Free wave equation: Difference between revisions
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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 [[ | 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> |
Revision as of 12:18, 23 June 2009
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.