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_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.


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