Laplace operator. Wave equation.

Question

Why does Laplace operator affect only x-coordinates but not time variable in wave equation. Does t variable in U(X, t) is different from x coordinates? I think they are equal.

Answer 1

Often when solving ODE or PDE with Laplace transform (in undergraduate courses), it is about the one-dimensional Laplace transform. In your case, the $t$ is considered constant (a snapshot in time) when we "squeeze" the space $x$ exponentially.

Indeed, $t$ (time) and $x$ (space) are in equal standing mathematically in terms of Laplace transform. You can actually do the Laplace transform on $t$ with $x$ unaffected.

There's of course multi-dimensional Laplace transform that affects more than one variable when operated.