# Free wave equation

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

${\displaystyle \Box f=0}$

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

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

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

In this case one can write down the solution as

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

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

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

for ${\displaystyle \,t=0\!}$, so that

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

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

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

Solution of the Cauchy problem in ${\displaystyle {\mathbb {R} }^{1+d}}$ can be given as follows You1966. We have

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

for ${\displaystyle \,t=0\!}$, but now ${\displaystyle \,x\in {\mathbb {R} }^{d}\!}$. One can write the solution as

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

when d is odd and

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

when d is even, being

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

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