# Free wave equation

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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 $\,x\in {\mathbb {R} }\!$ . 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 You1966. We have

$\,f=f_{0}(x),\ \partial _{t}f=0\!$ for $\,t=0\!$ , but now $\,x\in {\mathbb {R} }^{d}\!$ . One can write the solution as

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

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

$\,\phi (x,t)={\frac {1}{\Omega _{d}}}\int _{\Sigma (t)}f_{0}(x')d\Omega _{d}\!$ on the surface of the d-sphere centered at x and with radius t.