Free wave equation

The free wave equation on ${\mathbb {R} }^{1+d}$ is given by

$\Box f=0$

where f is a scalar or vector field on Minkowski space ${\mathbb {R} }^{1+d}$. In coordinates, this becomes

$-\partial _{tt}f+\Delta f=0.$

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 ${\mathbb {R} }^{1+1}$

In this case one can write down the solution as

$\,f(x,t)=g_{1}(x-t)+g_{2}(x+t)\!$

being $g_{1},\ g_{2}$ two arbitrary functions. This gives a complete solution to the Cauchy problem that can be cast as follows

$\,f=f_{0}(x)\ \partial _{t}f=f_{1}(x)\!$

for $\,t=0\!$, so that

$f(x,t)={\frac {1}{2}}[f_{0}(x+t)+f_{0}(x-t)]+{\frac {1}{2}}[F_{1}(x+t)+F_{1}(x-t)]$

being $\,F_{1}\!$ an arbitrarily chosen primitive of $\,f_{1}\!$.