Wave

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https://en.wikipedia.org/wiki/Wave 

Surface waves in water showing water ripples

 

Example of biological waves expanding over the brain cortex, an example of spreading depolarizations.

In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities, sometimes as described by a wave equation. In physical waves, at least two field quantities in the wave medium are involved. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction it is said to be a traveling wave; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero.

The types of waves most commonly studied in classical physics are mechanical and electromagnetic. 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 the local pressure and particle motion that propagate through the medium. Other examples of mechanical waves are seismic waves, gravity waves, surface waves, string vibrations (standing waves), and vortices. In an electromagnetic wave (such as light), coupling between the electric and magnetic fields which sustains propagation of a wave involving these fields according to Maxwell's equations. Electromagnetic waves can travel through a vacuum and through some dielectric media (at wavelengths where they are considered transparent). Electromagnetic waves, according to 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 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. Some, like the probability waves of quantum mechanics, may be completely static.

A physical wave 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 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 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 if those vectors are exactly in 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 the propagation direction is also referred to as the wave's polarization which can be an important attribute for waves having more than one single possible polarization.

Mathematical description

Single waves

A wave can be described just like a field, namely as a function ${\displaystyle F(x,t)}$ where ${\displaystyle x}$ is a position and ${\displaystyle t}$ is a time.

The value of ${\displaystyle x}$ 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 ${\displaystyle \mathbb {R} ^{3}}$. However, in many cases one can ignore one dimension, and let ${\displaystyle x}$ be a point of the Cartesian plane ${\displaystyle \mathbb {R} ^{2}}$. This is the case, for example, when studying vibrations of a drum skin. One may even restrict ${\displaystyle x}$ to a point of the Cartesian line ${\displaystyle \mathbb {R} }$ — that is, the set of real numbers. This is the case, for example, when studying vibrations in a violin string or recorder. The time ${\displaystyle t}$, on the other hand, is always assumed to be a scalar; that is, a real number.

The value of ${\displaystyle F(x,t)}$ can be any physical quantity of interest assigned to the point ${\displaystyle x}$ that may vary with time. For example, if ${\displaystyle F}$ represents the vibrations inside an elastic solid, the value of ${\displaystyle F(x,t)}$ is usually a vector that gives the current displacement from ${\displaystyle x}$ of the material particles that would be at the point ${\displaystyle x}$ in the absence of vibration. For an electromagnetic wave, the value of ${\displaystyle F}$ can be the electric field vector ${\displaystyle E}$, or the magnetic field vector ${\displaystyle H}$, or any related quantity, such as the Poynting vector ${\displaystyle E\times H}$. In fluid dynamics, the value of ${\displaystyle F(x,t)}$ could be the velocity vector of the fluid at the point ${\displaystyle x}$, or any scalar property like pressure, temperature, or density. In a chemical reaction, ${\displaystyle F(x,t)}$ could be the concentration of some substance in the neighborhood of point ${\displaystyle x}$ of the reaction medium.

For any dimension ${\displaystyle d}$ (1, 2, or 3), the wave's domain is then a subset ${\displaystyle D}$ of ${\displaystyle \mathbb {R} ^{d}}$, such that the function value ${\displaystyle F(x,t)}$ is defined for any point ${\displaystyle x}$ in ${\displaystyle D}$. For example, when describing the motion of a drum skin, one can consider ${\displaystyle D}$ to be a disk (circle) on the plane ${\displaystyle \mathbb {R} ^{2}}$ with center at the origin ${\displaystyle (0,0)}$, and let ${\displaystyle F(x,t)}$ be the vertical displacement of the skin at the point ${\displaystyle x}$ of ${\displaystyle D}$ and at time ${\displaystyle t}$.

Wave families

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 echos one could get from an airplane that may be approaching an airport.

In some of those situations, one may describe such a family of waves by a function ${\displaystyle F(A,B,\ldots ;x,t)}$ that depends on certain parameters ${\displaystyle A,B,\ldots }$, besides ${\displaystyle x}$ and ${\displaystyle t}$. Then one can obtain different waves — that is, different functions of ${\displaystyle x}$ and ${\displaystyle t}$ — by choosing different values for those parameters.

Sound pressure standing wave in a half-open pipe playing the 7th harmonic of the fundamental (n = 4)

For example, the sound pressure inside a recorder that is playing a "pure" note is typically a standing wave, that can be written as

${\displaystyle F(A,L,n,c;x,t)=A\left(\cos 2\pi x{\frac {2n-1}{4L}}\right)\left(\cos 2\pi ct{\frac {2n-1}{4L}}\right)}$

The parameter ${\displaystyle A}$ defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note); ${\displaystyle c}$ is the speed of sound; ${\displaystyle L}$ is the length of the bore; and ${\displaystyle n}$ is a positive integer (1,2,3,…) that specifies the number of nodes in the standing wave. (The position ${\displaystyle x}$ should be measured from the mouthpiece, and the time ${\displaystyle t}$ from any moment at which the pressure at the mouthpiece is maximum. The quantity ${\displaystyle \lambda =4L/(2n-1)}$ is the wavelength of the emitted note, and ${\displaystyle f=c/\lambda }$ 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 ${\displaystyle r}$ from the center of the skin to the strike point, and on the strength ${\displaystyle s}$ of the strike. Then the vibration for all possible strikes can be described by a function ${\displaystyle F(r,s;x,t)}$.

Sometimes the family of waves of interest has infinitely many parameters. For example, one may want to describe what happens to 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 ${\displaystyle h}$ such that ${\displaystyle h(x)}$ is the initial temperature at each point ${\displaystyle x}$ of the bar. Then the temperatures at later times can be expressed by a function ${\displaystyle F}$ that depends on the function ${\displaystyle h}$ (that is, a functional operator), so that the temperature at a later time is ${\displaystyle F(h;x,t)}$

Differential wave equations

Another way to describe and study a family of waves is to give a mathematical equation that, instead of explicitly giving the value of ${\displaystyle F(x,t)}$, only constrains how those values can change with time. Then the family of waves in question consists of all functions ${\displaystyle F}$ that satisfy those constraints — that is, all solutions of 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 ${\displaystyle F(x,t)}$ is the temperature inside a block of some homogeneous and isotropic solid material, its evolution is constrained by the partial differential equation

