The
wave equation in
Rn is a
partial differential equation, in fact the following one:
d d d d d d
---- u = ------- u + ... + ------- u
dtdt dx1 dx1 dxn dxn
with u:
Rn+1 ->
R stands for the
solution and initial
conditions:
u(0,x) = a(x)
d
--- u(0,x) = b(x)
dt
It has an
unique solution for all n. (The easiest way to see that would be the Cauchy-Kovaljevskaja theorem, but it's not here yet and it is spelled wrong anyway.)
The
equation describes how
waves spread in an
unlimited space, e.g. an
infinite string.
The
Huygens' Principle is connected to this equation.
And here is the solution for n=3:
1 d / / \ t /
u(t,x) = --- ----| t | a(x+tu) du | + --- |b(x+tv)dv
4Pi dt \ /S2 / 4Pi /S2
S2 is the
unit sphere in
R3.