Editing Wave (section)

=== 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.