${\displaystyle {\frac {\partial F}{\partial t}}(x,t)=\alpha \left({\frac {\partial ^{2}F}{\partial x_{1}^{2}}}(x,t)+{\frac {\partial ^{2}F}{\partial x_{2}^{2}}}(x,t)+{\frac {\partial ^{2}F}{\partial x_{3}^{2}}}(x,t)\right)+\beta Q(x,t)}$

where ${\displaystyle Q(p,f)}$ is the heat that is being generated per unit of volume and time in the neighborhood of ${\displaystyle x}$ at time ${\displaystyle t}$ (for example, by chemical reactions happening there); ${\displaystyle x_{1},x_{2},x_{3}}$ are the Cartesian coordinates of the point ${\displaystyle x}$; ${\displaystyle \partial F/\partial t}$ is the (first) derivative of ${\displaystyle F}$ with respect to ${\displaystyle t}$; and ${\displaystyle \partial ^{2}F/\partial x_{i}^{2}}$ is the second derivative of ${\displaystyle F}$ relative to ${\displaystyle x_{i}}$. (The symbol "${\displaystyle \partial }$" 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 a container of gas by a function ${\displaystyle F(x,t)}$ that gives the pressure at a point ${\displaystyle x}$ and time ${\displaystyle t}$ within that container. If the gas was initially at uniform temperature and composition, the evolution of ${\displaystyle F}$ is constrained by the formula

${\displaystyle {\frac {\partial ^{2}F}{\partial t^{2}}}(x,t)=\alpha \left({\frac {\partial ^{2}F}{\partial x_{1}^{2}}}(x,t)+{\frac {\partial ^{2}F}{\partial x_{2}^{2}}}(x,t)+{\frac {\partial ^{2}F}{\partial x_{3}^{2}}}(x,t)\right)+\beta P(x,t)}$

Here ${\displaystyle P(x,t)}$ is some extra compression force that is being applied to the gas near ${\displaystyle x}$ by some external process, such as a loudspeaker or piston right next to ${\displaystyle p}$.

This same differential equation describes the behavior of mechanical vibrations and electromagnetic fields in a homogeneous isotropic non-conducting solid. Note that this equation differs from that of heat flow only in that the left-hand side is ${\displaystyle \partial ^{2}F/\partial t^{2}}$, the second derivative of ${\displaystyle F}$ with respect to time, rather than the first derivative ${\displaystyle \partial F/\partial t}$. Yet this small change makes a huge difference on the set of solutions ${\displaystyle F}$. This differential equation is called "the" wave equation in mathematics, even though it describes only one very special kind of waves.

Wave in elastic medium

Main articles: Wave equation and D'Alembert's formula

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

Wavelength λ, can be measured between any two corresponding points on a waveform

 

Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that f(x,t) + g(x,t) = u(x,t)

• in the ${\displaystyle x}$ direction in space. For example, let the positive ${\displaystyle x}$ direction be to the right, and the negative ${\displaystyle x}$ direction be to the left.
• with constant amplitude ${\displaystyle u}$
• with constant velocity ${\displaystyle v}$, where ${\displaystyle v}$ is
  • independent of wavelength (no dispersion)
  • independent of amplitude (linear media, not nonlinear).
• with constant waveform, or shape

This wave can then be described by the two-dimensional functions

${\displaystyle u(x,t)=F(x-vt)}$ (waveform ${\displaystyle F}$ traveling to the right)

${\displaystyle u(x,t)=G(x+vt)}$ (waveform ${\displaystyle G}$ traveling to the left)

or, more generally, by d'Alembert's formula:

${\displaystyle u(x,t)=F(x-vt)+G(x+vt).}$

representing two component waveforms ${\displaystyle F}$ and ${\displaystyle G}$ traveling through the medium in opposite directions. A generalized representation of this wave can be obtained as the partial differential equation

${\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}.}$

General solutions are based upon Duhamel's principle.

Wave forms

Main article: Waveform

Sine, square, triangle and sawtooth waveforms.

The form or shape of F in d'Alembert's formula involves 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), 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.

Amplitude and modulation

Amplitude modulation can be achieved through f(x,t) = 1.00×sin(2π/0.10×(x−1.00×t)) and g(x,t) = 1.00×sin(2π/0.11×(x−1.00×t))only the resultant is visible to improve clarity of waveform.

 

Illustration of the envelope (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the carrier wave, which is being modulated.

 

Main article: Amplitude modulation

See also: Frequency modulation and Phase modulation

The amplitude of 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:

${\displaystyle u(x,t)=A(x,t)\sin \left(kx-\omega t+\phi \right),}$

where ${\displaystyle A(x,\ t)}$ is the amplitude envelope of the wave, ${\displaystyle k}$ is the wavenumber and ${\displaystyle \phi }$ is the phase. If the group velocity ${\displaystyle v_{g}}$ (see below) is wavelength-independent, this equation can be simplified as:

${\displaystyle u(x,t)=A(x-v_{g}t)\sin \left(kx-\omega t+\phi \right),}$

showing that 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.

Phase velocity and group velocity

Main articles: Phase velocity and Group velocity

See also: Envelope (waves) § Phase and group velocity

 

The red square moves with the phase velocity, while the green circles propagate with the group velocity

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 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

${\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.}$

A wave with the group and phase velocities going in different directions

Group velocity is a property of waves that have a defined envelope, measuring propagation through space (that is, phase velocity) of the overall shape of the waves' amplitudes – modulation or envelope of the wave.

Sine waves

Main article: Sine wave

Sinusoidal waves correspond to simple harmonic motion.

Mathematically, the most basic wave is the (spatially) one-dimensional sine wave (also called harmonic wave or sinusoid) with an amplitude ${\displaystyle u}$ described by the equation:

${\displaystyle u(x,t)=A\sin \left(kx-\omega t+\phi \right),}$

where

• ${\displaystyle A}$ is the maximum amplitude of the wave, maximum distance from the highest point of the disturbance in the medium (the crest) to the equilibrium point during one wave cycle. In the illustration to the right, this is the maximum vertical distance between the baseline and the wave.
• ${\displaystyle x}$ is the space coordinate
• ${\displaystyle t}$ is the time coordinate
• ${\displaystyle k}$ is the wavenumber
• ${\displaystyle \omega }$ is the angular frequency
• ${\displaystyle \phi }$ is the phase constant.

The units of the amplitude depend on the type of wave. Transverse mechanical waves (for example, a wave on a string) have an amplitude expressed as a distance (for example, meters), longitudinal mechanical waves (for example, sound waves) use units of pressure (for example, pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude in terms of its electric field (for example, volts/meter).

The wavelength ${\displaystyle \lambda }$ is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber ${\displaystyle k}$, the spatial frequency of the wave in radians per unit distance (typically per meter), can be associated with the wavelength by the relation

${\displaystyle k={\frac {2\pi }{\lambda }}.}$

The period ${\displaystyle T}$ is the time for one complete cycle of an oscillation of a wave. The frequency ${\displaystyle f}$ is the number of periods per unit time (per second) and is typically measured in hertz denoted as Hz. These are related by:

${\displaystyle f={\frac {1}{T}}.}$

In other words, the frequency and period of a wave are reciprocals.

The angular frequency ${\displaystyle \omega }$ represents the frequency in radians per second. It is related to the frequency or period by

${\displaystyle \omega =2\pi f={\frac {2\pi }{T}}.}$

The wavelength ${\displaystyle \lambda }$ of a sinusoidal waveform traveling at constant speed ${\displaystyle v}$ is given by:

${\displaystyle \lambda ={\frac {v}{f}},}$

where ${\displaystyle v}$ is called the phase speed (magnitude of the phase velocity) of the wave and ${\displaystyle f}$ is the wave's frequency.

Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean wave approaching shore, the incoming wave undulates with a varying local wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.

Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant systems, in the presence of dispersion the sine wave is the unique shape that will propagate unchanged but for phase and amplitude, making it easy to analyze. Due to the Kramers–Kronig relations, a linear medium with dispersion also exhibits loss, so the sine wave propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon the medium. The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time.

The sinusoid is defined for all times and distances, whereas in physical situations we usually deal with waves that exist for a limited span in space and duration in time. An arbitrary wave shape can be decomposed into an infinite set of sinusoidal waves by the use of Fourier analysis. As a result, the simple case of a single sinusoidal wave can be applied to more general cases. In particular, many media are linear, or nearly so, so the calculation of arbitrary wave behavior can be found by adding up responses to individual sinusoidal waves using the superposition principle to find the solution for a general waveform. When a medium is nonlinear, then the response to complex waves cannot be determined from a sine-wave decomposition.

Plane waves

Main article: Plane wave

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 ${\displaystyle {\hat {n}}}$ indicating the direction that the wave varies in, and a wave profile describing how the wave varies as a function of the displacement along that direction (${\displaystyle {\hat {n}}\cdot {\vec {x}}}$) and time (${\displaystyle t}$). Since the wave profile only depends on the position ${\displaystyle {\vec {x}}}$ in the combination ${\displaystyle {\hat {n}}\cdot {\vec {x}}}$, any displacement in directions perpendicular to ${\displaystyle {\hat {n}}}$ cannot affect the value of the field.

Plane waves are often used to model electromagnetic waves far from a source. For electromagnetic plane waves, the electric and magnetic fields themselves are transverse to the direction of propagation, and also perpendicular to each other.

Standing waves

Main articles: Standing wave, Acoustic resonance, Helmholtz resonator, and Organ pipe

Standing wave. The red dots represent the wave nodes

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 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 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.

• 

  One-dimensional standing waves; the fundamental mode and the first 5 overtones.

• 

  A two-dimensional standing wave on a disk; this is the fundamental mode.

• 

  A standing wave on a disk with two nodal lines crossing at the center; this is an overtone.

Physical properties

Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism

Waves exhibit common behaviors under a number of standard situations, for example:

Transmission and media

Main articles: Rectilinear propagation, Transmittance, and Transmission medium

Waves normally move in a straight line (that is, rectilinearly) through a transmission medium. Such media can be classified into one or more of the following categories:

• A bounded medium if it is finite in extent, otherwise an unbounded medium
• A linear medium if the amplitudes of different waves at any particular point in the medium can be added
• A uniform medium or homogeneous medium if its physical properties are unchanged at different locations in space
• An anisotropic medium if one or more of its physical properties differ in one or more directions
• An isotropic medium if its physical properties are the same in all directions

Absorption

Main articles: Absorption (acoustics) and Absorption (electromagnetic radiation)

Waves are usually defined in media which allow most or all of a wave's energy to propagate without loss. However materials may be characterized as "lossy" if they remove energy from a wave, usually converting it into heat. This is termed "absorption." A material which absorbs a wave's energy, either in transmission or reflection, is characterized by a refractive index which is complex. The amount of absorption will generally depend on the frequency (wavelength) of the wave, which, for instance, explains why objects may appear colored.

Reflection

Main article: Reflection (physics)

When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and line normal to the surface equals the angle made by the reflected wave and the same normal line.

Refraction

Main article: Refraction

Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results.

Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the phase velocity changes. Typically, refraction occurs when a wave passes from one medium into another. The amount by which a wave is refracted by a material is given by the refractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by Snell's law.

Diffraction

Main article: Diffraction

A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.

Interference

Main article: Interference (wave propagation)

Identical waves from two sources undergoing interference. Observed at the bottom one sees 5 positions where the waves add in phase, but in between which they are out of phase and cancel.

When waves in a linear medium (the usual case) cross each other in a region of space, they do not actually interact with each other, but continue on as if the other one weren't present. However at any point in that region the field quantities describing those waves add according to the superposition principle. If the waves are of the same frequency in a fixed phase relationship, then there will generally be positions at which the two waves are in phase and their amplitudes add, and other positions where they are out of phase and their amplitudes (partially or fully) cancel. This is called an interference pattern.

Polarization

Main article: Polarization (waves)

The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal directions. Transverse waves can be polarized, for instance. When polarization is used as a descriptor without qualification, it usually refers to the special, simple case of linear polarization. A transverse wave is linearly polarized if it oscillates in only one direction or plane. In the case of linear polarization, it is often useful to add the relative orientation of that plane, perpendicular to the direction of travel, in which the oscillation occurs, such as "horizontal" for instance, if the plane of polarization is parallel to the ground. Electromagnetic waves propagating in free space, for instance, are transverse; they can be polarized by the use of a polarizing filter.

Longitudinal waves, such as sound waves, do not exhibit polarization. For these waves there is only one direction of oscillation, that is, along the direction of travel.

Dispersion

Schematic of light being dispersed by a prism. Click to see animation.

 

Main articles: Dispersion relation, Dispersion (optics), and Dispersion (water waves)

A wave undergoes dispersion when either the phase velocity or the group velocity depends on the wave frequency. Dispersion is most easily seen by letting white light pass through a prism, the result of which is to produce the spectrum of colors of the rainbow. Isaac Newton performed experiments with light and prisms, presenting his findings in the Opticks (1704) that white light consists of several colors and that these colors cannot be decomposed any further.

Mechanical waves

Main article: Mechanical wave

Waves on strings

Main article: Vibrating string

The speed of a transverse wave traveling along a vibrating string (v) is directly proportional to the square root of the tension of the string (T) over the linear mass density (μ):

${\displaystyle v={\sqrt {\frac {T}{\mu }}},}$

where the linear density μ is the mass per unit length of the string.

Acoustic waves

Main article: Acoustic wave

Acoustic or sound waves travel at speed given by

${\displaystyle v={\sqrt {\frac {B}{\rho _{0}}}},}$

or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see speed of sound).

Water waves

Main article: Water waves

• Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.
• Sound – a mechanical wave that propagates through gases, liquids, solids and plasmas;
• Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect;
• Ocean surface waves, which are perturbations that propagate through water.

Seismic waves

Main article: Seismic waves

Seismic waves are waves of energy that travel through the Earth's layers, and are a result of earthquakes, volcanic eruptions, magma movement, large landslides and large man-made explosions that give out low-frequency acoustic energy.

Doppler effect

The Doppler effect (or the Doppler shift) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who described the phenomenon in 1842.

Shock waves

Formation of a shock wave by a plane.

Main article: Shock wave

A shock wave is a type of propagating disturbance. When a wave moves faster than the local speed of sound in a fluid, it is a shock wave. Like an ordinary wave, a shock wave carries energy and can propagate through a medium; however, it is characterized by an abrupt, nearly discontinuous change in pressure, temperature and density of the medium.

