Although the one-dimensional
classical wave equation arises in many contexts, it is the
interaction between time-varying electric and magnetic fields
predicted by Faraday's and Ampere's laws that provides the most
interesting phenomena. Consider the y-component of a
time-varying electric field in vacuum, 
. The line
integral along a path in the x,y plane
can be approximated by
where 
. Faraday's law
tells us that the induced
emf is equal to the rate of decrease of the magnetic flux, 
, through the closed path.
In the limit as 
we obtain
Similarly, the line integral of the magnetic field in the z direction,
, taken around a loop in the x,z plane (the minus sign
is chosen so that the right hand rule gives positive circulation
about the y axis)
is proportional to the displacement current
by ampere's law and yields the following equation:
In the limit as 
we obtain
Although good numerical techniques are available to solve eq:Faraday and eq:Ampere as a coupled system of first-order PDEs,[13] we will convert these equations into a single second-order equation. Taking the derivative of eq:Faraday with respect to x and eq:Ampere with respect to t and equating the mixed partial derivatives leads to the desired equation
where 
.
An interesting feature of eq:EME is that there is a second identical equation which governs the propagation of the magnetic field. Although the computer display shows only the transverse electric field, the magnetic field is just as important and, in fact, carries an equal amount of energy. Computer exercises will ask you to examine the relationships between these fields and the energy density.[3]
