In the standing wave equation, , how do you know to separate space and time wiggle to get a standing wave as opposed to a traveling wave? Where did this solution come from?
I’m not sure this kind of solution is exactly “derived”: it’s written down as a particular function we happen to know about that has the desired properties (sometimes called “solution by inspection”). It’s sort of like having a job to do and already having the right tool for it, so you don’t have to go out and manufacture one.
