Free wave equation: Difference between revisions

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<center><math>\, f(x,t)=g_1(x-t)+g_2(x+t)\!</math></center>
<center><math>\, f(x,t)=g_1(x-t)+g_2(x+t)\!</math></center>


being <math>g_1,\ g_2</math> two arbitrary functions and <math>\, x\in {\mathbb R}^1\!</math>. This gives a complete solution to the Cauchy problem that can be cast as follows
being <math>g_1,\ g_2</math> two arbitrary functions and <math>\, x\in {\mathbb R}\!</math>. This gives a complete solution to the Cauchy problem that can be cast as follows


<center><math>\, f=f_0(x),\ \partial_tf=f_1(x)\!</math></center>
<center><math>\, f=f_0(x),\ \partial_tf=f_1(x)\!</math></center>
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=== Solution in <math>{\mathbb R}^{1+d}</math> ===
=== Solution in <math>{\mathbb R}^{1+d}</math> ===


Solution of the Cauchy problem in <math>{\mathbb R}^{1+d}</math> can be given as follows. We have
Solution of the Cauchy problem in <math>{\mathbb R}^{1+d}</math> can be given as follows [[You1966]]. We have


<center><math>\, f=f_0(x),\ \partial_tf=0\!</math></center>
<center><math>\, f=f_0(x),\ \partial_tf=0\!</math></center>
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for <math>\, t=0\!</math>, but now <math>\, x\in {\mathbb R}^d\!</math>. One can write the solution as
for <math>\, t=0\!</math>, but now <math>\, x\in {\mathbb R}^d\!</math>. One can write the solution as


<center><math>f(x,t)=\frac{t\sqrt{\pi}}{\Gamma(n/2)}\left(\frac{\partial}{\partial t^2}\right)^{(n-1)/2}[t^{n-2}\phi(x,t)]</math></center>
<center><math>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)]</math></center>


when n is odd and
when d is odd and


<center><math>f(x,t)=\frac{2t}{\Gamma(n/2)}\left(\frac{\partial}{\partial t^2}\right)^{n/2}\int_0^t t_1^{n-2}\phi(x,t_1)\frac{t_1dt_1}{\sqrt{t^2-t_1^2}}</math></center>
<center><math>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}}</math></center>


when n is even, being
when d is even, being


<center><math>\, \phi(x,t)=\frac{1}{\Omega_n}\int_{\Sigma(t)} f_0(x')d\Omega_n\!</math></center>
<center><math>\, \phi(x,t)=\frac{1}{\Omega_d}\int_{\Sigma(t)} f_0(x')d\Omega_d\!</math></center>


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




{{stub}}
 
[[Category:Wave]]
[[Category:Wave]]
[[Category:Equations]]
[[Category:Equations]]

Latest revision as of 16:00, 1 July 2018

The free wave equation on is given by

where f is a scalar or vector field on Minkowski space . In coordinates, this becomes

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

In this case one can write down the solution as

being two arbitrary functions and . This gives a complete solution to the Cauchy problem that can be cast as follows

for , so that

being an arbitrarily chosen primitive of .

Solution in

Solution of the Cauchy problem in can be given as follows You1966. We have

for , but now . One can write the solution as

when d is odd and

when d is even, being

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