The

**wave equation** in

**R**^{n} 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:

**R**^{n+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

**R**^{3}.