Wave

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 ${\displaystyle t}$.

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 ${\displaystyle F(h;x,t)}$

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.

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).[9]

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.[10]

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.

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:[11][12][13]

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

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:[14]

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

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.[14][15]

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.

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 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.

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

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.

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.

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.

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.

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.

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.

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

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.[18]

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 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).

• 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 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.

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.[19] It is named after the Austrian physicist Christian Doppler, who described the phenomenon in 1842.

Formation of a shock wave by a plane.

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.[20]

• Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves[21]
• Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions.

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.

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-1⁄2 particles.

A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.[22]

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,[23] 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.[24] Gaussian wave packets also are used to analyze water waves.[25]

For example, a Gaussian wavefunction ψ might take the form:[26]

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

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,[27] 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.[28] 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