ∂ k {\displaystyle v_{\rm {g}}={\frac {\partial \omega }{\partial k}}} In almost all cases,

11782-435: The electric and magnetic fields themselves are transverse to the direction of propagation, and also perpendicular to each other. A standing wave, also known as a stationary wave , is a wave whose envelope remains in a constant position. This phenomenon arises as a result of interference between two waves traveling in opposite directions. The sum of two counter-propagating waves (of equal amplitude and frequency) creates

11919-406: The electric field parallel to this plane is termed p-like (parallel) and the component perpendicular to this plane is termed s-like (from senkrecht , German for 'perpendicular'). Polarized light with its electric field along the plane of incidence is thus denoted p-polarized , while light whose electric field is normal to the plane of incidence is called s-polarized . P -polarization

12056-426: The envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation . There are two velocities that are associated with waves, the phase velocity and the group velocity . Phase velocity is the rate at which the phase of the wave propagates in space : any given phase of

12193-495: The equation. This approach is extremely important in physics, because the constraints usually are a consequence of the physical processes that cause the wave to evolve. For example, if F ( x , t ) {\displaystyle F(x,t)} is the temperature inside a block of some homogeneous and isotropic solid material, its evolution is constrained by the partial differential equation where Q ( p , f ) {\displaystyle Q(p,f)}

12330-445: The formula Here P ( x , t ) {\displaystyle P(x,t)} is some extra compression force that is being applied to the gas near x {\displaystyle x} by some external process, such as a loudspeaker or piston right next to p {\displaystyle p} . This same differential equation describes the behavior of mechanical vibrations and electromagnetic fields in

12467-626: The laws of classical physics are approximately valid. In practice, physical objects ranging from those larger than atoms and molecules , to objects in the macroscopic and astronomical realm, can be well-described (understood) with classical mechanics. Beginning at the atomic level and lower, the laws of classical physics break down and generally do not provide a correct description of nature. Electromagnetic fields and forces can be described well by classical electrodynamics at length scales and field strengths large enough that quantum mechanical effects are negligible. Unlike quantum physics, classical physics

12604-414: The leftmost figure) and a vertically polarized wave of the same amplitude in the same phase . [REDACTED] [REDACTED] [REDACTED] Now if one were to introduce a phase shift in between those horizontal and vertical polarization components, one would generally obtain elliptical polarization as is shown in the third figure. When the phase shift is exactly ±90°, and the amplitudes are

12741-506: The length of the vector measured from the center of the sphere. Unpolarized light is light with a random, time-varying polarization . Natural light, like most other common sources of visible light, is produced independently by a large number of atoms or molecules whose emissions are uncorrelated . Unpolarized light can be produced from the incoherent combination of vertical and horizontal linearly polarized light, or right- and left-handed circularly polarized light. Conversely,

12878-415: The local pressure and particle motion that propagate through the medium. Other examples of mechanical waves are seismic waves , gravity waves , surface waves and string vibrations . In an electromagnetic wave (such as light), coupling between the electric and magnetic fields sustains propagation of waves involving these fields according to Maxwell's equations . Electromagnetic waves can travel through

13015-587: The medium in opposite directions. A generalized representation of this wave can be obtained as the partial differential equation 1 v 2 ∂ 2 u ∂ t 2 = ∂ 2 u ∂ x 2 . {\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}.} General solutions are based upon Duhamel's principle . The form or shape of F in d'Alembert's formula involves

13152-624: The medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. Wave propagation is any of the ways in which waves travel. With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves . Electromagnetic waves propagate in vacuum as well as in material media. Propagation of other wave types such as sound may occur only in

13289-419: The motion of a drum skin , one can consider D {\displaystyle D} to be a disk (circle) on the plane R 2 {\displaystyle \mathbb {R} ^{2}} with center at the origin ( 0 , 0 ) {\displaystyle (0,0)} , and let F ( x , t ) {\displaystyle F(x,t)} be the vertical displacement of

13426-499: The motion of bodies in free fall, and the propagation of light. Traditionally, light was reconciled with classical mechanics by assuming the existence of a stationary medium through which light propagated, the luminiferous aether , which was later shown not to exist. Mathematically, classical physics equations are those in which the Planck constant does not appear. According to the correspondence principle and Ehrenfest's theorem , as