See also: Sonic boom and Cherenkov radiation

Other

• Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves
• Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions.

Electromagnetic waves

Main article: Electromagnetic wave

Further information: Electromagnetic spectrum

An electromagnetic wave consists of two waves that are oscillations of the electric and magnetic fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. In the 19th century, James Clerk Maxwell showed that, in vacuum, the electric and magnetic fields satisfy the wave equation both with speed equal to that of the speed of light. From this emerged the idea that light is an electromagnetic wave. Electromagnetic waves can have different frequencies (and thus wavelengths), giving rise to various types of radiation such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and Gamma rays.

Quantum mechanical waves

Main article: Schrödinger equation

See also: Wave function

Schrödinger equation

The Schrödinger equation describes the wave-like behavior of particles in quantum mechanics. Solutions of this equation are wave functions which can be used to describe the probability density of a particle.

Dirac equation

The Dirac equation is a relativistic wave equation detailing electromagnetic interactions. Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The wave equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-½ particles.

A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.

de Broglie waves

Main articles: Wave packet and Matter wave

Louis de Broglie postulated that all particles with momentum have a wavelength

${\displaystyle \lambda ={\frac {h}{p}},}$

where h is Planck's constant, and p is the magnitude of the momentum of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT display have a de Broglie wavelength of about 10⁻¹³ m.

A wave representing such a particle traveling in the k-direction is expressed by the wave function as follows:

${\displaystyle \psi (\mathbf {r} ,\,t=0)=Ae^{i\mathbf {k\cdot r} },}$

where the wavelength is determined by the wave vector k as:

${\displaystyle \lambda ={\frac {2\pi }{k}},}$

and the momentum by:

${\displaystyle \mathbf {p} =\hbar \mathbf {k} .}$

However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet, a waveform often used in quantum mechanics to describe the wave function of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates on either side of the main wavelength value.

In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet. Gaussian wave packets also are used to analyze water waves.

For example, a Gaussian wavefunction ψ might take the form:

${\displaystyle \psi (x,\,t=0)=A\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}+ik_{0}x\right),}$

at some initial time t = 0, where the central wavelength is related to the central wave vector k₀ as λ₀ = 2π / k₀. It is well known from the theory of Fourier analysis, or from the Heisenberg uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian is itself a Gaussian. Given the Gaussian:

${\displaystyle f(x)=e^{-x^{2}/\left(2\sigma ^{2}\right)},}$

the Fourier transform is:

${\displaystyle {\tilde {f}}(k)=\sigma e^{-\sigma ^{2}k^{2}/2}.}$

The Gaussian in space therefore is made up of waves:

${\displaystyle f(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\ {\tilde {f}}(k)e^{ikx}\ dk;}$

that is, a number of waves of wavelengths λ such that kλ = 2 π.

The parameter σ decides the spatial spread of the Gaussian along the x-axis, while the Fourier transform shows a spread in wave vector k determined by 1/σ. That is, the smaller the extent in space, the larger the extent in k, and hence in λ = 2π/k.

Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles

Gravity waves

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. A ripple on a pond is one example.

Gravitational waves

Main article: Gravitational wave

Gravitational waves also travel through space. The first observation of gravitational waves was announced on 11 February 2016. Gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.

Earth's magnetic field

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Earth%27s_magnetic_field 

Computer simulation of the Earth's field in a period of normal polarity between reversals. The lines represent magnetic field lines, blue when the field points towards the center and yellow when away. The rotation axis of the Earth is centered and vertical. The dense clusters of lines are within the Earth's core.

Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from the Earth's interior out into space, where it interacts with the solar wind, a stream of charged particles emanating from the Sun. The magnetic field is generated by electric currents due to the motion of convection currents of a mixture of molten iron and nickel in the Earth's outer core: these convection currents are caused by heat escaping from the core, a natural process called a geodynamo. The magnitude of the Earth's magnetic field at its surface ranges from 25 to 65 μT (0.25 to 0.65 gauss). As an approximation, it is represented by a field of a magnetic dipole currently tilted at an angle of about 11 degrees with respect to Earth's rotational axis, as if there were an enormous bar magnet placed at that angle through the center of the Earth. The North geomagnetic pole actually represents the South pole of the Earth's magnetic field, and conversely the South geomagnetic pole corresponds to the north pole of Earth's magnetic field (because opposite magnetic poles attract and the north end of a magnet, like a compass needle, points toward the Earth's South magnetic field, i.e., the North geomagnetic pole near the Geographic North Pole). As of 2015, the North geomagnetic pole was located on Ellesmere Island, Nunavut, Canada.

While the North and South magnetic poles are usually located near the geographic poles, they slowly and continuously move over geological time scales, but sufficiently slowly for ordinary compasses to remain useful for navigation. However, at irregular intervals averaging several hundred thousand years, the Earth's field reverses and the North and South Magnetic Poles respectively, abruptly switch places. These reversals of the geomagnetic poles leave a record in rocks that are of value to paleomagnetists in calculating geomagnetic fields in the past. Such information in turn is helpful in studying the motions of continents and ocean floors in the process of plate tectonics.

The magnetosphere is the region above the ionosphere that is defined by the extent of the Earth's magnetic field in space. It extends several tens of thousands of kilometres into space, protecting the Earth from the charged particles of the solar wind and cosmic rays that would otherwise strip away the upper atmosphere, including the ozone layer that protects the Earth from the harmful ultraviolet radiation.

Significance

Earth's magnetic field deflects most of the solar wind, whose charged particles would otherwise strip away the ozone layer that protects the Earth from harmful ultraviolet radiation. One stripping mechanism is for gas to be caught in bubbles of magnetic field, which are ripped off by solar winds. Calculations of the loss of carbon dioxide from the atmosphere of Mars, resulting from scavenging of ions by the solar wind, indicate that the dissipation of the magnetic field of Mars caused a near total loss of its atmosphere.

The study of the past magnetic field of the Earth is known as paleomagnetism. The polarity of the Earth's magnetic field is recorded in igneous rocks, and reversals of the field are thus detectable as "stripes" centered on mid-ocean ridges where the sea floor is spreading, while the stability of the geomagnetic poles between reversals has allowed paleomagnetism to track the past motion of continents. Reversals also provide the basis for magnetostratigraphy, a way of dating rocks and sediments. The field also magnetizes the crust, and magnetic anomalies can be used to search for deposits of metal ores.

Humans have used compasses for direction finding since the 11th century A.D. and for navigation since the 12th century. Although the magnetic declination does shift with time, this wandering is slow enough that a simple compass can remain useful for navigation. Using magnetoreception, various other organisms, ranging from some types of bacteria to pigeons, use the Earth's magnetic field for orientation and navigation.

