# 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 $\, x\in {\mathbb R}\!$. This gives a complete solution to the Cauchy problem that can be cast as follows

$\, f=f_0(x),\ \partial_tf=f_1(x)\!$

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

$\, f=f_0(x),\ \partial_tf=0\!$

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

$f(x,t)=\frac{2t}{\Gamma(d/2)}\left(\frac{\partial}{\partial t^2}\right)^{d/2}\int_0^t t_1^{d-2}\phi(x,t_1)\frac{t_1dt_1}{\sqrt{t^2-t_1^2}}$

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.