Editing Wave (section)

== Mathematical description ==

=== Single waves ===
{{See also|Solitary wave (disambiguation){{!}}Solitary wave}}
A wave can be described just like a field, namely as a [[function (mathematics)|function]] <math>F(x,t)</math> where <math>x</math> is a position and <math>t</math> is a time.

The value of <math>x</math> is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a [[vector (mathematics)|vector]] in the [[analytic geometry|Cartesian three-dimensional space]] <math>\mathbb{R}^3</math>. However, in many cases one can ignore one dimension, and let <math>x</math> be a point of the Cartesian plane <math>\mathbb{R}^2</math>. This is the case, for example, when studying vibrations of a drum skin. One may even restrict <math>x</math> to a point of the Cartesian line <math>\R</math> – that is, the set of [[real number]]s. This is the case, for example, when studying vibrations in a [[string (music)|violin string]] or [[recorder (musical instrument)|recorder]]. The time <math>t</math>, on the other hand, is always assumed to be a [[scalar (physics)|scalar]]; that is, a real number.

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

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

=== Superposition ===
{{main|Superposition principle}}
Waves of the same type are often superposed and encountered simultaneously at a given point in space and time. The properties at that point are the sum of the properties of each component wave at that point. In general, the velocities are not the same, so the wave form will change over time and space.

=== Wave spectrum ===
{{See also|Wind wave#Spectrum|Electromagnetic spectrum|Spectrum (physical sciences)}}
{{Expand section|concept summary|date=May 2023}}

=== 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]] echoes 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 <math>F(A,B,\ldots;x,t)</math> that depends on certain [[parameter]]s <math>A,B,\ldots</math>, besides <math>x</math> and <math>t</math>. Then one can obtain different waves – that is, different functions of <math>x</math> and <math>t</math> – by choosing different values for those parameters.

[[File:Half-open pipe wave.gif|class=skin-invert-image|thumb|right|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 (musical instrument)|recorder]] that is playing a "pure" note is typically a [[standing wave]], that can be written as
: <math>F(A,L,n,c;x,t) = A \left(\cos 2\pi x\frac{2 n - 1}{4 L}\right) \left(\cos 2\pi c t\frac{2n - 1}{4 L}\right)</math>
The parameter <math>A</math> defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note); <math>c</math> is the speed of sound; <math>L</math> is the length of the bore; and <math>n</math> is a positive integer (1,2,3,...) that specifies the number of [[standing wave|nodes]] in the standing wave. (The position <math>x</math> should be measured from the [[wind instrument|mouthpiece]], and the time <math>t</math> from any moment at which the pressure at the mouthpiece is maximum. The quantity <math>\lambda = 4L/(2 n - 1)</math> is the [[wavelength]] of the emitted note, and <math>f = c/\lambda</math> 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 <math>r</math> from the center of the skin to the strike point, and on the strength <math>s</math> of the strike. Then the vibration for all possible strikes can be described by a function <math>F(r,s;x,t)</math>.

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 <math>h</math> such that <math>h(x)</math> is the initial temperature at each point <math>x</math> of the bar. Then the temperatures at later times can be expressed by a function <math>F</math> that depends on the function <math>h</math> (that is, a [[operator (mathematics)|functional operator]]), so that the temperature at a later time is <math>F(h;x,t)</math>

=== 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 <math>F(x,t)</math>, only constrains how those values can change with time. Then the family of waves in question consists of all functions <math>F</math> that satisfy those constraints – that is, all [[solution (mathematics)|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 <math>F(x,t)</math> is the temperature inside a block of some [[homogeneous]] and [[isotropic]] solid material, its evolution is constrained by the [[partial differential equation]]
: <math>\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)</math>
where <math>Q(p,f)</math> is the heat that is being generated per unit of volume and time in the neighborhood of <math>x</math> at time <math>t</math> (for example, by chemical reactions happening there); <math>x_1,x_2,x_3</math> are the Cartesian coordinates of the point <math>x</math>; <math>\partial F/\partial t</math> is the (first) derivative of <math>F</math> with respect to <math>t</math>; and <math>\partial^2 F/\partial x_i^2</math> is the second derivative of <math>F</math> relative to <math>x_i</math>. (The symbol "<math>\partial</math>" 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 [[heat diffusion|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 <math>F(x,t)</math> that gives the pressure at a point <math>x</math> and time <math>t</math> within that container. If the gas was initially at uniform temperature and composition, the evolution of <math>F</math> is constrained by the formula
: <math>\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)</math>
Here <math>P(x,t)</math> is some extra compression force that is being applied to the gas near <math>x</math> by some external process, such as a [[loudspeaker]] or [[piston]] right next to <math>p</math>.

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 <math>\partial^2 F/\partial t^2</math>, the second derivative of <math>F</math> with respect to time, rather than the first derivative <math>\partial F/\partial t</math>. Yet this small change makes a huge difference on the set of solutions <math>F</math>. This differential equation is called "the" [[wave equation]] in mathematics, even though it describes only one very special kind of waves.