Characteristics

At any location, the Earth's magnetic field can be represented by a three-dimensional vector. A typical procedure for measuring its direction is to use a compass to determine the direction of magnetic North. Its angle relative to true North is the declination (D) or variation. Facing magnetic North, the angle the field makes with the horizontal is the inclination (I) or magnetic dip. The intensity (F) of the field is proportional to the force it exerts on a magnet. Another common representation is in X (North), Y (East) and Z (Down) coordinates.

Common coordinate systems used for representing the Earth's magnetic field.

Intensity

The intensity of the field is often measured in gauss (G), but is generally reported in nanoteslas (nT), with 1 G = 100,000 nT. A nanotesla is also referred to as a gamma (γ). The Earth's field ranges between approximately 25,000 and 65,000 nT (0.25–0.65 G). By comparison, a strong refrigerator magnet has a field of about 10,000,000 nanoteslas (100 G).

A map of intensity contours is called an isodynamic chart. As the World Magnetic Model shows, the intensity tends to decrease from the poles to the equator. A minimum intensity occurs in the South Atlantic Anomaly over South America while there are maxima over northern Canada, Siberia, and the coast of Antarctica south of Australia.

The intensity of the magnetic field is subject to change over time. A 2021 paleomagnetic study from the University of Liverpool contributed to a growing body of evidence that the Earth's magnetic field cycles with intensity every 200 million years. The lead author stated that “Our findings, when considered alongside the existing datasets, support the existence of an approximately 200-million-year-long cycle in the strength of the Earth’s magnetic field related to deep Earth processes.”

Inclination

Main article: Magnetic dip

The inclination is given by an angle that can assume values between -90° (up) to 90° (down). In the northern hemisphere, the field points downwards. It is straight down at the North Magnetic Pole and rotates upwards as the latitude decreases until it is horizontal (0°) at the magnetic equator. It continues to rotate upwards until it is straight up at the South Magnetic Pole. Inclination can be measured with a dip circle.

An isoclinic chart (map of inclination contours) for the Earth's magnetic field is shown below.

Declination

Main article: Magnetic declination

Declination is positive for an eastward deviation of the field relative to true north. It can be estimated by comparing the magnetic north–south heading on a compass with the direction of a celestial pole. Maps typically include information on the declination as an angle or a small diagram showing the relationship between magnetic north and true north. Information on declination for a region can be represented by a chart with isogonic lines (contour lines with each line representing a fixed declination).

Geographical variation

Components of the Earth's magnetic field at the surface from the World Magnetic Model for 2015.

• 

  Intensity

• 

  Inclination

• 

  Declination

Dipolar approximation

Relationship between Earth's poles. A1 and A2 are the geographic poles; B1 and B2 are the geomagnetic poles; C1 (south) and C2 (north) are the magnetic poles.

See also: Dipole model of the Earth's magnetic field

Near the surface of the Earth, its magnetic field can be closely approximated by the field of a magnetic dipole positioned at the center of the Earth and tilted at an angle of about 11° with respect to the rotational axis of the Earth. The dipole is roughly equivalent to a powerful bar magnet, with its south pole pointing towards the geomagnetic North Pole. This may seem surprising, but the north pole of a magnet is so defined because, if allowed to rotate freely, it points roughly northward (in the geographic sense). Since the north pole of a magnet attracts the south poles of other magnets and repels the north poles, it must be attracted to the south pole of Earth's magnet. The dipolar field accounts for 80–90% of the field in most locations.

Magnetic poles

Main article: Geomagnetic pole

The movement of Earth's North Magnetic Pole across the Canadian arctic.

Historically, the north and south poles of a magnet were first defined by the Earth's magnetic field, not vice versa, since one of the first uses for a magnet was as a compass needle. A magnet's North pole is defined as the pole that is attracted by the Earth's North Magnetic Pole when the magnet is suspended so it can turn freely. Since opposite poles attract, the North Magnetic Pole of the Earth is really the south pole of its magnetic field (the place where the field is directed downward into the Earth).

The positions of the magnetic poles can be defined in at least two ways: locally or globally. The local definition is the point where the magnetic field is vertical. This can be determined by measuring the inclination. The inclination of the Earth's field is 90° (downwards) at the North Magnetic Pole and -90° (upwards) at the South Magnetic Pole. The two poles wander independently of each other and are not directly opposite each other on the globe. Movements of up to 40 kilometres (25 mi) per year have been observed for the North Magnetic Pole. Over the last 180 years, the North Magnetic Pole has been migrating northwestward, from Cape Adelaide in the Boothia Peninsula in 1831 to 600 kilometres (370 mi) from Resolute Bay in 2001. The magnetic equator is the line where the inclination is zero (the magnetic field is horizontal).

The global definition of the Earth's field is based on a mathematical model. If a line is drawn through the center of the Earth, parallel to the moment of the best-fitting magnetic dipole, the two positions where it intersects the Earth's surface are called the North and South geomagnetic poles. If the Earth's magnetic field were perfectly dipolar, the geomagnetic poles and magnetic dip poles would coincide and compasses would point towards them. However, the Earth's field has a significant non-dipolar contribution, so the poles do not coincide and compasses do not generally point at either.

Magnetosphere

An artist's rendering of the structure of a magnetosphere. 1) Bow shock. 2) Magnetosheath. 3) Magnetopause. 4) Magnetosphere. 5) Northern tail lobe. 6) Southern tail lobe. 7) Plasmasphere.

See also: Magnetosphere

Earth's magnetic field, predominantly dipolar at its surface, is distorted further out by the solar wind. This is a stream of charged particles leaving the Sun's corona and accelerating to a speed of 200 to 1000 kilometres per second. They carry with them a magnetic field, the interplanetary magnetic field (IMF).

The solar wind exerts a pressure, and if it could reach Earth's atmosphere it would erode it. However, it is kept away by the pressure of the Earth's magnetic field. The magnetopause, the area where the pressures balance, is the boundary of the magnetosphere. Despite its name, the magnetosphere is asymmetric, with the sunward side being about 10 Earth radii out but the other side stretching out in a magnetotail that extends beyond 200 Earth radii. Sunward of the magnetopause is the bow shock, the area where the solar wind slows abruptly.

Inside the magnetosphere is the plasmasphere, a donut-shaped region containing low-energy charged particles, or plasma. This region begins at a height of 60 km, extends up to 3 or 4 Earth radii, and includes the ionosphere. This region rotates with the Earth. There are also two concentric tire-shaped regions, called the Van Allen radiation belts, with high-energy ions (energies from 0.1 to 10 million electron volts (MeV)). The inner belt is 1–2 Earth radii out while the outer belt is at 4–7 Earth radii. The plasmasphere and Van Allen belts have partial overlap, with the extent of overlap varying greatly with solar activity.

As well as deflecting the solar wind, the Earth's magnetic field deflects cosmic rays, high-energy charged particles that are mostly from outside the Solar System. Many cosmic rays are kept out of the Solar System by the Sun's magnetosphere, or heliosphere. By contrast, astronauts on the Moon risk exposure to radiation. Anyone who had been on the Moon's surface during a particularly violent solar eruption in 2005 would have received a lethal dose.

