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 and $\,x\in {\mathbb {R} }^{1}\!$. 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 $\,f_{1}\!$.

Solution in ${\mathbb {R} }^{1+d}$

Solution of the Cauchy problem in ${\mathbb {R} }^{1+d}$ can be given as follows. We have

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

but now $\,x\in {\mathbb {R} }^{d}\!$. One can write the solution as

$f(x,t)={\frac {t{\sqrt {\pi }}}{\Gamma (n/2)}}\left({\frac {\partial }{\partial t^{2}}}\right)^{(n-1)/2}[t^{n-2}\phi (x,t)]$

when n is odd and

$f(x,t)={\frac {2t}{\Gamma (n/2)}}\left({\frac {\partial }{\partial t^{2}}}\right)^{n/2}\int _{0}^{t}t_{1}^{n-2}\phi (x,t_{1}){\frac {t_{1}dt_{1}}{\sqrt {t^{2}-t_{1}^{2}}}}$

when n is even, being

$\,\phi (x,t)={\frac {1}{\Omega _{n}}}\int _{\Sigma (t)}f_{0}(x')d\Omega _{n}\!$

on the surface of the d-sphere centered at x and with radius t.