13563-472: The original azimuth angle, and finally back to the original linearly polarized state (360° phase) where the cycle begins anew. In general the situation is more complicated and can be characterized as a rotation in the Poincaré sphere about the axis defined by the propagation modes. Examples for linear (blue), circular (red), and elliptical (yellow) birefringence are shown in the figure on the left. The total intensity and degree of polarization are unaffected. If

13700-413: The other hand, is always assumed to be a scalar ; that is, a real number. The value of F ( x , t ) {\displaystyle F(x,t)} can be any physical quantity of interest assigned to the point x {\displaystyle x} that may vary with time. For example, if F {\displaystyle F} represents the vibrations inside an elastic solid,

13837-475: The overall magnitude and phase of that wave. Specifically, the intensity of the light wave is proportional to the sum of the squared magnitudes of the two electric field components: I = ( | e x | 2 + | e y | 2 ) 1 2 η {\displaystyle I=\left(\left|e_{x}\right|^{2}+\left|e_{y}\right|^{2}\right)\,{\frac {1}{2\eta }}} However,

13974-609: The overall shape of the waves' amplitudes—modulation or envelope of the wave. 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 corresponds to uniform circular motion . Sine waves occur often in physics , including wind waves , sound waves, and light waves, such as monochromatic radiation . In engineering , signal processing , and mathematics , Fourier analysis decomposes general functions into

14111-447: The path length in the birefringent medium is sufficient, the two polarization components of a collimated beam (or ray ) can exit the material with a positional offset, even though their final propagation directions will be the same (assuming the entrance face and exit face are parallel). This is commonly viewed using calcite crystals , which present the viewer with two slightly offset images, in opposite polarizations, of an object behind

14248-475: The periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ (the wavelength of the wave). In a similar fashion, this periodicity of F implies a periodicity in time as well: F ( x − v ( t + T )) = F ( x − vt ) provided vT = λ , so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ / v . The amplitude of

14385-545: The phase of e x is zero; in other words e x is a real number while e y may be complex. Under these restrictions, e x and e y can be represented as follows: e x = 1 + Q 2 e y = 1 − Q 2 e i ϕ , {\displaystyle {\begin{aligned}e_{x}&={\sqrt {\frac {1+Q}{2}}}\\e_{y}&={\sqrt {\frac {1-Q}{2}}}\,e^{i\phi },\end{aligned}}} where

14522-458: The plane of incidence ( p and s polarizations, see below), that choice greatly simplifies the calculation of a wave's reflection from a surface. Any pair of orthogonal polarization states may be used as basis functions, not just linear polarizations. For instance, choosing right and left circular polarizations as basis functions simplifies the solution of problems involving circular birefringence (optical activity) or circular dichroism. For

14659-409: The plane wave approximation breaks down. An extreme example is radially or tangentially polarized light, at the focus of which the electric or magnetic field respectively is entirely longitudinal (along the direction of propagation). For longitudinal waves such as sound waves in fluids , the direction of oscillation is by definition along the direction of travel, so the issue of polarization

14796-457: The polarization of a coherent wave cannot be described simply using a Jones vector, as we have just done. Just considering electromagnetic waves, we note that the preceding discussion strictly applies to plane waves in a homogeneous isotropic non-attenuating medium, whereas in an anisotropic medium (such as birefringent crystals as discussed below) the electric or magnetic field may have longitudinal as well as transverse components. In those cases

14933-440: The polarization of an electromagnetic wave is determined by a quantum mechanical property of photons called their spin . A photon has one of two possible spins: it can either spin in a right hand sense or a left hand sense about its direction of travel. Circularly polarized electromagnetic waves are composed of photons with only one type of spin, either right- or left-hand. Linearly polarized waves consist of photons that are in

15070-456: The polarization of light, but some materials—those that exhibit birefringence , dichroism , or optical activity —affect light differently depending on its polarization. Some of these are used to make polarizing filters. Light also becomes partially polarized when it reflects at an angle from a surface. According to quantum mechanics , electromagnetic waves can also be viewed as streams of particles called photons . When viewed in this way,