Some of the charged particles do get into the magnetosphere. These spiral around field lines, bouncing back and forth between the poles several times per second. In addition, positive ions slowly drift westward and negative ions drift eastward, giving rise to a ring current. This current reduces the magnetic field at the Earth's surface. Particles that penetrate the ionosphere and collide with the atoms there give rise to the lights of the aurorae and also emit X-rays.

The varying conditions in the magnetosphere, known as space weather, are largely driven by solar activity. If the solar wind is weak, the magnetosphere expands; while if it is strong, it compresses the magnetosphere and more of it gets in. Periods of particularly intense activity, called geomagnetic storms, can occur when a coronal mass ejection erupts above the Sun and sends a shock wave through the Solar System. Such a wave can take just two days to reach the Earth. Geomagnetic storms can cause a lot of disruption; the "Halloween" storm of 2003 damaged more than a third of NASA's satellites. The largest documented storm occurred in 1859. It induced currents strong enough to short out telegraph lines, and aurorae were reported as far south as Hawaii.

Time dependence

Short-term variations

Background: a set of traces from magnetic observatories showing a magnetic storm in 2000.
Globe: map showing locations of observatories and contour lines giving horizontal magnetic intensity in μ T.

The geomagnetic field changes on time scales from milliseconds to millions of years. Shorter time scales mostly arise from currents in the ionosphere (ionospheric dynamo region) and magnetosphere, and some changes can be traced to geomagnetic storms or daily variations in currents. Changes over time scales of a year or more mostly reflect changes in the Earth's interior, particularly the iron-rich core.

Frequently, the Earth's magnetosphere is hit by solar flares causing geomagnetic storms, provoking displays of aurorae. The short-term instability of the magnetic field is measured with the K-index.

Data from THEMIS show that the magnetic field, which interacts with the solar wind, is reduced when the magnetic orientation is aligned between Sun and Earth – opposite to the previous hypothesis. During forthcoming solar storms, this could result in blackouts and disruptions in artificial satellites.

Secular variation

Main article: Geomagnetic secular variation

Estimated declination contours by year, 1590 to 1990 (click to see variation).

 

Strength of the axial dipole component of Earth's magnetic field from 1600 to 2020.

Changes in Earth's magnetic field on a time scale of a year or more are referred to as secular variation. Over hundreds of years, magnetic declination is observed to vary over tens of degrees. The animation shows how global declinations have changed over the last few centuries.

The direction and intensity of the dipole change over time. Over the last two centuries the dipole strength has been decreasing at a rate of about 6.3% per century. At this rate of decrease, the field would be negligible in about 1600 years. However, this strength is about average for the last 7 thousand years, and the current rate of change is not unusual.

A prominent feature in the non-dipolar part of the secular variation is a westward drift at a rate of about 0.2 degrees per year. This drift is not the same everywhere and has varied over time. The globally averaged drift has been westward since about 1400 AD but eastward between about 1000 AD and 1400 AD.

Changes that predate magnetic observatories are recorded in archaeological and geological materials. Such changes are referred to as paleomagnetic secular variation or paleosecular variation (PSV). The records typically include long periods of small change with occasional large changes reflecting geomagnetic excursions and reversals.

In July 2020 scientists report that analysis of simulations and a recent observational field model show that maximum rates of directional change of Earth's magnetic field reached ~10° per year – almost 100 times faster than current changes and 10 times faster than previously thought.

Studies of lava flows on Steens Mountain, Oregon, indicate that the magnetic field could have shifted at a rate of up to 6 degrees per day at some time in Earth's history, which significantly challenges the popular understanding of how the Earth's magnetic field works. This finding was later attributed to unusual rock magnetic properties of the lava flow under study, not rapid field change, by one of the original authors of the 1995 study.

Magnetic field reversals

Geomagnetic polarity during the late Cenozoic Era. Dark areas denote periods where the polarity matches today's polarity, light areas denote periods where that polarity is reversed.

Main article: Geomagnetic reversal

Although generally Earth's field is approximately dipolar, with an axis that is nearly aligned with the rotational axis, occasionally the North and South geomagnetic poles trade places. Evidence for these geomagnetic reversals can be found in basalts, sediment cores taken from the ocean floors, and seafloor magnetic anomalies. Reversals occur nearly randomly in time, with intervals between reversals ranging from less than 0.1 million years to as much as 50 million years. The most recent geomagnetic reversal, called the Brunhes–Matuyama reversal, occurred about 780,000 years ago. A related phenomenon, a geomagnetic excursion, takes the dipole axis across the equator and then back to the original polarity. The Laschamp event is an example of an excursion, occurring during the last ice age (41,000 years ago).

The past magnetic field is recorded mostly by strongly magnetic minerals, particularly iron oxides such as magnetite, that can carry a permanent magnetic moment. This remanent magnetization, or remanence, can be acquired in more than one way. In lava flows, the direction of the field is "frozen" in small minerals as they cool, giving rise to a thermoremanent magnetization. In sediments, the orientation of magnetic particles acquires a slight bias towards the magnetic field as they are deposited on an ocean floor or lake bottom. This is called detrital remanent magnetization.

Thermoremanent magnetization is the main source of the magnetic anomalies around mid-ocean ridges. As the seafloor spreads, magma wells up from the mantle, cools to form new basaltic crust on both sides of the ridge, and is carried away from it by seafloor spreading. As it cools, it records the direction of the Earth's field. When the Earth's field reverses, new basalt records the reversed direction. The result is a series of stripes that are symmetric about the ridge. A ship towing a magnetometer on the surface of the ocean can detect these stripes and infer the age of the ocean floor below. This provides information on the rate at which seafloor has spread in the past.

Radiometric dating of lava flows has been used to establish a geomagnetic polarity time scale, part of which is shown in the image. This forms the basis of magnetostratigraphy, a geophysical correlation technique that can be used to date both sedimentary and volcanic sequences as well as the seafloor magnetic anomalies.

Earliest appearance

Paleomagnetic studies of Paleoarchean lava in Australia and conglomerate in South Africa have concluded that the magnetic field has been present since at least about 3,450 million years ago.

Future

Variations in virtual axial dipole moment since the last reversal.

At present, the overall geomagnetic field is becoming weaker; the present strong deterioration corresponds to a 10–15% decline over the last 150 years and has accelerated in the past several years; geomagnetic intensity has declined almost continuously from a maximum 35% above the modern value achieved approximately 2,000 years ago. The rate of decrease and the current strength are within the normal range of variation, as shown by the record of past magnetic fields recorded in rocks.

The nature of Earth's magnetic field is one of heteroscedastic fluctuation. An instantaneous measurement of it, or several measurements of it across the span of decades or centuries, are not sufficient to extrapolate an overall trend in the field strength. It has gone up and down in the past for unknown reasons. Also, noting the local intensity of the dipole field (or its fluctuation) is insufficient to characterize Earth's magnetic field as a whole, as it is not strictly a dipole field. The dipole component of Earth's field can diminish even while the total magnetic field remains the same or increases.

