Changes

The '''wave equation''' is among the most well known, elegant, and important equations in all of mathematical physics.  A great many physical problems, usually relating to wave motion or vibration, turn into the wave equation when analyzed mathematically.  Some of these applications will be discussed below.

The '''wave equation''' is among the most well known, elegant, and important equations in all of mathematical physics.  A great many physical problems, usually relating to wave motion or vibration, turn into the wave equation when analyzed mathematically.  Some of these applications will be discussed below.

The wave equation concerns some "quantity" that is a function of both space and time.  That "quantity" might be air pressure, a magnetic field, the displacement of a string or membrane, or the abstract "wave function" of [[quantum mechanics]].  It is a [[partial differential equation]], since it involves [[partial derivative]]s.

The wave equation concerns some "quantity" that is a function of both space and time.  That "quantity" might be air pressure, a magnetic field, the displacement of a string or membrane, or the abstract "wave function" of [[quantum mechanics]].  It is a ''partial differential equation'', since it involves [[partial derivative]]s.

In one dimension, and denoting the "quantity" as <math>\psi</math> the equation is:

In one dimension, and denoting the "quantity" as <math>\psi</math>, the equation is:

:<math>\frac{\partial^2 \psi}{\partial t^2} = v^2\ \frac{\partial^2 \psi}{\partial x^2}</math>

:<math>\frac{\partial^2 \psi}{\partial t^2} = v^2\ \frac{\partial^2 \psi}{\partial x^2}</math>

for some constant <math>v</math>.  <math>v</math> is the velocity of the wave.

for some constant <math>v</math>.  <math>v</math> is the velocity of the wave.

The quantity <math>v</math> in the wave equation is the speed of propagation of the wave.  Dimensional analysis of the derivatives shows that it has the dimensions of velocity.  When dealing with electromagnetic waves, it is common to use <math>c</math>, the standard symbol for the speed of light.

The quantity <math>v</math> in the wave equation is the speed of propagation of the wave.  Dimensional analysis of the derivatives shows that it has the dimensions of velocity.  When dealing with electromagnetic waves, it is common to use <math>c</math>, the standard symbol for the speed of light.

==In Higher Dimensions and the Laplacian==

==In Higher Dimensions, and the Laplacian==

In two dimensions, and Cartesian coordinates, the wave equation is:

In two dimensions, and Cartesian coordinates, the wave equation is:

:<math>\frac{\partial^2 \psi}{\partial t^2} = v^2\ \left(\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2}\right)</math>

:<math>\frac{\partial^2 \psi}{\partial t^2} = v^2\ \left(\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2}\right)</math>