D'Alambert Equation: Solving for $\psi(\vec{r},t)$

[Petar Mali]

\Delta\psi(\vec{r},t)-\frac{1}{\upsilon^2}\frac{\partial^2\psi(\vec{r},t)}{\partial t^2}=-g(\vec{r},t)

How to get solution

\psi(\vec{r},t)=\frac{1}{r}F_1(t-\frac{r}{\upsilon})

where F_1 is any function of argument t-\frac{r}{\upsilon}.

 

[Petr Mugver]

Write the laplacian in spherical coordinates, and use separation of variables.

 

[arkajad]

Your \psi can not be a solution for a nontrivial g(r,t) on the rhs.

 

[HallsofIvy]

Well, not for a general g(r,t) but there are solutions for some specific, non-trivial functions.

 

[arkajad]

Well, not for a general g(r,t) but there are solutions for some specific, non-trivial functions.

For instance for g(r,t)=0. Where did you get the idea that your formula is a solution for a non-zero g?