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
Famous quotes containing the words surfaces, wave and/or equation:
“Footnotes are the finer-suckered surfaces that allow tentacular paragraphs to hold fast to the wider reality of the library.”
—Nicholson Baker (b. 1957)
“We are caught up Mr. Perry on a great wave whether we will or no, a great wave of expansion and progress. All these mechanical inventions—telephones, electricity, steel bridges, horseless vehicles—they are all leading somewhere. It’s up to us to be on the inside in the forefront of progress.”
—John Dos Passos (1896–1970)
“Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
—Anna Quindlen (b. 1952)