Free wave equation

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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.

Exact solutions

Being this a linear equation one can always write down a solution using Fourier series or transform. These solutions represent superpositions of traveling waves.

Solution in

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 in

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.