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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, x\in {\mathbb R}\!} . This gives a complete solution to the Cauchy problem that can be cast as follows

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, f=f_0(x),\ \partial_tf=f_1(x)\!}

for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, t=0\!} , so that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, F_1\!} an arbitrarily chosen primitive of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, f_1\!} .

Solution in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathbb R}^{1+d}}

Solution of the Cauchy problem in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathbb R}^{1+d}} can be given as follows You1966. We have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, f=f_0(x),\ \partial_tf=0\!}

for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, t=0\!} , but now Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, x\in {\mathbb R}^d\!} . One can write the solution as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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_1dt_1}{\sqrt{t^2-t_1^2}}}

when d is even, being

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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.