The Earth's magnetic north pole is drifting from northern Canada towards Siberia with a presently accelerating rate—10 kilometres (6.2 mi) per year at the beginning of the 20th century, up to 40 kilometres (25 mi) per year in 2003, and since then has only accelerated.

Physical origin

Main article: Dynamo theory

Earth's core and the geodynamo

The Earth's magnetic field is believed to be generated by electric currents in the conductive iron alloys of its core, created by convection currents due to heat escaping from the core. However the process is complex, and computer models that reproduce some of its features have only been developed in the last few decades.

A schematic illustrating the relationship between motion of conducting fluid, organized into rolls by the Coriolis force, and the magnetic field the motion generates.

The Earth and most of the planets in the Solar System, as well as the Sun and other stars, all generate magnetic fields through the motion of electrically conducting fluids. The Earth's field originates in its core. This is a region of iron alloys extending to about 3400 km (the radius of the Earth is 6370 km). It is divided into a solid inner core, with a radius of 1220 km, and a liquid outer core. The motion of the liquid in the outer core is driven by heat flow from the inner core, which is about 6,000 K (5,730 °C; 10,340 °F), to the core-mantle boundary, which is about 3,800 K (3,530 °C; 6,380 °F). The heat is generated by potential energy released by heavier materials sinking toward the core (planetary differentiation, the iron catastrophe) as well as decay of radioactive elements in the interior. The pattern of flow is organized by the rotation of the Earth and the presence of the solid inner core.

The mechanism by which the Earth generates a magnetic field is known as a dynamo. The magnetic field is generated by a feedback loop: current loops generate magnetic fields (Ampère's circuital law); a changing magnetic field generates an electric field (Faraday's law); and the electric and magnetic fields exert a force on the charges that are flowing in currents (the Lorentz force). These effects can be combined in a partial differential equation for the magnetic field called the magnetic induction equation,

${\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\eta \nabla ^{2}\mathbf {B} +\nabla \times (\mathbf {u} \times \mathbf {B} ),}$

where u is the velocity of the fluid; B is the magnetic B-field; and η=1/σμ is the magnetic diffusivity, which is inversely proportional to the product of the electrical conductivity σ and the permeability μ . The term ∂B/∂t is the time derivative of the field; ∇² is the Laplace operator and ∇× is the curl operator.

The first term on the right hand side of the induction equation is a diffusion term. In a stationary fluid, the magnetic field declines and any concentrations of field spread out. If the Earth's dynamo shut off, the dipole part would disappear in a few tens of thousands of years.

In a perfect conductor (${\displaystyle \sigma =\infty \;}$), there would be no diffusion. By Lenz's law, any change in the magnetic field would be immediately opposed by currents, so the flux through a given volume of fluid could not change. As the fluid moved, the magnetic field would go with it. The theorem describing this effect is called the frozen-in-field theorem. Even in a fluid with a finite conductivity, new field is generated by stretching field lines as the fluid moves in ways that deform it. This process could go on generating new field indefinitely, were it not that as the magnetic field increases in strength, it resists fluid motion.

The motion of the fluid is sustained by convection, motion driven by buoyancy. The temperature increases towards the center of the Earth, and the higher temperature of the fluid lower down makes it buoyant. This buoyancy is enhanced by chemical separation: As the core cools, some of the molten iron solidifies and is plated to the inner core. In the process, lighter elements are left behind in the fluid, making it lighter. This is called compositional convection. A Coriolis effect, caused by the overall planetary rotation, tends to organize the flow into rolls aligned along the north–south polar axis.

A dynamo can amplify a magnetic field, but it needs a "seed" field to get it started. For the Earth, this could have been an external magnetic field. Early in its history the Sun went through a T-Tauri phase in which the solar wind would have had a magnetic field orders of magnitude larger than the present solar wind. However, much of the field may have been screened out by the Earth's mantle. An alternative source is currents in the core-mantle boundary driven by chemical reactions or variations in thermal or electric conductivity. Such effects may still provide a small bias that are part of the boundary conditions for the geodynamo.

The average magnetic field in the Earth's outer core was calculated to be 25 gauss, 50 times stronger than the field at the surface.

Numerical models

Simulating the geodynamo by computer requires numerically solving a set of nonlinear partial differential equations for the magnetohydrodynamics (MHD) of the Earth's interior. Simulation of the MHD equations is performed on a 3D grid of points and the fineness of the grid, which in part determines the realism of the solutions, is limited mainly by computer power. For decades, theorists were confined to creating kinematic dynamo computer models in which the fluid motion is chosen in advance and the effect on the magnetic field calculated. Kinematic dynamo theory was mainly a matter of trying different flow geometries and testing whether such geometries could sustain a dynamo.

The first self-consistent dynamo models, ones that determine both the fluid motions and the magnetic field, were developed by two groups in 1995, one in Japan and one in the United States. The latter received attention because it successfully reproduced some of the characteristics of the Earth's field, including geomagnetic reversals.

Currents in the ionosphere and magnetosphere

Electric currents induced in the ionosphere generate magnetic fields (ionospheric dynamo region). Such a field is always generated near where the atmosphere is closest to the Sun, causing daily alterations that can deflect surface magnetic fields by as much as one degree. Typical daily variations of field strength are about 25 nanoteslas (nT) (one part in 2000), with variations over a few seconds of typically around 1 nT (one part in 50,000).

Measurement and analysis

Detection

The Earth's magnetic field strength was measured by Carl Friedrich Gauss in 1832 and has been repeatedly measured since then, showing a relative decay of about 10% over the last 150 years. The Magsat satellite and later satellites have used 3-axis vector magnetometers to probe the 3-D structure of the Earth's magnetic field. The later Ørsted satellite allowed a comparison indicating a dynamic geodynamo in action that appears to be giving rise to an alternate pole under the Atlantic Ocean west of South Africa.

Governments sometimes operate units that specialize in measurement of the Earth's magnetic field. These are geomagnetic observatories, typically part of a national Geological survey, for example, the British Geological Survey's Eskdalemuir Observatory. Such observatories can measure and forecast magnetic conditions such as magnetic storms that sometimes affect communications, electric power, and other human activities.

The International Real-time Magnetic Observatory Network, with over 100 interlinked geomagnetic observatories around the world, has been recording the Earth's magnetic field since 1991.

The military determines local geomagnetic field characteristics, in order to detect anomalies in the natural background that might be caused by a significant metallic object such as a submerged submarine. Typically, these magnetic anomaly detectors are flown in aircraft like the UK's Nimrod or towed as an instrument or an array of instruments from surface ships.

Commercially, geophysical prospecting companies also use magnetic detectors to identify naturally occurring anomalies from ore bodies, such as the Kursk Magnetic Anomaly.

