Characteristic Surfaces For The Wave Equation
Characteristic surfaces for the wave equation are level surfaces for solutions of the equation
There is little loss of generality if we set : in that case u satisfies
In vector notation, let
A family of solutions with planes as level surfaces is given by
where
If x and x0 are held fixed, the envelope of these solutions is obtained by finding a point on the sphere of radius 1/c where the value of u is stationary. This is true if is parallel to . Hence the envelope has equation
These solutions correspond to spheres whose radius grows or shrinks with velocity c. These are light cones in space-time.
The initial value problem for this equation consists in specifying a level surface S where u=0 for t=0. The solution is obtained by taking the envelope of all the spheres with centers on S, whose radii grow with velocity c. This envelope is obtained by requiring that
This condition will be satisfied if is normal to S. Thus the envelope corresponds to motion with velocity c along each normal to S. This is the Huygens' construction of wave fronts: each point on S emits a spherical wave at time t=0, and the wave front at a later time t is the envelope of these spherical waves. The normals to S are the light rays.
Read more about this topic: First-order Partial Differential Equation
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