Scalar Wave Equation in Two Space Dimensions
In two space dimensions, the wave equation is
We can use the three-dimensional theory to solve this problem if we regard u as a function in three dimensions that is independent of the third dimension. If
then the three-dimensional solution formula becomes
where α and β are the first two coordinates on the unit sphere, and dω is the area element on the sphere. This integral may be rewritten as an integral over the disc D with center (x,y) and radius ct:
It is apparent that the solution at (t,x,y) depends not only on the data on the light cone where
but also on data that are interior to that cone.
Read more about this topic: Wave Equation
Famous quotes containing the words wave, equation, space and/or dimensions:
“The wave of evil washes all our institutions alike.”
—Ralph Waldo Emerson (1803–1882)
“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)
“When my body leaves me
I’m lonesome for it.
but body
goes away to I don’t know where
and it’s lonesome to drift
above the space it
fills when it’s here.”
—Denise Levertov (b. 1923)
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)