Crustal magnetic anomalies

A model of short-wavelength features of Earth's magnetic field, attributed to lithospheric anomalies

Magnetometers detect minute deviations in the Earth's magnetic field caused by iron artifacts, kilns, some types of stone structures, and even ditches and middens in archaeological geophysics. Using magnetic instruments adapted from airborne magnetic anomaly detectors developed during World War II to detect submarines, the magnetic variations across the ocean floor have been mapped. Basalt — the iron-rich, volcanic rock making up the ocean floor — contains a strongly magnetic mineral (magnetite) and can locally distort compass readings. The distortion was recognized by Icelandic mariners as early as the late 18th century. More important, because the presence of magnetite gives the basalt measurable magnetic properties, these magnetic variations have provided another means to study the deep ocean floor. When newly formed rock cools, such magnetic materials record the Earth's magnetic field.

Statistical models

Each measurement of the magnetic field is at a particular place and time. If an accurate estimate of the field at some other place and time is needed, the measurements must be converted to a model and the model used to make predictions.

Spherical harmonics

See also: Multipole expansion

Schematic representation of spherical harmonics on a sphere and their nodal lines. P_{ℓ m} is equal to 0 along m great circles passing through the poles, and along ℓ-m circles of equal latitude. The function changes sign each ℓtime it crosses one of these lines.

 

Example of a quadrupole field. This can also be constructed by moving two dipoles together.

The most common way of analyzing the global variations in the Earth's magnetic field is to fit the measurements to a set of spherical harmonics. This was first done by Carl Friedrich Gauss. Spherical harmonics are functions that oscillate over the surface of a sphere. They are the product of two functions, one that depends on latitude and one on longitude. The function of longitude is zero along zero or more great circles passing through the North and South Poles; the number of such nodal lines is the absolute value of the order m. The function of latitude is zero along zero or more latitude circles; this plus the order is equal to the degree ℓ. Each harmonic is equivalent to a particular arrangement of magnetic charges at the center of the Earth. A monopole is an isolated magnetic charge, which has never been observed. A dipole is equivalent to two opposing charges brought close together and a quadrupole to two dipoles brought together. A quadrupole field is shown in the lower figure on the right.

Spherical harmonics can represent any scalar field (function of position) that satisfies certain properties. A magnetic field is a vector field, but if it is expressed in Cartesian components X, Y, Z, each component is the derivative of the same scalar function called the magnetic potential. Analyses of the Earth's magnetic field use a modified version of the usual spherical harmonics that differ by a multiplicative factor. A least-squares fit to the magnetic field measurements gives the Earth's field as the sum of spherical harmonics, each multiplied by the best-fitting Gauss coefficient g_m^ℓ or h_m^ℓ.

The lowest-degree Gauss coefficient, g₀⁰, gives the contribution of an isolated magnetic charge, so it is zero. The next three coefficients – g₁⁰, g₁¹, and h₁¹ – determine the direction and magnitude of the dipole contribution. The best fitting dipole is tilted at an angle of about 10° with respect to the rotational axis, as described earlier.

Radial dependence

Spherical harmonic analysis can be used to distinguish internal from external sources if measurements are available at more than one height (for example, ground observatories and satellites). In that case, each term with coefficient g_m^ℓ or h_m^ℓ can be split into two terms: one that decreases with radius as 1/r^ℓ+1 and one that increases with radius as r^ℓ. The increasing terms fit the external sources (currents in the ionosphere and magnetosphere). However, averaged over a few years the external contributions average to zero.

The remaining terms predict that the potential of a dipole source (ℓ=1) drops off as 1/r². The magnetic field, being a derivative of the potential, drops off as 1/r³. Quadrupole terms drop off as 1/r⁴, and higher order terms drop off increasingly rapidly with the radius. The radius of the outer core is about half of the radius of the Earth. If the field at the core-mantle boundary is fit to spherical harmonics, the dipole part is smaller by a factor of about 8 at the surface, the quadrupole part by a factor of 16, and so on. Thus, only the components with large wavelengths can be noticeable at the surface. From a variety of arguments, it is usually assumed that only terms up to degree 14 or less have their origin in the core. These have wavelengths of about 2,000 kilometres (1,200 mi) or less. Smaller features are attributed to crustal anomalies.

Global models

The International Association of Geomagnetism and Aeronomy maintains a standard global field model called the International Geomagnetic Reference Field. It is updated every five years. The 11th-generation model, IGRF11, was developed using data from satellites (Ørsted, CHAMP and SAC-C) and a world network of geomagnetic observatories. The spherical harmonic expansion was truncated at degree 10, with 120 coefficients, until 2000. Subsequent models are truncated at degree 13 (195 coefficients).

Another global field model, called the World Magnetic Model, is produced jointly by the United States National Centers for Environmental Information (formerly the National Geophysical Data Center) and the British Geological Survey. This model truncates at degree 12 (168 coefficients) with an approximate spatial resolution of 3,000 kilometers. It is the model used by the United States Department of Defense, the Ministry of Defence (United Kingdom), the United States Federal Aviation Administration (FAA), the North Atlantic Treaty Organization (NATO), and the International Hydrographic Organization as well as in many civilian navigation systems.

A third model, produced by the Goddard Space Flight Center (NASA and GSFC) and the Danish Space Research Institute, uses a "comprehensive modeling" approach that attempts to reconcile data with greatly varying temporal and spatial resolution from ground and satellite sources.

For users with higher accuracy needs, the United States National Centers for Environmental Information developed the Enhanced Magnetic Model (EMM), which extends to degree and order 790 and resolves magnetic anomalies down to a wavelength of 56 kilometers. It was compiled from satellite, marine, aeromagnetic and ground magnetic surveys. As of 2018, the latest version, EMM2017, includes data from The European Space Agency's Swarm satellite mission.

Effect of ocean tides

The oceans contribute to Earth's magnetic field. Seawater is an electrical conductor, and therefore interacts with the magnetic field. As the tides cycle around the ocean basins, the ocean water essentially tries to pull the geomagnetic field lines along. Because the salty water is slightly conductive, the interaction is relatively weak: the strongest component is from the regular lunar tide that happens about twice per day. Other contributions come from ocean swell, eddies, and even tsunamis.

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Sea level magnetic fields observed by satellites (NASA) 

The strength of the interaction depends also on the temperature of the ocean water. The entire heat stored in the ocean can now be inferred from observations of the Earth's magnetic field.

Biomagnetism

Main article: Magnetoreception

Animals, including birds and turtles, can detect the Earth's magnetic field, and use the field to navigate during migration. Some researchers have found that cows and wild deer tend to align their bodies north–south while relaxing, but not when the animals are under high-voltage power lines, suggesting that magnetism is responsible. Other researchers reported in 2011 that they could not replicate those findings using different Google Earth images.

Very weak electromagnetic fields disrupt the magnetic compass used by European robins and other songbirds, which use the Earth's magnetic field to navigate. Neither power lines nor cellphone signals are to blame for the electromagnetic field effect on the birds; instead, the culprits have frequencies between 2 kHz and 5 MHz. These include AM radio signals and ordinary electronic equipment that might be found in businesses or private homes.