15207-481: The polarization state is now fully parameterized by the value of Q (such that −1 < Q < 1 ) and the relative phase ϕ . In addition to transverse waves, there are many wave motions where the oscillation is not limited to directions perpendicular to the direction of propagation. These cases are far beyond the scope of the current article which concentrates on transverse waves (such as most electromagnetic waves in bulk media), but one should be aware of cases where

15344-403: The polarization state itself is independent of absolute phase . The basis vectors used to represent the Jones vector need not represent linear polarization states (i.e. be real ). In general any two orthogonal states can be used, where an orthogonal vector pair is formally defined as one having a zero inner product . A common choice is left and right circular polarizations, for example to model

15481-420: The presence of a magnetic field along the direction of propagation. When linearly polarized light is passed through such an object, it will exit still linearly polarized, but with the axis of polarization rotated. A combination of linear and circular birefringence will have as basis polarizations two orthogonal elliptical polarizations; however, the term "elliptical birefringence" is rarely used. One can visualize

15618-500: The product of these two basic types of transformations. In birefringent media there is no attenuation, but two modes accrue a differential phase delay. Well known manifestations of linear birefringence (that is, in which the basis polarizations are orthogonal linear polarizations) appear in optical wave plates /retarders and many crystals. If linearly polarized light passes through a birefringent material, its state of polarization will generally change, unless its polarization direction

15755-418: The propagation direction ( + z in this case) and η , one can just as well specify the wave in terms of just e x and e y describing the electric field. The vector containing e x and e y (but without the z component which is necessarily zero for a transverse wave) is known as a Jones vector . In addition to specifying the polarization state of the wave, a general Jones vector also specifies

15892-438: The propagation direction is also referred to as the wave's polarization , which can be an important attribute. A wave can be described just like a field, namely as a function F ( x , t ) {\displaystyle F(x,t)} where x {\displaystyle x} is a position and t {\displaystyle t} is a time. The value of x {\displaystyle x}

16029-472: The propagation direction. When considering light that is propagating parallel to the surface of the Earth, the terms "horizontal" and "vertical" polarization are often used, with the former being associated with the first component of the Jones vector, or zero azimuth angle. On the other hand, in astronomy the equatorial coordinate system is generally used instead, with the zero azimuth (or position angle, as it

16166-437: The radiation in one mode are known as polarizing filters or simply " polarizers ". This corresponds to g 2 = 0 in the above representation of the Jones matrix. The output of an ideal polarizer is a specific polarization state (usually linear polarization) with an amplitude equal to the input wave's original amplitude in that polarization mode. Power in the other polarization mode is eliminated. Thus if unpolarized light

16303-461: The remaining power in the unwanted polarization will be ( g 2 / g 1 ) of the power in the intended polarization. In addition to birefringence and dichroism in extended media, polarization effects describable using Jones matrices can also occur at (reflective) interface between two materials of different refractive index . These effects are treated by the Fresnel equations . Part of the wave

16440-441: The same, so the wave form will change over time and space. Sometimes one is interested in a single specific wave. More often, however, one needs to understand large set of possible waves; like all the ways that a drum skin can vibrate after being struck once with a drum stick , or all the possible radar echoes one could get from an airplane that may be approaching an airport . In some of those situations, one may describe such

16577-413: The same, then circular polarization is produced (fourth and fifth figures). Circular polarization can be created by sending linearly polarized light through a quarter-wave plate oriented at 45° to the linear polarization to create two components of the same amplitude with the required phase shift. The superposition of the original and phase-shifted components causes a rotating electric field vector, which

16714-420: The skin at the point x {\displaystyle x} of D {\displaystyle D} and at time t {\displaystyle t} . Waves of the same type are often superposed and encountered simultaneously at a given point in space and time. The properties at that point are the sum of the properties of each component wave at that point. In general, the velocities are not

16851-505: The solid and vibration along the direction of propagation. The differential propagation of transverse and longitudinal polarizations is important in seismology . Polarization can be defined in terms of pure polarization states with only a coherent sinusoidal wave at one optical frequency. The vector in the adjacent diagram might describe the oscillation of the electric field emitted by a single-mode laser (whose oscillation frequency would be typically 10 times faster). The field oscillates in

