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?