Free wave equation
The free wave equation on is given by
where f is a scalar or vector field on Minkowski space . In coordinates, this becomes
It is the prototype for many nonlinear wave equations.
One can add a mass term to create the Klein-Gordon equation.
Being this a linear equation one can always write down a solution using Fourier series or transform. These solutions represent superpositions of traveling waves.
In this case one can write down the solution as
being two arbitrary functions and . This gives a complete solution to the Cauchy problem that can be cast as follows
for , so that
being an arbitrarily chosen primitive of .
Solution of the Cauchy problem in can be given as follows You1966. We have
for , but now . One can write the solution as
when d is odd and
when d is even, being
on the surface of the d-sphere centered at x and with radius t.