16988-423: The sound pressure inside a recorder that is playing a "pure" note is typically a standing wave , that can be written as The parameter A {\displaystyle A} defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note); c {\displaystyle c} is the speed of sound; L {\displaystyle L}

17125-600: The string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves , such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves , gravitational waves , and transverse sound waves ( shear waves ) in solids. An electromagnetic wave such as light consists of

17262-463: The temperature in a metal bar when it is initially heated at various temperatures at different points along its length, and then allowed to cool by itself in vacuum. In that case, instead of a scalar or vector, the parameter would have to be a function h {\displaystyle h} such that h ( x ) {\displaystyle h(x)} is the initial temperature at each point x {\displaystyle x} of

17399-452: The two constituent linearly polarized states of unpolarized light cannot form an interference pattern , even if rotated into alignment ( Fresnel–Arago 3rd law ). A so-called depolarizer acts on a polarized beam to create one in which the polarization varies so rapidly across the beam that it may be ignored in the intended applications. Conversely, a polarizer acts on an unpolarized beam or arbitrarily polarized beam to create one which

17536-414: The two-dimensional functions or, more generally, by d'Alembert's formula : u ( x , t ) = F ( x − v t ) + G ( x + v t ) . {\displaystyle u(x,t)=F(x-vt)+G(x+vt).} representing two component waveforms F {\displaystyle F} and G {\displaystyle G} traveling through

17673-418: The value of F ( x , t ) {\displaystyle F(x,t)} is usually a vector that gives the current displacement from x {\displaystyle x} of the material particles that would be at the point x {\displaystyle x} in the absence of vibration. For an electromagnetic wave, the value of F {\displaystyle F} can be

17810-485: The velocity vector of the fluid at the point x {\displaystyle x} , or any scalar property like pressure , temperature , or density . In a chemical reaction, F ( x , t ) {\displaystyle F(x,t)} could be the concentration of some substance in the neighborhood of point x {\displaystyle x} of the reaction medium. For any dimension d {\displaystyle d} (1, 2, or 3),

17947-441: The wave (for example, the crest ) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and period T as v p = λ T . {\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.} Group velocity is a property of waves that have a defined envelope, measuring propagation through space (that is, phase velocity) of

18084-458: The wave varies in, and a wave profile describing how the wave varies as a function of the displacement along that direction ( n ^ ⋅ x → {\displaystyle {\hat {n}}\cdot {\vec {x}}} ) and time ( t {\displaystyle t} ). Since the wave profile only depends on the position x → {\displaystyle {\vec {x}}} in

18221-425: The wave's state of polarization is only dependent on the (complex) ratio of e y to e x . So let us just consider waves whose | e x | + | e y | = 1 ; this happens to correspond to an intensity of about 0.001 33   W /m in free space (where η = η 0 ). And because the absolute phase of a wave is unimportant in discussing its polarization state, let us stipulate that

18358-419: The wave's domain is then a subset D {\displaystyle D} of R d {\displaystyle \mathbb {R} ^{d}} , such that the function value F ( x , t ) {\displaystyle F(x,t)} is defined for any point x {\displaystyle x} in D {\displaystyle D} . For example, when describing

18495-503: The wave. DOP can be used to map the strain field in materials when considering the DOP of the photoluminescence . The polarization of the photoluminescence is related to the strain in a material by way of the given material's photoelasticity tensor . DOP is also visualized using the Poincaré sphere representation of a polarized beam. In this representation, DOP is equal to

18632-438: The wavenumber k , but both are related through the dispersion relationship : ω = Ω ( k ) . {\displaystyle \omega =\Omega (k).} In the special case Ω( k ) = ck , with c a constant, the waves are called non-dispersive, since all frequencies travel at the same phase speed c . For instance electromagnetic waves in vacuum are non-dispersive. In case of other forms of

18769-570: Was used above to show how different states of polarization are possible. The amplitude and phase information can be conveniently represented as a two-dimensional complex vector (the Jones vector ): e = [ a 1 e i θ 1 a 2 e i θ 2 ] . {\displaystyle \mathbf {e} ={\begin{bmatrix}a_{1}e^{i\theta _{1}}\\a_{2}e^{i\theta _{2}}\end{bmatrix}}.} Here

#739260